Ian Jauslin
c1b477a1b2
feature: compute the 2-point correlation function in easyeq.
feature: compute the Fourier transform of the 2-point correlation function
in anyeq and easyeq.
feature: compute the local maximum of the 2-point correlation function and
its Fourier transform.
feature: compute the compressibility for anyeq.
feature: allow for linear spacing of rho's.
feature: print the scattering length.
change: ux and uk now return real numbers.
fix: error in the computation of the momentum distribution: wrong
definition of delta functions.
fix: various minor bugs.
optimization: assign explicit types to variables.
4552 lines
181 KiB
TeX
4552 lines
181 KiB
TeX
%% Copyright 2021-2023 Ian Jauslin
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%%
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%% Licensed under the Apache License, Version 2.0 (the "License");
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%% you may not use this file except in compliance with the License.
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%% You may obtain a copy of the License at
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%%
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%% http://www.apache.org/licenses/LICENSE-2.0
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%%
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%% Unless required by applicable law or agreed to in writing, software
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%% distributed under the License is distributed on an "AS IS" BASIS,
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%% WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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%% See the License for the specific language governing permissions and
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%% limitations under the License.
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\documentclass[subsection_in_all]{ian}
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\usepackage{code}
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\usepackage{dsfont}
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\usepackage{largearray}
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\usepackage{dlmf}
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\def\Eta{H}
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\begin{document}
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\hbox{}
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\hfil{\bf\Large
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{\tt simplesolv}\par
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\medskip
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\hfil \large v0.4
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}
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\vfill
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\tableofcontents
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\vfill
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\eject
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\setcounter{page}1
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\pagestyle{plain}
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\indent
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{\tt simplesolv} is a tool to compute the solution of the equations of the ``Simplified approach'' to the repulsive Bose gas introduced in\-~\cite{CJL20,CJL21,CHe21}.
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This approach provides an approximation to various observables of the ground state of the Hamiltonian
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\begin{equation}
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H_N=-\frac12\sum_{i=1}^N\Delta_i+\sum_{1\leqslant i\leqslant j\leqslant N}v(|x_i-x_j|)
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\end{equation}
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in three dimensions, with periodic boundary conditions, in the thermodynamic limit $N\to\infty$ at fixed density $\rho$.
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\bigskip
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\indent
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{\tt simplesolv} is written in {\tt julia}.
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The source code is located in the {\tt src} directory of this bundle.
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Throughout the documentation, we will refer to the directory containing the {\tt src} directory as the ``installation directory'', and will denote it by the bash variable {\tt\$SIMPLESOLV} (so that the main julia file, for instance, is located at {\tt \$SIMPLESOLV/src/main.jl}).
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\vskip20pt
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\section{Basic usage}
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\indent
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Denoting the location of the installation directory by {\tt\$SIMPLESOLV}, {\tt simplesolv} is run by calling
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\begin{code}
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julia \$SIMPLESOLV/src/main.jl [args] <command>
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\end{code}
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where the optional arguments {\tt [args]} take the form {\tt [-p params] [-U potential] [-M method] [-s savefile]}.
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\bigskip
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\indent
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A few commands support multithreaded execution.
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To enable {\tt julia} to run on several processors, it should be run with the {\tt -p} option.
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For example, to run on 8 CPUs, run
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\begin{code}
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julia -p 8 \$SIMPLESOLV/src/main.jl [args] <command>
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\end{code}
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\bigskip
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\indent
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{\tt command} specifies which computation is to be carried out, such as {\tt energy} to compute the ground state energy, or {\tt condensate\_fraction} for the uncondensed fraction.
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The list of available commands depends on the \refname{sec:methods}{{\tt method}} argument, which specifies one of the available methods to solve the equation at hand.
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The available methods are (see section\-~\ref{sec:methods} for further details)
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\begin{itemize}
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\item \refname{sec:easyeq}{{\tt easyeq}} ({\bf default}) for the Simple or Medium equation, or any interpolation between them, with a soft potential using the Newton algorithm,
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\item \refname{sec:anyeq}{{\tt anyeq}} for any equation in the ``Simplified approach'' using the Newton algorithm.
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\item \refname{sec:simpleq-Kv}{{\tt simpleq-Kv}} for the Simple equation using explicit expressions involving $\mathfrak Kv$ (see\-~(\ref{Kv})),
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\item \refname{sec:simpleq-hardcore}{{\tt simpleq-hardcore}} for the Simple equation with a hard core potential using the Newton algorithm,
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\item \refname{sec:simpleq-iteration}{{\tt simpleq-iteration}} for the Simple equation with a soft potential using the iteration defined in\-~\cite{CJL20}.
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\end{itemize}
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Each method is described in detail below, along with the list of commands ({\tt command}) and parameters ({\tt params}) compatible with them.
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In addition, the scattering length can be displayed using the {\tt scattering\_length} command:
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\begin{code}
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julia \$SIMPLESOLV/src/main.jl scattering\_length
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\end{code}
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\bigskip
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\indent
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{\tt params} should be a `{\tt;}' separated list of entries, each of which is of the form {\tt key=value}.
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For example {\tt -p "rho=1e-6;v\_a=2"}.
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(Note that you should not end the list of parameters by a `{\tt;}', otherwise {\tt simplesolv} will interpret that as there being an empty parameter entry, which it cannot split into a key and value, and will fail.)
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\bigskip
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\indent
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\refname{sec:potentials}{{\tt potential}} specifies which potential $v$ should be used, from the following list (see section\-~\ref{sec:potentials} for further details).
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\begin{itemize}
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\item \refname{sec:exp}{{\tt exp}} ({\bf default}) for $v(|x|)=ae^{-|x|}$,
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\item \refname{sec:tent}{{\tt tent}} for $v(|x|)=\mathds 1_{|x|<b}a\frac{2\pi}{3}(1-\frac{|x|}b)^2(\frac{|x|}b+2)$,
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\item \refname{sec:expcry}{{\tt expcry}} for $v(|x|)=e^{-|x|}-ae^{-b|x|}$,
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\item \refname{sec:npt}{{\tt npt}} for $v(|x|)=x^2e^{-|x|}$,
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\item \refname{sec:alg}{{\tt alg}} for $v(|x|)=\frac1{1+\frac14|x|^4}$,
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\item \refname{sec:algwell}{{\tt algwell}} for $v(|x|)=\frac{1+a|x|^4}{(1+|x|^2)^4}$.
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\item \refname{sec:exact}{{\tt exact}} for $v(|x|)=\frac{12c( |x|^6b^6(2e-b^2) +b^4|x|^4(9e-7b^2) +4b^2|x|^2(3e-2b^2) +(5e+16b^2))}{(1+b^2|x|^2)^2(4+b^2|x|^2)^2((1+b^2|x|^2)^2-c)}$
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\end{itemize}
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The parameters in the potential can be set using the {\tt params} argument: to set $a$ set {\tt v\_a}, to set $b$ set {\tt v\_b}, to set $c$ set {\tt v\_c}, and to set $e$ set {\tt v\_e}.
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\bigskip
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\indent
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{\tt savefile} can be used to accelerate the computation of observables in the \refname{eq:anyeq_compleq}{{\tt compleq}} equation.
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Indeed, as is discussed in section\-~\ref{sec:anyeq}, the computation of \refname{eq:anyeq_compleq}{{\tt compleq}} is based on the computation of a large matrix, which can be pre-computed, saved in a file using the \refname{command:anyeq_save_Abar}{{\tt save\_Abar}} command, and reused by specifying that file in the {\tt savefile} argument.
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\section{Methods}\label{sec:methods}
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\indent
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In this section, we describe the different computation methods.
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\bigskip
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\subsection{\tt easyeq}\label{sec:easyeq}
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\indent
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This method is used to solve a family of equations, called {\tt easyeq}, that interpolate between the Simple equation and the Medium equation:
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\begin{equation}
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-\Delta u
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=
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v(1-u)-2\rho K+\rho^2 L
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\label{easyeq}
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\end{equation}
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with
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\begin{equation}
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K:=\beta_K u\ast S+(1-\beta_K)\frac{2e}\rho u
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,\quad
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L:=\beta_Lu\ast u\ast S+(1-\beta_L)\frac{2e}\rho u\ast u
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\label{easyeq_KL}
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\end{equation}
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\begin{equation}
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S:=(1-u)v
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,\quad
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e:=\frac\rho2\int dx\ (1-u(|x|))v(|x|)
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.
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\label{easyeq_Se}
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\end{equation}
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for a soft potential $v$ at density $\rho>0$.
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\bigskip
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\indent
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\makelink{eq:easyeq_simpleq}{}
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\makelink{eq:easyeq_medeq}{}
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The special choice $\beta_K=\beta_L=0$ is called the Simple equation ({\tt simpleq}), and the choice $\beta_K=\beta_L=1$ is called the Medium equation ({\tt medeq})
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\bigskip
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\subsubsection{Usage}
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\indent
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Unless otherwise noted, this method takes the following parameters (specified via the {\tt [-p params]} flag, as a `{\tt;}' separated list of entries).
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\begin{itemize}
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\item\makelink{param:easyeq_rho}{}
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{\tt rho} ({\tt Float64}, default: $10^{-6}$): density $\rho$.
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\item\makelink{param:easyeq_tolerance}{}
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{\tt tolerance} ({\tt Float64}, default: $10^{-11}$): maximal size of final step in Newton iteration.
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\item\makelink{param:easyeq_maxiter}{}
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{\tt maxiter} ({\tt Int64}, default: 21): maximal number of iterations of the Newton algorithm before giving up.
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\item\makelink{param:easyeq_order}{}
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{\tt order} ({\tt Int64}, default: 100): order used for all Gauss quadratures (denoted by $N$ below).
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\item\makelink{param:easyeq_minlrho_init}{}
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{\tt minlrho\_init} ({\tt Float64}, default: $-6$): to initialize the Newton algorithm, we first compute the solution for a smaller $\rho$, {\tt minlrho} is the minimal value for $\log_{10}\rho$ to start this initialization process.
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\item\makelink{param:easyeq_nlrho_init}{}
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{\tt nlrho\_init} ({\tt Int64}, default: 0): number of steps in the initialization process described above. Set to 0 to disable the incremental initialization process.
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\item\makelink{param:easyeq_bK}{}
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{\tt bK}, {\tt bL} ({\tt Float64}, default: 1, 1): the values of $\beta_K$ and $\beta_L$.
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\item\makelink{param:easyeq_eq}{}
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{\tt eq} ({\tt String}, default: ``{\tt simpleq}'', acceptable values: ``{\tt simpleq}'', ``{\tt medeq}''): A shortcut to select either the \refname{eq:easyeq_simpleq}{Simple equation} ($\beta_K=\beta_L=0$) or the \refname{eq:easyeq_medeq}{Medium equation} ($\beta_L=\beta_K=1$). When this option is set, neither {\tt bK} nor {\tt bL} should be set.
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\end{itemize}
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\bigskip
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The available {\tt commands} are the following.
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\begin{itemize}
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\item {\tt energy}\makelink{command:easyeq_energy}{}:
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compute the energy $e$ at a given $\rho$.\par
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\underline{Output}: [$e$] [Newton error $\epsilon$].
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\item {\tt energy\_rho}\makelink{command:easyeq_energy_rho}{}:
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compute the energy $e$ as a function of $\rho$.
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The Newton algorithm is initialized with the hardcore scattering solution (\ref{easyeq_init}) for the lowest $\rho$, and with the previously computed $\rho$ for the larger densities.\par
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\underline{Disabled parameters}: \refname{param:easyeq_rho}{{\tt rho}}.\par
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\underline{Extra parameters}:
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\begin{itemize}
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\item\makelink{param:easyeq_minlrho}{}
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{\tt minlrho} ({\tt Float64}, default: $10^{-6}$): minimal value for $\log_{10}\rho$.
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\item\makelink{param:easyeq_maxlrho}{}
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{\tt maxlrho} ({\tt Float64}, default: $10^{2}$): maximal value for $\log_{10}\rho$.
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\item\makelink{param:easyeq_nlrho}{}
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{\tt nlrho} ({\tt Int64}, default: $100$): number of values for $\rho$ (spaced logarithmically).
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\item\makelink{param:easyeq_minrho}{}
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{\tt minrho} ({\tt Float64}, default: $10^{-6}$): minimal value for $\rho$.
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\item\makelink{param:easyeq_maxrho}{}
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{\tt maxrho} ({\tt Float64}, default: $10^{2}$): maximal value for $\rho$.
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\item\makelink{param:easyeq_nrho}{}
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{\tt nrho} ({\tt Int64}, default: $0$): number of values for $\rho$ (spaced linearly). If {\tt nrho} is $\neq0$, then the linear spacing will be used, and \refname{param:easyeq_minlrho}{{\tt minlrho}}, \refname{param:easyeq_maxlrho}{{\tt maxlrho}}, \refname{param:easyeq_nlrho}{{\tt nlrho}} will be ignored. Otherwise, the logarithmic spacing will be used and \refname{param:easyeq_minrho}{{\tt minrho}}, \refname{param:easyeq_maxrho}{{\tt maxrho}} will be ignored.
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\item\makelink{param:easyeq_rhos}{}
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{\tt rhos} ({\tt Array\{Float64\}}): list of values for $\rho$, specified as a `{\tt,}' separated list.
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This parameter takes precedence over \refname{param:easyeq_minlrho}{{\tt minlrho}}, \refname{param:easyeq_maxlrho}{{\tt maxlrho}}, \refname{param:easyeq_nlrho}{{\tt nlrho}}, \refname{param:easyeq_minrho}{{\tt minrho}}, \refname{param:easyeq_maxrho}{{\tt maxrho}}, \refname{param:easyeq_nrho}{{\tt nrho}}.
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\end{itemize}
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\underline{Output} (one line for each value of $\rho$): [$\rho$] [$e$] [Newton error $\epsilon$].
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\item\makelink{command:easyeq_condensate_fraction}{}
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{\tt condensate\_fraction}: compute the uncondensed fraction $\eta$ at a given $\rho$.\par
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\underline{Output}: [$\eta$] [Newton error $\epsilon$].
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\item\makelink{command:easyeq_condensate_fraction_rho}{}
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{\tt condensate\_fraction\_rho}: compute the uncondensed fraction $\eta$ as a function of $\rho$.
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The Newton algorithm is initialized with the hardcore scattering solution (\ref{easyeq_init}) for the lowest $\rho$, and with the previously computed $\rho$ for the larger densities.\par
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\underline{Disabled parameters}: same as \refname{command:easyeq_energy_rho}{{\tt energy\_rho}}.\par
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\underline{Extra parameters}: same as \refname{command:easyeq_energy_rho}{{\tt energy\_rho}}.\par
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\underline{Output} (one line for each value of $\rho$): [$\rho$] [$\eta$] [Newton error $\epsilon$].
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\item\makelink{command:easyeq_uk}{}
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{\tt uk}: compute the Fourier transform $\hat u(|k|)$.
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The values $|k|$ at which $\hat u$ is computed are those coming from the Gauss quadratures, and cannot be set.\par
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\underline{Output} (one line for each value of $|k|$): [$|k|$] [$\hat u(|k|)$]
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\item\makelink{command:easyeq_ux}{}
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{\tt ux}: compute $u$ as a function of $|x|$.\par
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\underline{Extra parameters}:
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\begin{itemize}
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\item\makelink{param:easyeq_xmin}{}
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{\tt xmin} ({\tt Float64}, default: 0): minimum of the range of $|x|$ to be printed.
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\item\makelink{param:easyeq_xmax}{}
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{\tt xmax} ({\tt Float64}, default: 100): maximum of the range of $|x|$ to be printed.
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\item\makelink{param:easyeq_nx}{}
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{\tt nx} ({\tt Int64}, default: 100): number of points to print (linearly spaced).
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\end{itemize}
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\underline{Output} (one line for each value of $x$): [$|x|$] [$u(|x|)$]
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\item\makelink{command:easyeq_uux}{}
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{\tt uux}: compute $2u-\rho u\ast u$ as a function of $|x|$.\par
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\underline{Extra parameters}: Same as \refname{command:easyeq_ux}{{\tt ux}}.\par
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\underline{Output} (one line for each value of $x$): [$|x|$] [$2u(|x|)-\rho u\ast u(|x|)$]
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\item\makelink{command:easyeq_2pt}{}
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{\tt 2pt}: compute the spherically averaged two-point correlation function $C_2(|x|)$ at a given $\rho$.\par
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\underline{Extra parameters}: same as \refname{command:easyeq_ux}{{\tt ux}}, plus\par
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\begin{itemize}
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\item\makelink{param:easyeq_window_L}{}
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{\tt window\_L} ({\tt Float64}, default: $10^3$): size of the Hann window used to numerically invert the Fourier transform in the computation of the two-point correlation function, see\-~(\ref{hann}).
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\end{itemize}
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\underline{Output} (one line for each value of $|x|$): [$|x|$] [$C_2(|x|)$]
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\item\makelink{command:easyeq_2pt_max}{}
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{\tt 2pt\_max}: compute the maximum of the spherically averaged two-point correlation function $C_2(|x|)$ at a given $\rho$.\par
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\underline{Extra parameters}: \refname{param:easyeq_window_L}{{\tt window\_L}} plus
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\begin{itemize}
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\item\makelink{param:easyeq_dx}{}
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{\tt dx} ({\tt Float64}, default: $10^{-7}$): step used to numerically approximate derivatives.
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\item\makelink{param:easyeq_x0}{}
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{\tt x0} ({\tt Float64}, default: $1$): initial guess for the maximum is $\rho^{-1/3}\mathrm{\tt x0}$.
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\item\makelink{param:easyeq_maxstep}{}
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{\tt maxstep} ({\tt Float64}, default: $\infty$): maximal size of single step in maximization algorithm.
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\item\makelink{param:easyeq_tolerance_max}{}
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{\tt tolerance\_max} ({\tt Float64}, default: \refname{param:easyeq_tolerance}{{\tt tolerance}}): same as \refname{param:easyeq_tolerance}{{\tt tolerance}}, used for the Newton algorithm underlying the maximization algorithm.
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\end{itemize}
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\underline{Output}: [$|x_{\mathrm{max}}|$] [$C_2(|x_{\mathrm{max}}|)$]
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\item\makelink{command:easyeq_2pt_max_rho}{}
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{\tt 2pt\_max\_rho}: compute the maximum of the spherically averaged two-point correlation function $C_2(|x|)$ for a range of $\rho$.\par
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\underline{Extra parameters}: same as \refname{command:easyeq_2pt_max}{{\tt easyeq\_2pt\_max}} plus those of \refname{command:easyeq_energy_rho}{{\tt energy\_rho}}.\par
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\underline{Output} (one line for each value of $\rho$): [$\rho$] [$|x_{\mathrm{max}}|$] [$C_2(|x_{\mathrm{max}}|)$]\par
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\underline{Multithread support}: yes, different values of $\rho$ split up among workers.
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\item\makelink{command:easyeq_2pt_fourier}{}
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{\tt 2pt\_fourier}: compute the spherically averaged Fourier transform of the two-point correlation function $\hat C_2(|k|)$ at a given $\rho$.\par
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\underline{Extra parameters}:
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\begin{itemize}
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\item\makelink{param:easyeq_kmin}{}
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{\tt kmin} ({\tt Float64}, default: 0): minimum of the range of $|k|$ to be printed.
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\item\makelink{param:easyeq_kmax}{}
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{\tt kmax} ({\tt Float64}, default: 10): maximum of the range of $|k|$ to be printed.
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\item\makelink{param:easyeq_nk}{}
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{\tt nk} ({\tt Int64}, default: 100): number of $|k|$'s to be printed.
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\item\makelink{param:easyeq_2pt_fourier_window_L}{}
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{\tt window\_L} ({\tt Float64}, default: 1000): what is actually computed is $\hat C_2$ convolved with a Gaussian of variance $1/\sqrt L$ with $L=\mathrm{\tt window\_L}$, see\-~(\ref{easyeq_gaussian}).
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\end{itemize}
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\underline{Output} (one line for each value of $|k|$): [$|k|$] [$\hat C_2(|k|)$].\par
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\underline{Multithread support}: yes, different values of $k$ are split up among workers.
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\item\makelink{command:easyeq_2pt_fourier_max}{}
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{\tt 2pt\_fourier\_max}: compute the maximum of the spherically averaged Fourier transformed two-point correlation function $\hat C_2(|k|)$.\par
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\underline{Extra parameters}: \refname{param:easyeq_2pt_fourier_window_L}{{\tt window\_L}}, \refname{param:easyeq_maxstep}{{\tt maxstep}} plus
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\begin{itemize}
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\item\makelink{param:easyeq_dk}{}
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{\tt dk} ({\tt Float64}, default: $10^{-7}$): step used to numerically approximate derivatives.
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\item\makelink{param:easyeq_k0}{}
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{\tt k0} ({\tt Float64}, default: $1$): initial guess for the maximum is $\rho^{1/3}\mathrm{\tt k0}$.
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\end{itemize}
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\underline{Output}: [$|k_{\mathrm{max}}|$] [$\hat C_2(|k_{\mathrm{max}}|)$]\par
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\item\makelink{command:easyeq_2pt_fourier_max_rho}{}
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{\tt 2pt\_fourier\_max\_rho}: compute the maximum of the spherically averaged Fourier transformed two-point correlation function $\hat C_2(|k|)$ for a range of $\rho$.\par
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\underline{Extra parameters}: same as those of \refname{command:easyeq_2pt_fourier_max}{{\tt 2pt\_fourier\_max}} plus those of \refname{command:easyeq_energy_rho}{{\tt energy\_rho}}.\par
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\underline{Output} (one line for each value of $\rho$): [$\rho$] [$|k_{\mathrm{max}}|$] [$\hat C_2(|k_{\mathrm{max}}|)$]\par
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\underline{Multithread support}: yes, different values of $\rho$ split up among workers.
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\item\makelink{command:easyeq_momentum_distribution}{}
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|
{\tt momentum\_distribution}: compute the momentum distribution $\mathcal M(|k|)$ at a given $\rho$. The momentum distribution is computed for $k=k_{l,j}$ (see\-~(\ref{klj})).\par
|
|
\underline{Extra parameters}:
|
|
\begin{itemize}
|
|
\item\makelink{param:easyeq_momentum_kmin}{}
|
|
{\tt kmin} ({\tt Float64}, default: 0): minimum of the range of $|k|$ to be printed.
|
|
|
|
\item\makelink{param:easyeq_momentum_kmax}{}
|
|
{\tt kmax} ({\tt Float64}, default: 10): maximum of the range of $|k|$ to be printed.
|
|
|
|
\item\makelink{param:easyeq_momentum_window_L}{}
|
|
{\tt window\_L} ({\tt Float64}, default: 1000): what is actually computed is $\mathfrak M_k$ convolved with a Gaussian of variance $1/\sqrt L$ where $L=\sqrt{\mathrm{\tt window\_L}}/k^2$, see\-~(\ref{easyeq_gaussian_momt}).
|
|
\end{itemize}
|
|
\underline{Output} (one line for each value of $|k|$): [$|k|$] [$\mathcal M(|k|)$]
|
|
|
|
\end{itemize}
|
|
|
|
\subsubsection{Description}
|
|
\subsubsubsection{Fourier space formulation}
|
|
The computation is carried out in Fourier space.
|
|
We take the convention that the Fourier transform of a function $f(|x|)$ is
|
|
\begin{equation}
|
|
\hat f(|k|)=\int_{\mathbb R^3} dx\ e^{ikx}f(|x|)
|
|
=\frac{4\pi}{|k|}\int_0^\infty dr\ r\sin(|k|r)f(r)
|
|
.
|
|
\end{equation}
|
|
In Fourier space, (\ref{easyeq}) becomes
|
|
\begin{equation}
|
|
k^2\hat u=\hat S-2\rho\hat A_K\hat u+\rho^2\hat A_L\hat u^2
|
|
\end{equation}
|
|
\begin{equation}
|
|
\hat A_K:=\beta_K\hat S+(1-\beta_K)\frac{2e}\rho
|
|
,\quad
|
|
\hat A_L:=\beta_L\hat S+(1-\beta_L)\frac{2e}\rho
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\hat S(|k|)=\int_{\mathbb R^3} dx\ e^{ikx}(1-u(|x|))v(|x|)
|
|
=\hat v(k)-\hat u\hat\ast\hat v(k)
|
|
,\quad
|
|
\hat f\hat\ast\hat g(|k|):=\int_{\mathbb R^3}\frac{dp}{(2\pi)^3}\ \hat f(|k-p|)\hat g(|p|)
|
|
.
|
|
\label{hatS_easyeq}
|
|
\end{equation}
|
|
We write this as a quadratic equation for $\hat u$, and solve it, keeping the solution that decays as $|k|\to\infty$:
|
|
\begin{equation}
|
|
\rho\hat u(|k|)=
|
|
\frac{A_K(|k|)}{A_L(|k|)}\left(\xi(|k|)+1-\sqrt{(\xi(|k|)+1)^2-\frac{A_L(|k|)}{A_K^2(|k|)}\hat S(|k|)}\right)
|
|
\end{equation}
|
|
that is,
|
|
\begin{equation}
|
|
\rho\hat u(|k|)=
|
|
\frac{\hat S(|k|)}{2A_K(|k|)(\xi(|k|)+1)}\Phi\left({\textstyle\frac{A_L(|k|)}{A_K^2(|k|)}\frac{\hat S(|k|)}{(\xi(|k|)+1)^2}}\right)
|
|
\label{uk_easyeq}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\xi(|k|):=\frac{k^2}{2\rho\hat A_K}
|
|
,\quad
|
|
\Phi(x):=\frac{2(1-\sqrt{1-x})}x
|
|
.
|
|
\end{equation}
|
|
We can write this as a root finding problem:
|
|
\begin{equation}
|
|
\Omega(u):=
|
|
\rho\hat u(|k|)-
|
|
\frac{\hat S(|k|)}{2A_K(|k|)(\xi(|k|)+1)}\Phi\left({\textstyle\frac{A_L(|k|)}{A_K^2(|k|)}\frac{\hat S(|k|)}{(\xi(|k|)+1)^2}}\right)
|
|
=0
|
|
.
|
|
\label{Omega_easyeq}
|
|
\end{equation}
|
|
Furthermore, using bipolar coordinates (see lemma\-~\ref{lemma:bipolar}), we write $\hat S$ as
|
|
\begin{equation}
|
|
\hat S(|k|)=\hat v(|k|)-\frac1{8\pi^3}\int_0^\infty dt\ t\hat u(t)\Eta(|k|,t)
|
|
,\quad
|
|
\Eta(y,t):=\frac{2\pi}y\int_{|y-t|}^{y+t}ds\ s\hat v(s)
|
|
.
|
|
\label{easyeq_Eta}
|
|
\end{equation}
|
|
By a simple change of variables,
|
|
\begin{equation}
|
|
\Eta(y,t)=
|
|
4\pi\left(\mathds 1_{y>t}\frac{t}y+\mathds 1_{y\leqslant t}\right)
|
|
\int_0^1 ds\ ((y+t)s+|y-t|(1-s))v((y+t)s+|y-t|(1-s))
|
|
.
|
|
\label{Eta}
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subsubsubsection{Evaluating integrals}
|
|
To compute these integrals numerically, we will use Gauss-Legendre quadratures:
|
|
\begin{equation}
|
|
\int_0^1 dt\ f(t)\approx\frac12\sum_{i=1}^N w_if\left({\textstyle\frac{r_i+1}2}\right)
|
|
\end{equation}
|
|
where $w_i$ and $r_i$ are the Gauss-Legendre {\it weights} and {\it abcissa}.
|
|
The order $N$ corresponds to the parameter \refname{param:easyeq_order}{{\tt order}}.
|
|
The error made by the quadrature is estimated in appendix\-~\ref{appGL}.
|
|
We compactify the integrals to the interval $(0,1)$: $y:=\frac1{t+1}$:
|
|
\begin{equation}
|
|
\hat S(|k|)=\hat v(|k|)-\frac1{8\pi^3}\int_0^1 dy\ \frac{(1-y)\hat u(\frac{1-y}y)\Eta(|k|,\frac{1-y}y)}{y^3}
|
|
\end{equation}
|
|
so, using the Gauss-Legendre quadrature, we approximate
|
|
\begin{equation}
|
|
\hat S(|k|)\approx\hat v(|k|)-\frac1{16\pi^3}\sum_{j=1}^N w_j \frac{(1-y_j)\hat u(\frac{1-y_j}{y_j})\Eta(|k|,\frac{1-y_j}{y_j})}{y_j^3}
|
|
,\quad
|
|
y_j:=\frac{r_j+1}2
|
|
.
|
|
\end{equation}
|
|
This suggests a natural discretization of Fourier space: let
|
|
\begin{equation}
|
|
k_i:=\frac{1-r_i}{1+r_i}\equiv\frac{1-y_i}{y_i}
|
|
.
|
|
\label{easyeq_ki}
|
|
\end{equation}
|
|
Thus, defining
|
|
\begin{equation}
|
|
\mathbb U_i:=\rho\hat u(k_i)
|
|
,\quad
|
|
\hat v_i:=\hat v(k_i)
|
|
\end{equation}
|
|
and approximate\-~(\ref{uk_easyeq}):
|
|
\begin{equation}
|
|
\mathbb U_i=\frac{\mathbb T_i}{2(\mathbb X_i+1)}\Phi\left({\textstyle\mathbb B_i\frac{\mathbb T_i}{(\mathbb X_i+1)^2}}\right)
|
|
\label{bbU_easyeq}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\mathbb A_{K,i}:=\beta_K\mathbb S_i+(1-\beta_K)\mathbb E
|
|
,\quad
|
|
\mathbb B_{i}:=\frac{\beta_L\mathbb S_i+(1-\beta_L)\mathbb E}{\mathbb A_{K,i}}
|
|
,\quad
|
|
\mathbb T_i:=\frac{\mathbb S_i}{\mathbb A_{K,i}}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\mathbb S_i:=\hat v_i-\frac1{16\pi^3\rho}\sum_{j=1}^N w_j\frac{(1-y_j)\mathbb U_j\Eta(k_i,k_j)}{y_j^3}
|
|
,\quad
|
|
\mathbb E:=\hat v(0)-\frac1{16\pi^3\rho}\sum_{j=1}^N w_j\frac{(1-y_j)\mathbb U_j\Eta(0,k_j)}{y_j^3}
|
|
\label{bbSE_easyeq}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\mathbb X_i:=\frac{k_i^2}{2\rho\mathbb A_{K,i}}
|
|
.
|
|
\end{equation}
|
|
This is a discrete equation for the vector $(\mathbb U_i)_{i=1}^N$.
|
|
\bigskip
|
|
|
|
\subsubsubsection{Main algorithm to compute $\mathbb U$}\par\penalty10000
|
|
\medskip\penalty10000
|
|
\point
|
|
We rewrite\-~(\ref{bbU_easyeq}) as a root finding problem:
|
|
\begin{equation}
|
|
\Xi_i(\mathbb U):=\mathbb U_i-\frac{\mathbb T_i}{2(\mathbb X_i+1)}\Phi\left({\textstyle\mathbb B_i\frac{\mathbb T_i}{(\mathbb X_i+1)^2}}\right)
|
|
=0
|
|
\label{root_easyeq}
|
|
\end{equation}
|
|
which we solve using the Newton algorithm, that is, we define a sequence of $\mathbb U$'s:
|
|
\begin{equation}
|
|
\mathbb U^{(n+1)}=\mathbb U^{(n)}-(D\Xi(\mathbb U^{(n)}))^{-1}\Xi(\mathbb U^{(n)})
|
|
\end{equation}
|
|
where $D\Xi$ is the Jacobian of $\Xi$:
|
|
\begin{equation}
|
|
(D\Xi)_{i,j}:=\frac{\partial\Xi_i}{\partial\mathbb U_j}
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
For small values of $\rho$, we initialize the algorithm with the hardcore scattering solution
|
|
\begin{equation}
|
|
\hat u_0(k)=\frac{4\pi a_0}{k^2}
|
|
\label{easyeq_init}
|
|
\end{equation}
|
|
where $a_0$ is the scattering length of the potential $v$ (or an approximation thereof, which need not be very good).
|
|
Thus,
|
|
\begin{equation}
|
|
\mathbb U^{(0)}_i=\frac{4\pi a_0}{k_i^2}
|
|
.
|
|
\end{equation}
|
|
This is a good approximation for small $\rho$.
|
|
For larger $\rho$, we choose $\mathbb U^{(0)}$ as the solution of {\tt easyeq} for a slightly smaller $\rho$, and proceed inductively (using the parameters \refname{param:easyeq_minlrho_init}{{\tt minlrho\_init}} and \refname{param:easyeq_nlrho_init}{{\tt nlrho\_init}}).
|
|
\bigskip
|
|
|
|
\point
|
|
We are left with computing the Jacobian of $\Xi$:
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\frac{\partial\Xi_i}{\partial\mathbb U_j}
|
|
=\delta_{j,i}
|
|
-\frac1{2(\mathbb X_i+1)}\left(
|
|
\partial_j\mathbb T_i-\frac{\mathbb T_i\partial_j\mathbb X_i}{\mathbb X_i+1}
|
|
\right)\Phi\left({\textstyle\mathbb B_i\frac{\mathbb T_i}{(\mathbb X_i+1)^2}}\right)
|
|
\\[0.5cm]\hfill
|
|
-\frac{\mathbb T_i}{2(\mathbb X_i+1)^3}\left(
|
|
\mathbb B_i\partial_j\mathbb T_i+\mathbb T_i\partial_j\mathbb B_i-2\frac{\mathbb B_i\mathbb T_i\partial_j\mathbb X_i}{\mathbb X_i+1}
|
|
\right)\partial\Phi\left({\textstyle\mathbb B_i\frac{\mathbb T_i}{(\mathbb X_i+1)^2}}\right)
|
|
\end{largearray}
|
|
\label{jacobian_easyeq}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\partial_j\mathbb B_i=
|
|
(\beta_L(1-\beta_K)-\beta_K(1-\beta_L))
|
|
\frac{\mathbb E\partial_j\mathbb S_i-\mathbb S_i\partial_j\mathbb E}{(\beta_K\mathbb S_i+(1-\beta_K)\mathbb E)^2}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_j\mathbb T_i=
|
|
(1-\beta_K)\frac{\mathbb E\partial_j\mathbb S_i-\mathbb S_i\partial_j\mathbb E}{(\beta_K\mathbb S_i+(1-\beta_K)\mathbb E)^2}
|
|
,\quad
|
|
\partial_j\mathbb A_{K,i}=\beta_K\partial_j\mathbb S_i+(1-\beta_K)\partial_j\mathbb E
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_j\mathbb S_i:=-\frac1{16\pi^3\rho}w_j\frac{(1-y_j)\Eta(k_i,k_j)}{y_j^3}
|
|
,\quad
|
|
\partial_j\mathbb E:=-\frac1{16\pi^3\rho}\sum_{j=1}^N w_j\frac{(1-y_j)\Eta(0,k_j)}{y_j^3}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_j\mathbb X_i:=-\frac{k_i^2}{2\rho\mathbb A_{K,i}^2}\partial_j\mathbb A_{K,i}
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
We iterate the Newton algorithm until the Newton relative error $\epsilon$ becomes smaller than the \refname{param:easyeq_tolerance}{{\tt tolerance}} parameter.
|
|
The Newton error is defined as
|
|
\begin{equation}
|
|
\epsilon=\frac{\|\mathbb U^{(n+1)}-\mathbb U^{(n)}\|_2}{\|\mathbb U^{(n)}\|_2}
|
|
\end{equation}
|
|
where $\|\cdot\|_2$ is the $l_2$ norm.
|
|
The energy thus obtained is
|
|
\begin{equation}
|
|
e=\frac\rho2\mathds E
|
|
\end{equation}
|
|
the Fourier transform $\hat u$ of the solution is
|
|
\begin{equation}
|
|
\hat u(k_i)\approx\frac{\mathbb U_i}\rho
|
|
\end{equation}
|
|
and the solution $u$ in real space is obtained by inverting the Fourier transform
|
|
\begin{equation}
|
|
\begin{array}{>\displaystyle r@{\ }>\displaystyle l}
|
|
u(|x|)=\int\frac{dk}{8\pi^3}\ e^{-ikx}\hat u(|k|)
|
|
=&\frac1{2\pi^2|x|}\int_0^1dy\ \frac{(1-y)\sin(|x|\frac{1-y}y)\hat u(\frac{1-y}y)}{y^3}
|
|
\\[0.5cm]
|
|
\approx&
|
|
\frac1{4\pi^2|x|}\sum_{j=1}^Nw_j (1+k_j)^2k_j\sin(|x|k_j)\hat u(k_j)
|
|
.
|
|
\end{array}
|
|
\label{ux}
|
|
\end{equation}
|
|
To compute $2u-\rho u\ast u$, we replace $\hat u$ with $2\hat u-\rho\hat u^2$ in the previous equation.
|
|
\bigskip
|
|
|
|
\subsubsubsection{Condensate fraction}
|
|
Finally, to compute the uncondensed fraction, we solve the modified {\tt easyeq} (see\-~\cite{CJL21})
|
|
\begin{equation}
|
|
(-\Delta+2\mu)u_\mu=v(1-u_\mu)-2\rho K+\rho^2L
|
|
\label{easyeq_mu}
|
|
\end{equation}
|
|
where $K$ and $L$ are defined as in\-~(\ref{easyeq_KL})-(\ref{easyeq_Se}) in which $u$ is replaced with $u_\mu$.
|
|
The uncondensed fraction is then
|
|
\begin{equation}
|
|
\eta=\partial_\mu e|_{\mu=0}
|
|
=-\frac\rho2\int dx\ v(|x|)\partial_\mu u_\mu(|x|)|_{\mu=0}
|
|
.
|
|
\end{equation}
|
|
To compute the energy in the presence of the parameter $\mu$, we proceed in the same way as for $\mu=0$, the only difference being that $k^2$ should formally be replaced by $k^2+2\mu$.
|
|
In other words, we consider $\mathbb U_i=u_\mu(|k_i|)$ and define $\Xi(\mathbb U,\mu)$ in the same way as in\-~(\ref{root_easyeq}), except that $\mathbb X_i$ should be replaced by
|
|
\begin{equation}
|
|
\frac{k_i^2+2\mu}{2\rho\mathbb A_{K,i}}
|
|
.
|
|
\end{equation}
|
|
We then solve
|
|
\begin{equation}
|
|
\Xi(\mathbb U,\mu)=0
|
|
.
|
|
\end{equation}
|
|
By differentiating this identity with respect to $\mu$, we find $\partial_\mu u_\mu$:
|
|
\begin{equation}
|
|
D\Xi\partial_\mu \mathbb U=-\partial_\mu\Xi
|
|
\end{equation}
|
|
and the uncondensed fraction is
|
|
\begin{equation}
|
|
\eta
|
|
=\frac12\int dx\ v(x)(D\Xi)^{-1}\partial_\mu\Xi|_{\mu=0}
|
|
\end{equation}
|
|
thus
|
|
\begin{equation}
|
|
\eta=\frac1{16\pi^3}\int_0^1 dy\ \frac{(1-y)\Eta(0,\frac{1-y}y)(D\Xi)^{-1}\partial_\mu\Xi|_{\mu=0}(\frac{1-y}y)}{y^3}
|
|
\end{equation}
|
|
which we approximate using a Gauss-Legendre quadrature:
|
|
\begin{equation}
|
|
\eta\approx\frac1{32\pi^3}\sum_{j=1}^N w_j(1-k_j)^2k_j\Eta(0,k_j)\sum_{i=1}^N(D\Xi)^{-1}_{j,i}\partial_\mu\Xi_i|_{\mu=0}
|
|
.
|
|
\label{eta_easyeq}
|
|
\end{equation}
|
|
We then compute, using\-~(\ref{jacobian_easyeq}),
|
|
\begin{equation}
|
|
\partial_\mu\Xi_i|_{\mu=0}
|
|
=
|
|
\frac{\mathbb T_i}{2\rho\mathbb A_{K,i}(\mathbb X_i+1)^2}\Phi\left({\textstyle\mathbb B_i\frac{\mathbb T_i}{(\mathbb X_i+1)^2}}\right)
|
|
+\frac{\mathbb B_i\mathbb T_i^2}{\rho\mathbb A_{K,i}(\mathbb X_i+1)^4}\partial\Phi\left({\textstyle\mathbb B_i\frac{\mathbb T_i}{(\mathbb X_i+1)^2}}\right)
|
|
.
|
|
\end{equation}
|
|
|
|
\subsubsubsection{Correlation function (spherical average)}
|
|
The two-point correlation function is
|
|
\begin{equation}
|
|
c_2(x):=
|
|
2\rho\frac{\delta e}{\delta v(x)}
|
|
\end{equation}
|
|
and its spherical average is
|
|
\begin{equation}
|
|
C_2(|x|):=\frac1{4\pi|x|^2}\int dy\ \delta(|x|-|y|)c_2(y)
|
|
.
|
|
\end{equation}
|
|
In Fourier space,
|
|
\begin{equation}
|
|
c_2(x)=
|
|
2\rho
|
|
\int dk\ e^{ikx}\frac{\delta e}{\delta\hat v(k)}
|
|
\end{equation}
|
|
so
|
|
\begin{equation}
|
|
C_2(|x|)=
|
|
2\rho\int dk\ \left(\frac1{4\pi|x|^2}\int dy\ \delta(|x|-|y|)e^{iky}\right)\frac{\delta e}{\delta\hat v(k)}
|
|
=2\rho\int dk\ \frac{\sin(|k||x|)}{|k||x|}\frac{\delta e}{\delta\hat v(k)}
|
|
.
|
|
\label{easyeq_C2fourier}
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
We can compute this quantity by considering a modified {\tt easyeq} in Fourier space, by formally replacing $\hat v$ with
|
|
\begin{equation}
|
|
\hat v+\lambda g(|k|)
|
|
,\quad
|
|
g(|k|):=\frac{\sin(|k||x|)}{|k||x|}
|
|
.
|
|
\label{2pt_addv}
|
|
\end{equation}
|
|
Indeed, if $e_\lambda$ denotes the energy of this modified equation,
|
|
\begin{equation}
|
|
\partial_\lambda e_\lambda|_{\lambda=0}
|
|
=\int dk\ \frac{\delta e}{\delta\hat v(k)}\partial_\lambda({\textstyle \hat v(k)+\lambda g(|k|)})
|
|
=\int dk\ g(|k|)\frac{\delta e}{\delta\hat v(k)}
|
|
.
|
|
\end{equation}
|
|
So, denoting the solution of the modified equation by $u_\lambda$,
|
|
\begin{equation}
|
|
C_2(x)=2\rho\partial_\lambda e_\lambda|_{\lambda=0}
|
|
=\rho^2g(0)-\rho^2\int\frac{dk}{(2\pi)^3}\ (g(k)\hat u(k)+\hat v(k)\partial_\lambda\hat u_\lambda(k)|_{\lambda=0})
|
|
.
|
|
\end{equation}
|
|
We compute $\partial_\lambda u_\lambda|_{\lambda=0}$ in the same way as the uncondensed fraction: we define $\Xi(\mathbb U,\lambda)$ by formally adding $\lambda g(|k|)$ to $\hat v$, solve $\Xi(\mathbb U,\lambda)=0$, and differentiate:
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb U|_{\lambda=0}=-(D\Xi)^{-1}\partial_\lambda\Xi|_{\lambda=0}
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
We compute $\partial_\lambda\Xi|_{\lambda=0}$:
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\partial_\lambda\Xi_i
|
|
=
|
|
-\frac1{2(\mathbb X_i+1)}\left(
|
|
\partial_\lambda\mathbb T_i-\frac{\mathbb T_i\partial_\lambda\mathbb X_i}{\mathbb X_i+1}
|
|
\right)\Phi\left({\textstyle\mathbb B_i\frac{\mathbb T_i}{(\mathbb X_i+1)^2}}\right)
|
|
\\[0.5cm]\hfill
|
|
-\frac{\mathbb T_i}{2(\mathbb X_i+1)^3}\left(
|
|
\mathbb B_i\partial_\lambda\mathbb T_i+\mathbb T_i\partial_\lambda\mathbb B_i-2\frac{\mathbb B_i\mathbb T_i\partial_\lambda\mathbb X_i}{\mathbb X_i+1}
|
|
\right)\partial\Phi\left({\textstyle\mathbb B_i\frac{\mathbb T_i}{(\mathbb X_i+1)^2}}\right)
|
|
\end{largearray}
|
|
\label{dXi_2pt_easyeq}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb B_i=
|
|
(\beta_L(1-\beta_K)-\beta_K(1-\beta_L))
|
|
\frac{\mathbb E\partial_\lambda\mathbb S_i-\mathbb S_i\partial_\lambda\mathbb E}{(\beta_K\mathbb S_i+(1-\beta_K)\mathbb E)^2}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb T_i=
|
|
(1-\beta_K)\frac{\mathbb E\partial_\lambda\mathbb S_i-\mathbb S_i\partial_\lambda\mathbb E}{(\beta_K\mathbb S_i+(1-\beta_K)\mathbb E)^2}
|
|
,\quad
|
|
\partial_\lambda\mathbb A_{K,i}=\beta_K\partial_\lambda\mathbb S_i+(1-\beta_K)\partial_\lambda\mathbb E
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb S_i:=g(k_i)
|
|
-\frac1{16\pi^3\rho}\sum_{j=1}^N w_j\frac{(1-y_j)\mathbb U_j\partial_\lambda\Eta(k_i,k_j)}{y_j^3}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_\lambda \mathbb E:=g(0)-\frac1{16\pi^3\rho}\sum_{j=1}^N w_j\frac{(1-y_j)\mathbb U_j\partial_\lambda\Eta(0,k_j)}{y_j^3}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb X_i:=-\frac{k_i^2}{2\rho\mathbb A_{K,i}^2}\partial_\lambda\mathbb A_{K,i}
|
|
\end{equation}
|
|
where $\partial\lambda\Eta$ is computed similarly to $\Eta$\-~(\ref{Eta}) but with $\hat v$ replaced by $g$:
|
|
\begin{equation}
|
|
\partial_\lambda\Eta(y,t)=
|
|
4\pi\left(\mathds 1_{y>t}\frac{t}y+\mathds 1_{y\leqslant t}\right)
|
|
\int_0^1 ds\ ((y+t)s+|y-t|(1-s))g((y+t)s+|y-t|(1-s))
|
|
.
|
|
\label{GG}
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
In order to invert the Fourier transform in\-~(\ref{easyeq_C2fourier}) numerically, we will use a Hann window (see appendix\-~\ref{appendix:hann})
|
|
\begin{equation}
|
|
H_L(k):=\mathds 1_{|k|<\frac L2}\cos^2({\textstyle\frac{\pi|k|}{L}})
|
|
.
|
|
\label{hann}
|
|
\end{equation}
|
|
The parameter $L$ is set using \refname{param:easyeq_window_L}{{\tt window\_L}}.
|
|
The computation is changed only in that $g$ is changed to $H_L(k)\frac{\sin(|k||x|)}{|k||x|}$.
|
|
\bigskip
|
|
|
|
\point
|
|
To compute the maximum of $C_2$, we use a modified Newton algorithm.
|
|
The initial guess for the maximum is $|x_0|=\rho^{-\frac13}$\refname{param:easyeq_x0}{{\tt x0}}.
|
|
The modified Newton algorithm is an iteration:
|
|
\begin{equation}
|
|
x_{n+1}=x_n+\frac{\partial C_2(|x_n|)}{|\partial^2C_2(|x_n|)|}
|
|
\label{easyeq_newton_2pt}
|
|
\end{equation}
|
|
in which the derivatives are approximated using finite differences:
|
|
\begin{equation}
|
|
\partial C_2(x)\approx \frac{C_2(|x|+dx)-C_2(|x|)}{dx}
|
|
,\quad
|
|
\partial^2 C_2(x)\approx \frac{C_2(|x|+dx)+C_2(|x|-dx)-2C_2(|x|)}{dx^2}
|
|
.
|
|
\label{easyeq_dx_2pt}
|
|
\end{equation}
|
|
This is a modification of the usual Newton iteration $x_n+\partial C_2/\partial^2C_2$ which is designed to follow the direction of the gradient, and thus to move toward a local maximum.
|
|
In addition, if $|\partial C_2|/|\partial^2 C_2|$ is larger than \refname{param:easyeq_maxstep}{{\tt maxstep}}, then the step is replaced with $\pm$\refname{param:easyeq_maxstep}{{\tt maxstep}}.
|
|
This prevents the algorithm from stepping over a maximum and land on another, further away.
|
|
This is useful if one has a good idea of where the global maximum is, and does not want to get trapped in a smaller local maximum.
|
|
\bigskip
|
|
|
|
\indent
|
|
The algorithm is run for a maximum of \refname{param:easyeq_maxiter}{{\tt maxiter}} iterations, or until $|x_{n+1}-x_n|$ is smaller than \refname{param:easyeq_tolerance}{{\tt tolerance}}.
|
|
If the maximal number of iterations is reached, or if the solution found is not a local maximum, then the algorithm fails, and returns $+\infty$.
|
|
The point thus computed is therefore a local maximum, but it is not guaranteed to be the global maximum.
|
|
|
|
\subsubsubsection{Fourier transform of two-point correlation (spherical average)}
|
|
The Fourier transform of the two-point correlation function is
|
|
\begin{equation}
|
|
\hat c_2(q):=
|
|
2\rho\frac{\delta e}{\delta v(q)}
|
|
\end{equation}
|
|
and its spherical average is
|
|
\begin{equation}
|
|
\hat C_2(|q|):=\frac1{4\pi|q|^2}\int dk\ \delta(|q|-|k|)c_2(k)
|
|
=
|
|
\frac\rho{2\pi|q|^2}\int dk\ \delta(|q|-|k|)\frac{\delta e}{\delta\hat v(k)}
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
To compute $\frac{\delta e}{\delta\hat v(q)}$, one idea would be to proceed in the same way as for the two-point correlation function, by replacing $\hat v$ with
|
|
\begin{equation}
|
|
\hat v+\lambda g(|k|)
|
|
,\quad
|
|
g(|k|):=\frac1{4\pi|q|^2}\delta(|q|-|k|)
|
|
\end{equation}
|
|
where $\delta$ is the Dirac-delta function distribution (compare this with\-~(\ref{2pt_addv})).
|
|
However, the $\delta$ function causes all sorts of problems with the quadratures.
|
|
\bigskip
|
|
|
|
\point
|
|
Instead, we approximate $\hat C_2$ by convolving it with a normalized Gaussian: let
|
|
\begin{equation}
|
|
\Gamma_L(|k|):=\left(\frac{L}{2\pi}\right)^{\frac32}e^{-\frac L2k^2}
|
|
\label{easyeq_gaussian}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\hat{\mathfrak C}_2(|q|)
|
|
:=
|
|
\int dp\ \hat C_2(|q-p|)\Gamma_L(|p|)
|
|
=\int dk\ \int dp\ \frac\rho{2\pi|q-p|^2}\delta(|q-p|-|k|)\frac{\delta e}{\delta\hat v(k)}\Gamma_L(|p|)
|
|
\end{equation}
|
|
which by lemma\-~\ref{lemma:bipolar} is
|
|
\begin{equation}
|
|
\hat{\mathfrak C}_2(|q|)
|
|
=\int dk\
|
|
\frac\rho{|q|}
|
|
\frac{\delta e}{\delta\hat v(k)}
|
|
\int_0^\infty dt\ \int_{||q|-t|}^{|q|+t}ds\ s\frac{\delta(t-|k|)}t\Gamma_L(s)
|
|
\end{equation}
|
|
that is
|
|
\begin{equation}
|
|
\hat{\mathfrak C}_2(|q|)
|
|
=\int dk\
|
|
\frac{\delta e}{\delta\hat v(k)}
|
|
\frac{\rho}{|q||k|}
|
|
\int_{||q|-|k||}^{|q|+|k|}ds\ s\Gamma_L(s)
|
|
\end{equation}
|
|
which is the directional derivative of $e$ with respect to $\hat v$ in the direction of $2\rho g$ with
|
|
\begin{equation}
|
|
g(|k|):=\frac1{2|q||k|}\int_{||q|-|k||}^{|q|+|k|}ds\ s\Gamma_L(s)
|
|
=
|
|
\frac1{2|k|rL}(\Gamma_L(|k|-r)-\Gamma_L(|k|+r))
|
|
.
|
|
\label{easyeq_2pt_fourier_g}
|
|
\end{equation}
|
|
Note that
|
|
\begin{equation}
|
|
g(0):=\Gamma_L(|q|)
|
|
.
|
|
\end{equation}
|
|
To compute this derivative, we replace $\hat v$ with
|
|
\begin{equation}
|
|
\hat v+\lambda g(|k|)
|
|
\end{equation}
|
|
so, denoting the solution of the modified equation by $u_\lambda$, for $q\neq 0$,
|
|
\begin{equation}
|
|
\hat{\mathfrak C}_2(|q|)=2\rho\partial_\lambda e_\lambda|_{\lambda=0}
|
|
=\rho^2\left(
|
|
-\int\frac{dk}{(2\pi)^3}\ g(|k|)\hat u(|k|)
|
|
-\int\frac{dk}{(2\pi)^3}\ \hat v(|k|)\partial_\lambda\hat u_\lambda(|k|)|_{\lambda=0}
|
|
\right)
|
|
.
|
|
\end{equation}
|
|
To compute $\partial_\lambda\hat u_\lambda|_{\lambda=0}$, we differentiate $\Xi(\mathbb U,\lambda)=0$:
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb U|_{\lambda=0}=-(D\Xi)^{-1}\partial_\lambda\Xi|_{\lambda=0}
|
|
.
|
|
\end{equation}
|
|
The computation of $\partial_\lambda\Xi|_{\lambda=0}$ is identical to\-~(\ref{dXi_2pt_easyeq}), but with the $g$ defined in\-~(\ref{easyeq_2pt_fourier_g}).
|
|
\bigskip
|
|
|
|
\point
|
|
To compute the maximum of $\hat C_2$, we proceed as for $C_2$, see\-~(\ref{easyeq_newton_2pt})-(\ref{easyeq_dx_2pt}).
|
|
The only difference is that the algorithm is initialized with $|k_0|=\rho^{\frac13}$\refname{param:easyeq_k0}{{\tt k0}}.
|
|
|
|
\subsubsubsection{Momentum distribution}
|
|
To compute the momentum distribution (see\-~\cite{CHe21}), we add a parameter $\lambda$ to {\tt easyeq}:
|
|
\begin{equation}
|
|
-\Delta u_\lambda(|x|)
|
|
=
|
|
(1-u_\lambda(|x|))v(|x|)-2\rho K(|x|)+\rho^2 L(|x|)
|
|
-2\lambda \hat u_0(q)\cos(q\cdot x)
|
|
\end{equation}
|
|
($\hat u_0\equiv\hat u_\lambda|_{\lambda=0}$).
|
|
The momentum distribution is then
|
|
\begin{equation}
|
|
\mathcal M(q)=\partial_\lambda e|_{\lambda=0}
|
|
=-\frac\rho2\int\frac{dk}{(2\pi)^3}\ \hat v(k)\partial_\lambda\hat u_\lambda(k)|_{\lambda=0}
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
Note that the Fourier transform of $2\lambda\hat u_0(q)\cos(q\cdot x)$ is
|
|
\begin{equation}
|
|
-(2\pi)^3\lambda\hat u_0(q)(\delta(q+k)+\delta(q-k))
|
|
.
|
|
\label{fouriermomentum}
|
|
\end{equation}
|
|
The presence of delta functions does not play well with the quadratures.
|
|
To get around this, we instead compute a regularization of $\mathcal M(q)$ by convolving it with a peaked spherically symmetric function.
|
|
Let $\Gamma_L$ denote the Gaussian with variance $1/\sqrt L$
|
|
\begin{equation}
|
|
\Gamma_L(|k|):=\left(\frac{L}{2\pi}\right)^{\frac32}e^{-\frac L2k^2}
|
|
.
|
|
\label{easyeq_gaussian_momt}
|
|
\end{equation}
|
|
In fact, we will scale $L$ with $k$, and set $L$ to
|
|
\begin{equation}
|
|
L=\sqrt{\mathrm{\refname{param:easyeq_momentum_window_L}{{\tt window\_L}}}}/k^2
|
|
.
|
|
\end{equation}
|
|
To compute
|
|
\begin{equation}
|
|
\mathfrak M(q):=\mathcal M\ast\Gamma_L(q)
|
|
\end{equation}
|
|
we solve the equation
|
|
\begin{equation}
|
|
-\Delta u_\lambda(|x|)
|
|
=
|
|
(1-u_\lambda(|x|))v(|x|)-2\rho K(|x|)+\rho^2 L(|x|)
|
|
-2\lambda\int dk\ \hat u_0(k)\cos(k\cdot x)\Gamma_L(q-k)
|
|
.
|
|
\end{equation}
|
|
Note that the Fourier transform of
|
|
\begin{equation}
|
|
-2\lambda\int dk\ \hat u_0(k)\cos(k\cdot x)\Gamma_L(q-k)
|
|
\end{equation}
|
|
is
|
|
\begin{equation}
|
|
-(2\pi)^3\lambda\hat u_0(q)(\Gamma_L(k+q)+\Gamma_L(k-q))
|
|
.
|
|
\end{equation}
|
|
Since the ground state is unique, $\mathcal M$ is spherically symmetric.
|
|
The term $\Gamma_L(k\pm q)$ is not, so we take its spherical average (which will not change the final result): by lemma\-~\ref{lemma:bipolar},
|
|
\begin{equation}
|
|
-\frac1{4\pi r^2}\int dq\ \delta(|q|-r)(2\pi)^3\lambda\hat u_0(q)(\Gamma_L(k+q)+\Gamma_L(k-q))
|
|
=
|
|
-\frac{(2\pi)^3}{|k|r}\lambda\hat u_0(r)\int_{||k|-r|}^{|k|+r} ds\ s\Gamma_L(s)
|
|
.
|
|
\end{equation}
|
|
In this setup, the approximation of the delta function is thus
|
|
\begin{equation}
|
|
\tilde\delta(|k|,r):=
|
|
\frac1{2|k|r}\int_{||k|-r|}^{|k|+r} ds\ s\Gamma_L(s)
|
|
=
|
|
\frac{1}{2|k|rL}(\Gamma_L(|k|-r)-\Gamma_L(|k|+r))
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
To compute the momentum distribution at $q$, we define $\Xi(\mathbb U,\lambda)$ by replacing $\mathbb T$ with
|
|
\begin{equation}
|
|
\mathbb T_{i}=
|
|
\frac1{\mathbb A_{K,i}}
|
|
\left(
|
|
\mathbb S_i
|
|
-2(2\pi)^3\lambda \hat u(|q|)\tilde\delta(k_i,|q|)
|
|
\right)
|
|
.
|
|
\end{equation}
|
|
Then we solve $\Xi(\mathbb U,\lambda)=0$, and differentiate:
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb U|_{\lambda=0}=-(D\Xi)^{-1}\partial_\lambda\Xi|_{\lambda=0}
|
|
.
|
|
\end{equation}
|
|
Finally,
|
|
\begin{equation}
|
|
\partial_\lambda\Xi_{i}|_{\lambda=0}
|
|
=
|
|
-\partial_\lambda\mathbb T_{i}|_{\lambda=0}\left(
|
|
\frac1{2(\mathbb X_{i}+1)}
|
|
\Phi\left(\mathbb B_i{\textstyle\frac{\mathbb T_{i}}{(\mathbb X_{i}+1)^2}}\right)
|
|
+\frac{\mathbb B_i\mathbb T_{i}}{2(\mathbb X_{i}+1)^3}
|
|
\partial\Phi\left(\mathbb B_i{\textstyle\frac{\mathbb T_{i}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
\right)
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb T_{i}|_{\lambda=0}=
|
|
-\frac{2(2\pi)^3}{\mathbb A_{K,i}} \hat u(|q|)\tilde\delta(k_{i},|q|)
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subsection{\tt anyeq}\label{sec:anyeq}
|
|
\indent
|
|
This method is used to solve any of the equations in the Simplified approach.
|
|
Specifically, it solves the equation
|
|
\begin{equation}
|
|
-\Delta u
|
|
=
|
|
v(1-u)-2\rho K+\rho^2 L
|
|
\label{anyeq}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
K=\gamma_K(1-\alpha_Ku)\left(\beta_K u\ast S+(1-\beta_K)\frac{2e}\rho u\right)
|
|
\label{anyeq_K}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
L:=L_1+L_2+L_3
|
|
\end{equation}
|
|
\begin{equation}
|
|
L_1:=(1-\alpha_{L,1}u)\left(\beta_{L,1}u\ast u\ast S+(1-\beta_{L,1})\frac{2e}\rho u\ast u\right)
|
|
\end{equation}
|
|
\begin{equation}
|
|
L_2:=-\gamma_{L,2}(1-\alpha_{L,2}u)\left(\beta_{L,2}2u\ast(u(u\ast S))+(1-\beta_{L,2})\frac{4e}\rho u\ast u^2\right)
|
|
\end{equation}
|
|
\begin{equation}
|
|
L_3:=\gamma_{L,3}(1-\alpha_{L,3}u)\left(\beta_{L,3}\frac12\int dydz\ u(y)u(z-x)u(z)u(y-x)S(z-y)+(1-\beta_{L,3})\frac e\rho u^2\ast u^2\right)
|
|
\end{equation}
|
|
\begin{equation}
|
|
e=\frac\rho2\int dx\ (1-u(|x|))v(|x|)
|
|
.
|
|
\label{anyeq_e}
|
|
\end{equation}
|
|
The parameters $\alpha_\cdot$, $\beta_\cdot$ and $\gamma_\cdot$ can be set to turn\-~(\ref{anyeq}) into any of the approximations of the Simplified approach.
|
|
For ease of use, there are several predefined equations, given in the following table.
|
|
\bigskip
|
|
|
|
\def\arraystretch{1.5}
|
|
\setlength\tabcolsep{5pt}
|
|
\makelink{table:anyeq_eqs}{}
|
|
\hfil\begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|c|c|}
|
|
\hline
|
|
&$\alpha_K$ &$\beta_K$ &$\gamma_K$ &$\alpha_{L,1}$ &$\beta_{L,1}$ &$\alpha_{L,2}$ &$\beta_{L,2}$ &$\gamma_{L,2}$ &$\alpha_{L,3}$ &$\beta_{L,3}$ &$\gamma_{L,3}$\\
|
|
\hline
|
|
\makelink{eq:anyeq_compleq}{}{\tt compleq}&1&1&1&1&1&1&1&1&1&1&1\\
|
|
\makelink{eq:anyeq_bigeq}{} {\tt bigeq }&1&1&1&1&1&1&1&1&-&-&0\\
|
|
\makelink{eq:anyeq_fulleq}{} {\tt fulleq }&1&1&1&1&1&1&1&1&1&0&1\\
|
|
\makelink{eq:anyeq_medeq}{} {\tt medeq }&0&1&1&0&1&-&-&0&-&-&0\\
|
|
\makelink{eq:anyeq_simpleq}{}{\tt simpleq}&0&0&1&0&0&-&-&0&-&-&0\\
|
|
\hline
|
|
\end{tabular}
|
|
\bigskip
|
|
|
|
Note that there is no $\gamma_{L,1}$, whose computation would be rather different.
|
|
Note, in addition, that {\tt simpleq} and {\tt medeq} coincide with their definitions in\-~(\ref{easyeq}).
|
|
The method used to solve this equation is very different from \refname{sec:easyeq}{{\tt easyeq}}, and is significantly longer to run.
|
|
\bigskip
|
|
|
|
\subsubsection{Usage}
|
|
\indent
|
|
Unless otherwise noted, this method takes the following parameters (specified via the {\tt [-p params]} flag, as a `{\tt;}' separated list of entries).
|
|
\begin{itemize}
|
|
\item\makelink{param:anyeq_rho}{}
|
|
{\tt rho} ({\tt Float64}, default: $10^{-6}$): density $\rho$.
|
|
|
|
\item\makelink{param:anyeq_tolerance}{}
|
|
{\tt tolerance} ({\tt Float64}, default: $10^{-11}$): maximal size of final step in Newton iteration.
|
|
|
|
\item\makelink{param:anyeq_maxiter}{}
|
|
{\tt maxiter} ({\tt Int64}, default: 21): maximal number of iterations of the Newton algorithm before giving up.
|
|
|
|
\item\makelink{param:anyeq_P}{}
|
|
{\tt P} ({\tt Int64}, default: 11): order of all Chebyshev polynomial expansions (denoted by $P$ below).
|
|
|
|
\item\makelink{param:anyeq_N}{}
|
|
{\tt N} ({\tt Int64}, default: 12): order of all Gauss quadratures (denoted by $N$ below).
|
|
|
|
\item\makelink{param:anyeq_J}{}
|
|
{\tt J} ({\tt Int64}, default: 10): number of splines (denoted by $J$ below).
|
|
|
|
\item\makelink{param:anyeq_nlrho_init}{}
|
|
{\tt nlrho\_init} ({\tt Int64}, default: 0): we initialize the Newton algorithm using the solution of \refname{eq:easyeq_medeq}{{\tt medeq}}, computed using the methods in \refname{sec:easyeq}{{\tt easyeq}}. If {\tt nlrho\_init} is $\neq 0$, then the solution of \refname{eq:easyeq_medeq}{{\tt medeq}} is first computed for {\tt nlrho} smaller values of $\rho$ starting from $10^{\mathrm{\refname{param:anyeq_minlrho_init}{{\tt minlrho\_init}}}}$. This is useful when $\rho$ is too large for the solution of \refname{eq:easyeq_medeq}{{\tt medeq}} to be computed directly.
|
|
|
|
\item\makelink{param:anyeq_minlrho_init}{}
|
|
{\tt minlrho\_init} ({\tt Float64}, default: $-6$): see \refname{param:anyeq_nlrho_init}{{\tt nlrho\_init}}.
|
|
|
|
\item\makelink{param:anyeq_aK}{}
|
|
{\tt aK}, {\tt bK}, {\tt gK}, {\tt aL1}, {\tt bL1}, {\tt aL2}, {\tt bL2}, {\tt gL2}, {\tt aL3}, {\tt bL3}, {\tt gL3} ({\tt Float64}, default: 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0): the values of $\alpha_K$, $\beta_K$, $\gamma_K$, $\alpha_{L,1}$, $\beta_{L,1}$, $\alpha_{L,2}$, $\beta_{L,2}$, $\gamma_{L,2}$, $\alpha_{L,3}$, $\beta_{L,3}$, $\gamma_{L,3}$.
|
|
|
|
\item\makelink{param:anyeq_eq}{}
|
|
{\tt eq} ({\tt String}, default: ``{\tt bigeq}'', acceptable values: ``{\tt compleq}'', ``{\tt bigeq}'', ``{\tt fulleq}'', ``{\tt medeq}'', ``{\tt simpleq}''): A shortcut to select any of the equations defined in the \refname{table:anyeq_eqs}{table above}. When this option is set, none of
|
|
\refname{param:anyeq_aK}{\tt aK}, \refname{param:anyeq_aK}{\tt bK}, \refname{param:anyeq_aK}{\tt gK}, \refname{param:anyeq_aK}{\tt aL1}, \refname{param:anyeq_aK}{\tt bL1}, \refname{param:anyeq_aK}{\tt aL2}, \refname{param:anyeq_aK}{\tt bL2}, \refname{param:anyeq_aK}{\tt gL2}, \refname{param:anyeq_aK}{\tt aL3}, \refname{param:anyeq_aK}{\tt bL3}, \refname{param:anyeq_aK}{\tt gL3} should be set.
|
|
\end{itemize}
|
|
\bigskip
|
|
|
|
The available {\tt commands} are the following.
|
|
|
|
\begin{itemize}
|
|
\item\makelink{command:anyeq_energy}{}
|
|
{\tt energy}: compute the energy $e$ at a given $\rho$.\par
|
|
\underline{Output}: [$e$] [Newton error $\epsilon$].
|
|
|
|
\item\makelink{command:anyeq_energy_rho}{}
|
|
{\tt energy\_rho}: compute the energy $e$ as a function of $\rho$.
|
|
The Newton algorithm is initialized with the solution of \refname{eq:easyeq_medeq}{{\tt medeq}}.\par
|
|
\underline{Disabled parameters}: \refname{param:anyeq_rho}{{\tt rho}}.\par
|
|
\underline{Extra parameters}:
|
|
\begin{itemize}
|
|
\item\makelink{param:anyeq_minlrho}{}
|
|
{\tt minlrho} ({\tt Float64}, default: $10^{-6}$): minimal value for $\log_{10}\rho$.
|
|
|
|
\item\makelink{param:anyeq_maxlrho}{}
|
|
{\tt maxlrho} ({\tt Float64}, default: $10^{2}$): maximal value for $\log_{10}\rho$.
|
|
|
|
\item\makelink{param:anyeq_nlrho}{}
|
|
{\tt nlrho} ({\tt Int64}, default: $100$): number of values for $\rho$ (spaced logarithmically).
|
|
|
|
\item\makelink{param:anyeq_minrho}{}
|
|
{\tt minrho} ({\tt Float64}, default: $10^{-6}$): minimal value for $\rho$.
|
|
|
|
\item\makelink{param:anyeq_maxrho}{}
|
|
{\tt maxrho} ({\tt Float64}, default: $10^{2}$): maximal value for $\rho$.
|
|
|
|
\item\makelink{param:anyeq_nrho}{}
|
|
{\tt nrho} ({\tt Int64}, default: $0$): number of values for $\rho$ (spaced linearly). If {\tt nrho} is $\neq0$, then the linear spacing will be used, and \refname{param:anyeq_minlrho}{{\tt minlrho}}, \refname{param:anyeq_maxlrho}{{\tt maxlrho}}, \refname{param:anyeq_nlrho}{{\tt nlrho}} will be ignored. Otherwise, the logarithmic spacing will be used and \refname{param:anyeq_minrho}{{\tt minrho}}, \refname{param:anyeq_maxrho}{{\tt maxrho}} will be ignored.
|
|
|
|
\item\makelink{param:anyeq_rhos}{}
|
|
{\tt rhos} ({\tt Array\{Float64\}}): list of values for $\rho$, specified as a `{\tt,}' separated list.
|
|
This parameter takes precedence over \refname{param:anyeq_minlrho}{{\tt minlrho}}, \refname{param:anyeq_maxlrho}{{\tt maxlrho}}, \refname{param:anyeq_nlrho}{{\tt nlrho}}, \refname{param:anyeq_minrho}{{\tt minrho}}, \refname{param:anyeq_maxrho}{{\tt maxrho}}, \refname{param:anyeq_nrho}{{\tt nrho}}.
|
|
\end{itemize}
|
|
\underline{Output} (one line for each value of $\rho$): [$\rho$] [$e$] [Newton error $\epsilon$].\par
|
|
\underline{Multithread support}: yes, different values of $\rho$ split up among workers.
|
|
|
|
\item\makelink{command:anyeq_energy_rho_init_prevrho}{}
|
|
{\tt energy\_rho\_init\_prevrho}: compute the energy $e$ as a function of $\rho$.
|
|
The Newton algorithm is initialized with the solution of \refname{eq:easyeq_medeq}{{\tt medeq}} for the lowest $\rho$, and with the previously computed $\rho$ for the larger densities.\par
|
|
\underline{Disabled parameters}: same as \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par
|
|
\underline{Extra parameters}: same as \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par
|
|
\underline{Output} (one line for each value of $\rho$): [$\rho$] [$e$] [Newton error $\epsilon$].
|
|
|
|
\item\makelink{command:anyeq_energy_rho_init_nextrho}{}
|
|
{\tt energy\_rho\_init\_nextrho}: same as \refname{command:anyeq_energy_rho_init_prevrho}{{\tt energy\_rho\_init\_prevrho}} except that the energy is computed for decreasing densities instead of increasing ones.
|
|
The Newton algorithm is initialized with the solution of \refname{eq:easyeq_medeq}{{\tt medeq}} for the largest $\rho$, and with the previously computed $\rho$ for the smaller densities.\par
|
|
\underline{Disabled parameters}: same as \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par
|
|
\underline{Extra parameters}: same as \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par
|
|
\underline{Output} (one line for each value of $\rho$): [$\rho$] [$e$] [Newton error $\epsilon$].
|
|
|
|
\item\makelink{command:anyeq_condensate_fraction}{}
|
|
{\tt condensate\_fraction}: compute the uncondensed fraction $\eta$ at a given $\rho$.\par
|
|
\underline{Output}: [$\eta$] [Newton error $\epsilon$].
|
|
|
|
\item\makelink{command:anyeq_condensate_fraction_rho}{}
|
|
{\tt condensate\_fraction\_rho}: compute the uncondensed fraction $\eta$ as a function of $\rho$.
|
|
The Newton algorithm is initialized with the solution of \refname{eq:easyeq_medeq}{{\tt medeq}}.\par
|
|
\underline{Disabled parameters}: same as \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par
|
|
\underline{Extra parameters}: same as \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par
|
|
\underline{Output} (one line for each value of $\rho$): [$\rho$] [$\eta$] [Newton error $\epsilon$].\par
|
|
\underline{Multithread support}: yes, different values of $\rho$ split up among workers.
|
|
|
|
\item\makelink{command:anyeq_uk}{}
|
|
{\tt uk}: compute the Fourier transform $\hat u(|k|)$.
|
|
The values $|k|$ at which $\hat u$ is computed are those coming from the Gauss quadratures, and cannot be set.\par
|
|
\underline{Output} (one line for each value of $|k|$): [$|k|$] [$\hat u(|k|)$]
|
|
|
|
\item\makelink{command:anyeq_ux}{}
|
|
{\tt ux}: compute $u$ as a function of $|x|$.\par
|
|
\underline{Extra parameters}:
|
|
\begin{itemize}
|
|
\item\makelink{param:anyeq_xmin}{}
|
|
{\tt xmin} ({\tt Float64}, default: 0): minimum of the range of $|x|$ to be printed.
|
|
|
|
\item\makelink{param:anyeq_xmax}{}
|
|
{\tt xmax} ({\tt Float64}, default: 100): maximum of the range of $|x|$ to be printed.
|
|
|
|
\item\makelink{param:anyeq_nx}{}
|
|
{\tt nx} ({\tt Int64}, default: 100): number of points to print (linearly spaced).
|
|
|
|
\end{itemize}
|
|
\underline{Output} (one line for each value of $x$): [$|x|$] [$u(|x|)$]
|
|
|
|
\item\makelink{command:anyeq_2pt}{}
|
|
{\tt 2pt}: compute the spherically averaged two-point correlation function $C_2(|x|)$ at a given $\rho$.\par
|
|
\underline{Extra parameters}: same as \refname{command:anyeq_ux}{{\tt ux}}, plus\par
|
|
\begin{itemize}
|
|
\item\makelink{param:anyeq_window_L}{}
|
|
{\tt window\_L} ({\tt Float64}, default: $10^3$): size of the Hann window used to numerically invert the Fourier transform in the computation of the tow-point correlation function, see\-~(\ref{hann}).
|
|
\end{itemize}
|
|
\underline{Output} (one line for each value of $|x|$): [$|x|$] [$C_2(|x|)$]
|
|
|
|
\item\makelink{command:anyeq_2pt_max}{}
|
|
{\tt 2pt\_max}: compute the maximum of the spherically averaged two-point correlation function $C_2(|x|)$ at a given $\rho$.\par
|
|
\underline{Extra parameters}: \refname{param:anyeq_window_L}{{\tt window\_L}} plus
|
|
\begin{itemize}
|
|
\item\makelink{param:anyeq_dx}{}
|
|
{\tt dx} ({\tt Float64}, default: $10^{-7}$): step used to numerically approximate derivatives.
|
|
|
|
\item\makelink{param:anyeq_x0}{}
|
|
{\tt x0} ({\tt Float64}, default: $1$): initial guess for the maximum is $\rho^{-1/3}\mathrm{\tt x0}$.
|
|
|
|
\item\makelink{param:anyeq_maxstep}{}
|
|
{\tt maxstep} ({\tt Float64}, default: $\infty$): maximal size of single step in maximization algorithm.
|
|
|
|
\item\makelink{param:easyeq_tolerance_max}{}
|
|
{\tt tolerance\_max} ({\tt Float64}, default: \refname{param:easyeq_tolerance}{{\tt tolerance}}): same as \refname{param:easyeq_tolerance}{{\tt tolerance}}, used for the Newton algorithm underlying the maximization algorithm.
|
|
\end{itemize}
|
|
\underline{Output}: [$|x_{\mathrm{max}}|$] [$C_2(|x_{\mathrm{max}}|)$]
|
|
|
|
\item\makelink{command:anyeq_2pt_max_rho}{}
|
|
{\tt 2pt\_max\_rho}: compute the maximum of the spherically averaged two-point correlation function $C_2(|x|)$ for a range of $\rho$.\par
|
|
\underline{Extra parameters}: same as \refname{command:anyeq_2pt_max}{{\tt anyeq\_2pt\_max}} plus those of \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par
|
|
\underline{Output} (one line for each value of $\rho$): [$\rho$] [$|x_{\mathrm{max}}|$] [$C_2(|x_{\mathrm{max}}|)$]\par
|
|
\underline{Multithread support}: yes, different values of $\rho$ split up among workers.
|
|
|
|
\item\makelink{command:anyeq_2pt_fourier}{}
|
|
{\tt 2pt\_fourier}: compute the spherically averaged Fourier transform of the two-point correlation function $\hat C_2(|k|)$ at a given $\rho$.\par
|
|
\underline{Extra parameters}:
|
|
\begin{itemize}
|
|
\item\makelink{param:anyeq_kmin}{}
|
|
{\tt kmin} ({\tt Float64}, default: 0): minimum of the range of $|k|$ to be printed.
|
|
|
|
\item\makelink{param:anyeq_kmax}{}
|
|
{\tt kmax} ({\tt Float64}, default: 10): maximum of the range of $|k|$ to be printed.
|
|
|
|
\item\makelink{param:anyeq_nk}{}
|
|
{\tt nk} ({\tt Int64}, default: 100): number of $|k|$'s to be printed.
|
|
|
|
\item\makelink{param:anyeq_2pt_fourier_window_L}{}
|
|
{\tt window\_L} ({\tt Float64}, default: 1000): what is actually computed is $\hat C_2$ convolved with a Gaussian of variance $1/\sqrt L$ where $L=\sqrt{\mathrm{\tt window\_L}}$, see\-~(\ref{anyeq_gaussian}).
|
|
\end{itemize}
|
|
\underline{Output} (one line for each value of $|k|$): [$|k|$] [$\hat C_2(|k|)$].\par
|
|
\underline{Multithread support}: yes, different values of $k$ are split up among workers.
|
|
|
|
\item\makelink{command:anyeq_2pt_fourier_max}{}
|
|
{\tt 2pt\_fourier\_max}: compute the maximum of the spherically averaged Fourier transformed two-point correlation function $\hat C_2(|k|)$.\par
|
|
\underline{Extra parameters}: \refname{param:anyeq_2pt_fourier_window_L}{{\tt window\_L}}, \refname{param:anyeq_maxstep}{{\tt maxstep}} plus
|
|
\begin{itemize}
|
|
\item\makelink{param:anyeq_dk}{}
|
|
{\tt dk} ({\tt Float64}, default: $10^{-7}$): step used to numerically approximate derivatives.
|
|
|
|
\item\makelink{param:anyeq_k0}{}
|
|
{\tt k0} ({\tt Float64}, default: $1$): initial guess for the maximum is $\rho^{1/3}\mathrm{\tt k0}$.
|
|
\end{itemize}
|
|
\underline{Output}: [$|k_{\mathrm{max}}|$] [$\hat C_2(|k_{\mathrm{max}}|)$]\par
|
|
|
|
\item\makelink{command:anyeq_2pt_fourier_max_rho}{}
|
|
{\tt 2pt\_fourier\_max\_rho}: compute the maximum of the spherically averaged Fourier transformed two-point correlation function $\hat C_2(|k|)$ for a range of $\rho$.\par
|
|
\underline{Extra parameters}: same as those of \refname{command:anyeq_2pt_fourier_max}{{\tt 2pt\_fourier\_max}} plus those of \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par
|
|
\underline{Output} (one line for each value of $\rho$): [$\rho$] [$|k_{\mathrm{max}}|$] [$\hat C_2(|k_{\mathrm{max}}|)$]\par
|
|
\underline{Multithread support}: yes, different values of $\rho$ split up among workers.
|
|
|
|
\item\makelink{command:anyeq_momentum_distribution}{}
|
|
{\tt momentum\_distribution}: compute the momentum distribution $\mathcal M(|k|)$ at a given $\rho$. The momentum distribution is computed for $k=k_{l,j}$ (see\-~(\ref{klj})).\par
|
|
\underline{Extra parameters}:
|
|
\begin{itemize}
|
|
\item\makelink{param:anyeq_momentum_kmin}{}
|
|
{\tt kmin} ({\tt Float64}, default: 0): minimum of the range of $|k|$ to be printed.
|
|
|
|
\item\makelink{param:anyeq_momentum_kmax}{}
|
|
{\tt kmax} ({\tt Float64}, default: 10): maximum of the range of $|k|$ to be printed.
|
|
|
|
\item\makelink{param:anyeq_momentum_window_L}{}
|
|
{\tt window\_L} ({\tt Float64}, default: 1000): what is actually computed is $\mathfrak M_k$ convolved with a Gaussian of variance $1/\sqrt L$ where $L=\sqrt{\mathrm{\tt window\_L}}/k^2$, see\-~(\ref{anyeq_gaussian_momt}).
|
|
\end{itemize}
|
|
\underline{Output} (one line for each value of $|k|$): [$|k|$] [$\mathcal M(|k|)$]
|
|
|
|
\item\makelink{command:anyeq_compressibility_rho}{}
|
|
{\tt compressibility\_rho}: compute the compressibility $\chi$ as a function of $\rho$.
|
|
The Newton algorithm is initialized with the solution of \refname{eq:easyeq_medeq}{{\tt medeq}}.
|
|
\underline{Disabled parameters}: same as \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par
|
|
\underline{Extra parameters}: same as \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par
|
|
\underline{Output} (one line for each value of $\rho$): [$\rho$] [$\chi$].
|
|
|
|
\item \makelink{command:anyeq_save_Abar}{}
|
|
{\tt save\_Abar}: compute the matrix $\bar A$. This matrix is used to compute observables for \refname{eq:anyeq_compleq}{{\tt compleq}}. This command is useful to output the value of $\bar A$ to a file once and for all, and use this file to run commands without recomputing $\bar A$.\par
|
|
\underline{Disabled parameters}: \refname{param:anyeq_rho}{{\tt rho}}, \refname{param:anyeq_tolerance}{{\tt tolerance}}, \refname{param:anyeq_maxiter}{{\tt maxiter}}, \refname{param:anyeq_minlrho_init}{{\tt minlrho\_init}}, \refname{param:anyeq_nlrho_init}{{\tt nlrho\_init}}.\par
|
|
\underline{Output}: [$\bar A$] (the output is not designed to be human-readable; it is obtained through nested {\tt for} loops; for details, see the code).\par
|
|
\underline{Multithread support}: yes, the first indices are split up among workers, which produces $NJ$ jobs.
|
|
|
|
\end{itemize}
|
|
|
|
\subsubsection{Description}
|
|
\subsubsubsection{Fourier space formulation}
|
|
The computation is carried out in Fourier space.
|
|
We take the convention that the Fourier transform of a function $f(|x|)$ is
|
|
\begin{equation}
|
|
\hat f(|k|)=\int_{\mathbb R^3} dx\ e^{ikx}f(|x|)
|
|
=\frac{4\pi}{|k|}\int_0^\infty dr\ \frac{\sin(|k|r)}rf(r)
|
|
.
|
|
\label{anyeq_fourier}
|
|
\end{equation}
|
|
We define a Fourier-space convolution:
|
|
\begin{equation}
|
|
\hat f\hat\ast\hat g(|k|):=\int_{\mathbb R^3}\frac{dp}{(2\pi)^3}\ \hat f(|k-p|)\hat g(|p|)
|
|
.
|
|
\label{anyeq_hatast}
|
|
\end{equation}
|
|
In Fourier space, (\ref{anyeq}) becomes
|
|
\begin{equation}
|
|
k^2\hat u(|k|)=\hat S(|k|)-2\rho\hat K(|k|)+\rho^2\hat L(|k|)
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\hat S(|k|)=\int_{\mathbb R^3} dx\ e^{ikx}(1-u(|x|))v(|x|)
|
|
=\hat v(k)-\hat u\hat\ast\hat v(k)
|
|
\end{equation}
|
|
\begin{equation}
|
|
\rho\hat K=
|
|
\gamma_K(\beta_K\rho\hat S+(1-\beta_K)2e)\hat u
|
|
-
|
|
\gamma_K\alpha_K\hat u\hat\ast((\beta_K\rho\hat S+(1-\beta_K)2e)\hat u)
|
|
\end{equation}
|
|
\begin{equation}
|
|
\rho\hat L_1=
|
|
(\beta_{L,1}\rho\hat S+(1-\beta_{L,1})2e)\hat u^2
|
|
-
|
|
\alpha_{L,1}\hat u\hat\ast((\beta_{L,1}\rho\hat S+(1-\beta_{L,1})2e)\hat u^2)
|
|
\end{equation}
|
|
\begin{equation}
|
|
\rho\hat L_2=
|
|
-\gamma_{L,2}(\beta_{L,2}2\rho(\hat u\hat\ast(\hat u\hat S))+(1-\beta_{L,2})4e\hat u\hat\ast\hat u)\hat u
|
|
+
|
|
\gamma_{L,2}\alpha_{L,2}\hat u\hat\ast((\beta_{L,2}2\rho(\hat u\hat\ast(\hat u\hat S))+(1-\beta_{L,2})4e\hat u\hat\ast\hat u)\hat u)
|
|
\end{equation}
|
|
\begin{equation}
|
|
\rho\hat L_3=
|
|
\gamma_{L,3}(1-\beta_{L,3})e(\hat u\hat\ast\hat u)^2
|
|
-
|
|
\gamma_{L,3}\alpha_{L,3}(1-\beta_{L,3})\hat u\hat\ast(e(\hat u\hat\ast\hat u)^2)
|
|
+
|
|
\gamma_{L,3}\beta_{L,3}\hat l_3
|
|
-
|
|
\frac1{2\rho}\gamma_{L,3}\alpha_{L,3}\beta_{L,3}\hat u\hat\ast\hat l_3
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\hat l_3(|k|):=
|
|
\frac1{\rho^2}\int\frac{dq}{(2\pi)^3}\frac{dq'}{(2\pi)^3}\ \rho\hat u(|k-q'|)\rho\hat u(|q|)\rho\hat u(|q'|)\rho\hat u(|k-q|)\hat S(|q'-q|)
|
|
\label{l3}
|
|
\end{equation}
|
|
Therefore,
|
|
\begin{equation}
|
|
\Omega(u)=0
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\Omega(u):=
|
|
\rho^2\hat u(|k|)^2\sigma_{L,1}(|k|)
|
|
-\left(\frac{k^2}\rho+2\sigma_K(|k|)+2f_1(|k|)\right)\rho\hat u(|k|)
|
|
+\left(\hat S(|k|)+\frac12f_2(|k|)+G(|k|)\right)
|
|
\label{Omega}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\sigma_K:=\frac1{\rho}\gamma_K(\beta_K\rho\hat S+(1-\beta_K)2e)
|
|
,\quad
|
|
\sigma_{L,1}:=\frac1{\rho}(\beta_{L,1}\rho\hat S+(1-\beta_{L,1})2e)
|
|
\end{equation}
|
|
\begin{equation}
|
|
f_1:=\frac1{\rho^2}\gamma_{L,2}(\beta_{L,2}\rho(\rho\hat u\hat\ast(\rho\hat u\hat S))+(1-\beta_{L,2})2e\rho\hat u\hat\ast\rho\hat u)
|
|
\end{equation}
|
|
\begin{equation}
|
|
f_2:=\gamma_{L,3}(1-\beta_{L,3})\frac{2e}{\rho^3}(\rho\hat u\hat\ast\rho\hat u)^2
|
|
+\gamma_{L,3}\beta_{L,3}\hat l_3
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
G:=
|
|
\frac2{\rho^2}\gamma_K\alpha_K\rho\hat u\hat\ast((\beta_K\rho\hat S+(1-\beta_K)2e)\rho\hat u)
|
|
-
|
|
\frac1{\rho^2}\alpha_{L,1}\rho\hat u\hat\ast((\beta_{L,1}\rho\hat S+(1-\beta_{L,1})2e)\rho\hat u^2)
|
|
\\[0.5cm]\hfill
|
|
+
|
|
\frac2{\rho}\alpha_{L,2}\rho\hat u\hat\ast(f_1\rho\hat u)
|
|
-
|
|
\frac1{2\rho}\alpha_{L,3}\rho\hat u\hat\ast f_2
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
Therefore,
|
|
\begin{equation}
|
|
\rho\hat u=1+\xi
|
|
-\sqrt{(1+\xi)^2-(1+\zeta)}
|
|
\label{u}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\xi:=\frac1{\sigma_{L,1}}\left(\sigma_K-\sigma_{L,1}+\frac{k^2}{2\rho}+f_1\right)
|
|
,\quad
|
|
\zeta:=\frac1{\sigma_{L,1}}\left(\hat S-\sigma_{L,1}+\frac12f_2+G\right)
|
|
\end{equation}
|
|
We rewrite\-~(\ref{u}) as
|
|
\begin{equation}
|
|
U=
|
|
\frac{1+\zeta}{2(\xi+1)}\ \Phi\left({\textstyle\frac{1+\zeta}{(\xi+1)^2}}\right)
|
|
\label{anyeq_hatu}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
U:=\rho\hat u
|
|
\end{equation}
|
|
\begin{equation}
|
|
\Phi(x):=\frac{2(1-\sqrt{1-x})}x
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subsubsubsection{Evaluating integrals}
|
|
To evaluate integrals numerically, we will split integration intervals over splines and use Gauss-Legendre quadratures.
|
|
More specifically, to compute integrals of the form
|
|
\begin{equation}
|
|
\int \frac{dk}{(2\pi)^3}\ f(U(k),k)
|
|
=
|
|
\frac1{2\pi^2}\int_0^\infty dt\ t^2f(U(t),t)
|
|
\end{equation}
|
|
we first compactify space to $(-1,1]$ by changing variables to $\tau=\frac{1-t}{1+t}$:
|
|
\begin{equation}
|
|
\int \frac{dk}{(2\pi)^3}\ f(U(k),k)
|
|
=
|
|
\frac1{\pi^2}\int_{-1}^1 d\tau\ \frac{(1-\tau)^2}{(1+\tau)^4}f({\textstyle U(\frac{1-\tau}{1+\tau}),\frac{1-\tau}{1+\tau}})
|
|
.
|
|
\end{equation}
|
|
We then split $(-1,1]$ into $J$ sub-intervals (given by the parameter \refname{param:anyeq_J}{{\tt J}}), called {\it splines}:
|
|
\begin{equation}
|
|
(-1,1]=\bigcup_{j=0}^{J-1}(\tau_j,\tau_{j+1}]
|
|
.
|
|
\end{equation}
|
|
The $\tau$ are taken to be equally spaced, but the code is designed in such a way that this could be changed easily in the future:
|
|
\begin{equation}
|
|
\tau_j=-1+\frac{2j}J
|
|
.
|
|
\end{equation}
|
|
In these terms,
|
|
\begin{equation}
|
|
\int \frac{dk}{(2\pi)^3}\ f(U(k),k)
|
|
=
|
|
\sum_{l=0}^{J-1}
|
|
\int_{\tau_l}^{\tau_{l+1}} d\tau\ \frac{(1-\tau)^2}{(1+\tau)^4}g(U({\textstyle\frac{1-\tau}{1+\tau},\frac{1-\tau}{1+\tau}})
|
|
.
|
|
\end{equation}
|
|
We then change variables to $r$:
|
|
\begin{equation}
|
|
\tau=
|
|
-\frac{\tau_{l+1}-\tau_l}2\sin({\textstyle\frac{\pi r}2})+\frac{\tau_{l+1}+\tau_l}2
|
|
=:\frac{1-\vartheta_l(r)}{1+\vartheta_l(r)}
|
|
\label{chebyshevvars}
|
|
\end{equation}
|
|
(the reason for this specific change of variables will become clear at the end of this paragraph and in the next one)
|
|
and find
|
|
\begin{equation}
|
|
\int \frac{dk}{(2\pi)^3}\ f(U(k),k)
|
|
=
|
|
\sum_{l=0}^{J-1}
|
|
\frac{\tau_{l+1}-\tau_l}{16\pi}\int_{-1}^1 dr\ \cos({\textstyle\frac{\pi r}2})(1+\vartheta_l(r))^2\vartheta_l^2(r)f(U(\vartheta_l(r)),\vartheta_l(r))
|
|
.
|
|
\label{anyeq_preintegration}
|
|
\end{equation}
|
|
We then approximate the integral using a Gauss-Legendre quadrature of order $N$ (see appendix\-~\ref{appGL}):
|
|
\begin{equation}
|
|
\int \frac{dk}{(2\pi)^3}\ f(k)
|
|
\approx
|
|
\sum_{l=0}^{J-1}
|
|
\frac{\tau_{l+1}-\tau_l}{16\pi}
|
|
\sum_{j=1}^Nw_j
|
|
\cos({\textstyle\frac{\pi x_j}2})(1+\vartheta_l(x_j))^2\vartheta_l^2(x_j)f(\mathbb U_{l,j},\vartheta_l(x_j))
|
|
\label{anyeq_integration}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\mathbb U_{l,j}:=U(k_{l,j})
|
|
,\quad
|
|
k_{l,j}:=\frac{2+(\tau_{l+1}-\tau_l)\sin(\frac{\pi x_j}2)-(\tau_{l+1}+\tau_\alpha)}{2-(\tau_{l+1}-\tau_l)\sin(\frac{\pi x_j}2)+(\tau_{l+1}+\tau_l)}
|
|
.
|
|
\end{equation}
|
|
The choice of the change of variables\-~(\ref{chebyshevvars}) is made so that $U$ is evaluated at $k_{l,j}$, which are the momenta that appear naturally in the Chebyshev polynomial expansion below\-~(\ref{klj}).
|
|
In this way, we can compute arbitrary integrals of functions of $U$ by just computing the values of $\mathbb U_{l,j}$.
|
|
\bigskip
|
|
|
|
\subsubsubsection{Chebyshev polynomial expansion}
|
|
In the case of \refname{sec:easyeq}{{\tt easyeq}}, we saw that using Gauss quadratures reduced the computation to evaluating $U$ at a fixed set of discrete momenta.
|
|
This is not the case here, due to the presence of more complicated convolutions of $U$.
|
|
Instead, we will approximate $U$ using polynomial approximations.
|
|
We will now discuss this approximation for an arbitrary function $a(|k|)$, as we will need it for other functions than simply $U$.
|
|
\bigskip
|
|
|
|
\point
|
|
As in the previous paragraph, we compactify $[0,\infty)$ to $(-1,1]$ via the change of variables $r\mapsto\frac{1-r}{1+r}$, and split the interval into splines.
|
|
Inside each spline, we use a Chebyshev polynomial expansion.
|
|
However, we need to be careful in the first spline, which encapsulates the behavior at $|k|\to\infty$: $U$ decays to 0 as $|k|\to\infty$, and this behavior cannot be reproduced by a polynomial expansion.
|
|
To avoid this, if $a\sim|k|^{-\nu_a}$, we expand $|k|^{\nu_a}a$ instead of $a$.
|
|
Actually, to simplify expressions in the presence of the compactification, we will expand $2^{-\nu_a}(1+|k|)^{\nu_a}a$, which makes no conceptual difference.
|
|
In the case of $a=U$, we will use $\nu=2$, which we know holds for the Simple equation\-~\cite{CJL20}.
|
|
Putting all this together, we write, for $\tau\in(-1,1]$, if $|k|\equiv\frac{1-\tau}{1+\tau}$,
|
|
\begin{equation}
|
|
2^{-\nu_a}(1+|k|)^{\nu_a}a(|k|)\equiv
|
|
\frac1{(1+\tau)^{\nu_a}}a({\textstyle\frac{1-\tau}{1+\tau}})
|
|
=\sum_{l=0}^{J-1}\mathds 1_{\tau_l<\tau\leqslant\tau_{l+1}}\sum_{n=0}^\infty \mathbf F_{l,n}^{(\nu_a)}(a) T_n({\textstyle\frac{2\tau-(\tau_l+\tau_{l+1})}{\tau_{l+1}-\tau_l}})
|
|
\label{a_chebyshev_spline}
|
|
\end{equation}
|
|
in which $\mathds 1_{\tau_l<\tau\leqslant\tau_{l+1}}\in\{0,1\}$ is equal to 1 if and only if $\tau_l<\tau\leqslant\tau_{l+1}$, and
|
|
\begin{equation}
|
|
\mathbf F_{l,n}^{(\nu_a)}(a):=
|
|
\frac{2-\delta_{n,0}}{\pi}
|
|
\int_0^{\pi} d\theta\
|
|
\frac{\cos(n\theta)}{(1+\frac{\tau_{l+1}-\tau_l}2\cos(\theta)+\frac{\tau_{l+1}+\tau_l}2)^{\nu_a}}
|
|
a({\textstyle\frac{2-(\tau_{l+1}-\tau_l)\cos(\theta)-(\tau_{l+1}+\tau_\alpha)}{2+(\tau_{l+1}-\tau_l)\cos(\theta)+(\tau_{l+1}+\tau_l)}})
|
|
\label{bfF}
|
|
\end{equation}
|
|
(see appendix\-~\ref{appChebyshev} for a discussion of the Chebyshev polynomial expansion and the error of its truncations).
|
|
\bigskip
|
|
|
|
\point
|
|
In order to compute an approximation for $a$ using\-~(\ref{a_chebyshev_spline}), we will truncate the sum over $n$ to a finite value $P$ (given by the parameter \refname{param:anyeq_P}{{\tt P}}).
|
|
In addition, to compute the integral in\-~(\ref{bfF}), we will use a Gauss-Legendre quadrature of order $N$ (given by the parameter \refname{param:anyeq_N}{{\tt N}}), see appendix\-~\ref{appGL}:
|
|
\begin{equation}
|
|
\mathbf F_{l,n}^{(\nu_a)}(a)\approx\mathfrak F_{l,n}^{(\nu_a)}(\mathfrak a):=
|
|
\frac{2-\delta_{n,0}}2\sum_{j=1}^N w_j \cos({\textstyle\frac{n\pi(1+x_j)}2})\frac{\mathfrak a_{l,j}}{(1-\frac{\tau_{l+1}-\tau_l}2\sin(\frac{\pi x_j}2)+\frac{\tau_{l+1}+\tau_l}2)^{\nu_a}}
|
|
\label{frakF}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\mathfrak a_{l,j}:=a(k_{l,j})
|
|
,\quad
|
|
k_{l,j}:=\frac{2+(\tau_{l+1}-\tau_l)\sin(\frac{\pi x_j}2)-(\tau_{l+1}+\tau_\alpha)}{2-(\tau_{l+1}-\tau_l)\sin(\frac{\pi x_j}2)+(\tau_{l+1}+\tau_l)}
|
|
\label{klj}
|
|
\end{equation}
|
|
and $(x_j,w_j)$ are the abcissa and weights for Gauss-Legendre quadratures.
|
|
\bigskip
|
|
|
|
\point
|
|
All in all, we approximate
|
|
\begin{equation}
|
|
a({\textstyle\frac{1-\tau}{1+\tau}})
|
|
\approx(1+\tau)^{\nu_a}\sum_{l=0}^{J-1}\mathds 1_{\tau_l<\tau\leqslant\tau_{l+1}}\sum_{n=0}^P \mathfrak F_{l,n}^{(\nu_a)}(\mathfrak a) T_n({\textstyle\frac{2\tau-(\tau_l+\tau_{l+1})}{\tau_{l+1}-\tau_l}})
|
|
\label{approxchebyshev}
|
|
\end{equation}
|
|
with $\mathfrak F$ defined in\-~(\ref{frakF}).
|
|
Furthermore, using the Chebyshev polynomial expansion and Gauss-Legendre quadratures, we can compute all the observables we are interested in by computing $\mathbb U_{l,j}\equiv U(k_{l,j})$.
|
|
With this in mind, we will represent the function $U$ as a vector of dimension $NJ$ whose components are $\mathbb U_{l,j}$.
|
|
\bigskip
|
|
|
|
\subsubsubsection{Convolutions}
|
|
Using the Chebyshev polynomial approximation, we can compute convolutions as follows.
|
|
\medskip
|
|
|
|
\point
|
|
First, we rewrite the convolution as a two-dimensional integral, using bipolar coordinates (see lemma\-~\ref{lemma:bipolar}):
|
|
\begin{equation}
|
|
(a\hat\ast b)(|k|)=\frac1{4\pi^2|k|}\int_0^\infty dt\ ta(t)\int_{||k|-t|}^{|k|+t}ds\ sb(s)
|
|
\label{conv_approx1}
|
|
\end{equation}
|
|
We change variables to compactify the integrals $\tau=\frac{1-t}{1+t}$, $\sigma=\frac{1-s}{1+s}$:
|
|
\begin{equation}
|
|
(a\hat\ast b)(|k|)=\frac1{4\pi^2|k|}\int_{-1}^1 d\tau\ \frac{2(1-\tau)}{(1+\tau)^3}a({\textstyle\frac{1-\tau}{1+\tau}})\int_{\alpha_-(|k|,\tau)}^{\alpha_+(|k|,\tau)}d\sigma\ \frac{2(1-\sigma)}{(1+\sigma)^3}b({\textstyle\frac{1-\sigma}{1+\sigma}})
|
|
\label{convvar}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\alpha_-(|k|,\tau):=\frac{1-|k|-\frac{1-\tau}{1+\tau}}{1+|k|+\frac{1-\tau}{1+\tau}}
|
|
,\quad
|
|
\alpha_+(|k|,\tau):=\frac{1-||k|-\frac{1-\tau}{1+\tau}|}{1+||k|-\frac{1-\tau}{1+\tau}|}
|
|
.
|
|
\label{anyeq_alpha}
|
|
\end{equation}
|
|
Therefore, using the approximation\-~(\ref{approxchebyshev}), if $\mathfrak a_{l,j}:=a(k_{l,j})$ and $\mathfrak b_{l,j}:=b(k_{l,j})$ (\ref{klj}),
|
|
\begin{equation}
|
|
(a\hat\ast b)(|k_{l,j}|)\approx
|
|
(\mathfrak a\odot \mathfrak b)_{l,j}:=
|
|
\sum_{n,m=0}^P
|
|
\sum_{l',l''=0}^{J-1}
|
|
\mathfrak F_{l',n}^{(\nu_a)}(\mathfrak a)\mathfrak F_{l'',m}^{(\nu_b)}(\mathfrak b)
|
|
A_{l',n;l'',m}^{(\nu_a,\nu_b)}(|k_{l,j}|)
|
|
\label{odot}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
A_{l',n;l'',m}^{(\nu_a,\nu_b)}(|k|):=
|
|
\frac1{4\pi^2|k_{l,j}|}
|
|
\int_{\tau_{l'}}^{\tau_{l'+1}} d\tau\ \frac{2(1-\tau)}{(1+\tau)^{3-\nu_a}}T_n({\textstyle\frac{2\tau-(\tau_{l'}+\tau_{l'+1})}{\tau_{l'+1}-\tau_{l'}}})
|
|
\cdot\\[0.5cm]\hfill\cdot
|
|
\mathds 1_{\alpha_-(|k|,\tau)<\tau_{l''+1}}
|
|
\mathds 1_{\alpha_+(|k|,\tau)>\tau_{l''}}
|
|
\int_{\max(\tau_{l''},\alpha_-(|k|,\tau))}^{\min(\tau_{l''+1},\alpha_+(|k|,\tau))}d\sigma\ \frac{2(1-\sigma)}{(1+\sigma)^{3-\nu_b}}T_m({\textstyle\frac{2\sigma-(\tau_{l''}+\tau_{l''+1})}{\tau_{l''+1}-\tau_{l''}}})
|
|
\end{largearray}
|
|
\label{conv_approx2}
|
|
\end{equation}
|
|
(we need the indicator functions to ensure that the bounds of the integral are correct).
|
|
Note that $A$ is independent of $a$ and $b$, and can be computed once and for all at the beginning of execution for all values of $k_{l,j}$ (\ref{klj}).
|
|
We then compute the integrals using Gauss-Legendre quadratures (as is proved in the next paragraph, the integrand is non singular provided $\nu_a,\nu_b\geqslant 2$).
|
|
\bigskip
|
|
|
|
\point
|
|
Note that these integrals are not singular as long as $\nu_a,\nu_b\geqslant 2$: indeed (since the only possible problems occur at $-1$, it suffices to consider the case with only one spline), $\alpha_-,\alpha_+>-1$ for $\tau>-1$, and
|
|
\begin{equation}
|
|
\alpha_\pm(k,\tau)=-1+(1+\tau)\pm\frac k2(1+\tau)^2+O(1+\tau)^3
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\int_{\alpha_-(|k|,\tau)}^{\alpha_+(|k|,\tau)}d\sigma\ \left|\frac{2(1-\sigma)}{(1+\sigma)^{3-\nu_b}}T_m(\sigma)\right|
|
|
\leqslant
|
|
4\int_{\alpha_-(|k|,\tau)}^{\alpha_+(|k|,\tau)}d\sigma\ \frac1{(1+\sigma)^{3-\nu_b}}
|
|
\end{equation}
|
|
which, if $\nu_b=2$, yields
|
|
\begin{equation}
|
|
4\log\left(\frac{1+\alpha_+(|k|,\tau)}{1+\alpha_-(|k|,\tau)}\right)
|
|
=4|k|(1+\tau)+O(1+\tau)^2
|
|
\end{equation}
|
|
and if $\nu_b>2$, yields
|
|
\begin{equation}
|
|
\frac4{\nu_b-2}\left((1+\alpha_+(|k|,\tau))^{\nu_b-2}-(1+\alpha_-(|k|,\tau))^{\nu_b-2}\right)
|
|
=
|
|
\frac4{\nu_b-2}|k|(1+\tau)^{\nu_b-1}(1+O(1+\tau))
|
|
.
|
|
\end{equation}
|
|
Therefore,
|
|
\begin{equation}
|
|
\left|\frac{2(1-\tau)}{(1+\tau)^{3-\nu_a}}T_n(\tau)\int_{\alpha_-(|k|,\tau)}^{\alpha_+(|k|,\tau)}d\sigma\ \frac{2(1-\sigma)}{(1+\sigma)^{3-\nu_b}}T_m(\sigma)\right|
|
|
\leqslant
|
|
\frac8{c_{\nu_b}}
|
|
|k|(1-\tau)(1+\tau)^{\nu_a+\nu_b-4}(1+O(1+\tau))
|
|
\end{equation}
|
|
where $c_{2}:=1$ and $c_{\nu_b}:=\nu_b-2$ if $\nu_b>2$.
|
|
The integrand is, therefore, not singular as long as $\nu_a,\nu_b\geqslant 2$.
|
|
\bigskip
|
|
|
|
\point
|
|
Evaluating convolutions at $k=0$ is not immediate, as the formula for $A(0)$ involves a bit of a computation.
|
|
To compute $A(0)$, we expand
|
|
\begin{equation}
|
|
\alpha_-(|k|,\tau)=\tau-\frac{|k|(1+\tau)^2}2+O(k^2)
|
|
,\quad
|
|
\alpha_+(|k|,\tau)=\tau+\frac{|k|(1+\tau)^2}2+O(k^2)
|
|
\end{equation}
|
|
so
|
|
\begin{equation}
|
|
A_{\zeta,n;\zeta',m}^{(\nu_a,\nu_b)}(0)=\mathds 1_{\zeta=\zeta'}
|
|
\frac1{\pi^2}
|
|
\int_{\tau_\zeta}^{\tau_{\zeta+1}} d\tau\ \frac{(1-\tau)^2}{(1+\tau)^{4-\nu_a-\nu_b}}T_n({\textstyle\frac{2\tau-(\tau_\zeta+\tau_{\zeta+1})}{\tau_{\zeta+1}-\tau_\zeta}})
|
|
T_m({\textstyle\frac{2\tau-(\tau_{\zeta}+\tau_{\zeta+1})}{\tau_{\zeta+1}-\tau_{\zeta}}})
|
|
\end{equation}
|
|
which we then compute using a Gauss-Legendre quadrature.
|
|
We can then evaluate convolutions at $k=0$:
|
|
\begin{equation}
|
|
(a\hat\ast b)(0)\approx\left<\mathfrak a\mathfrak b\right>
|
|
:=
|
|
\frac1{4\pi^2|k_{l,j}|}
|
|
\sum_{n,m=0}^P
|
|
\sum_{l,l'=0}^{J-1}
|
|
\mathfrak F_{l,n}^{(\nu_a)}(\mathfrak a)\mathfrak F_{l',m}^{(\nu_b)}(\mathfrak b)
|
|
A_{l,n;l',m}^{(\nu_a,\nu_b)}(0)
|
|
\label{avg}
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
|
|
\point
|
|
Let us now compute some choices of $\mathfrak a,\mathfrak b$ more explicitly.
|
|
\medskip
|
|
|
|
\subpoint
|
|
Let us start with $\mathds 1_{l,n}\hat\ast a$ where $\mathds 1_{l,n}$ is the vector which has 0's everywhere except at position $(l,n)$.
|
|
We have
|
|
\begin{equation}
|
|
\mathfrak F_{l',m}^{(\nu_1)}(\mathds 1_{l,n})=\delta_{l',l}\frac{2-\delta_{m,0}}2w_n\frac{\cos({\textstyle\frac{m\pi(1+x_n)}2})}{(1-\frac{\tau_{l+1}-\tau_l}2\sin(\frac{\pi x_j}2)+\frac{\tau_{l+1}+\tau_l}2)^{\nu_1}}
|
|
\end{equation}
|
|
so
|
|
\begin{equation}
|
|
(\mathds 1_{n,l''}\odot\mathfrak a)_{m,l}=
|
|
\sum_{l,p}\sum_{l'}\frac{2-\delta_{l,0}}2w_n\frac{\cos({\textstyle\frac{l\pi(1+x_n)}2})}{(1-\frac{\tau_{l''+1}-\tau_{l''}}2\sin(\frac{\pi x_n}2)+\frac{\tau_{l''+1}+\tau_{l''}}2)^{\nu_1}}A_{l'',l;l',p}^{(\nu_1,\nu_a)}(k_{l,m})\mathfrak F_{l',p}^{(\nu_a)}(\mathfrak a)
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subpoint
|
|
We now turn to $a\hat\ast\hat v$.
|
|
Let
|
|
\begin{equation}
|
|
\Upsilon(t,|k|):=
|
|
\int_{||k|-t|}^{|k|+t}ds\ \frac{s\hat v(s)}{|k|}
|
|
\label{Upsilon}
|
|
\end{equation}
|
|
We have
|
|
\begin{equation}
|
|
a\hat\ast\hat v=\frac 1{4\pi^2}\int_0^\infty dt\ ta(t)\Upsilon(t,|k|)
|
|
.
|
|
\label{avUpsilon}
|
|
\end{equation}
|
|
Following the integration scheme\-~(\ref{anyeq_integration}),
|
|
\begin{equation}
|
|
(\mathfrak a\odot\mathfrak v)_{l,i}=
|
|
\frac1{32\pi}\sum_{l'=0}^{J-1}(\tau_{l'+1}-\tau_{l'})\sum_{j=1}^Nw_j \cos({\textstyle\frac{\pi x_j}2})(1+\vartheta_{l'}(x_j))^2\vartheta_{l'}(x_j)\mathfrak a_{l',j}\Upsilon(\vartheta(x_j),k_{l,i})
|
|
.
|
|
\label{adotv}
|
|
\end{equation}
|
|
At $k=0$,
|
|
\begin{equation}
|
|
\left<\mathfrak a\mathfrak v\right>=
|
|
\frac1{32\pi}\sum_{l'=0}^{J-1}(\tau_{l'+1}-\tau_{l'})\sum_{j=1}^Nw_j \cos({\textstyle\frac{\pi x_j}2})(1+\vartheta_{l'}(x_j))^2\vartheta_{l'}(x_j)\mathfrak a_{l',j}\Upsilon(\vartheta(x_j),0)
|
|
\label{<av>}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\Upsilon(t,0)=2tv(t)
|
|
.
|
|
\end{equation}
|
|
The quantities $\Upsilon(\vartheta(x_n),k_{l,i})$ and $\Upsilon(\vartheta(x_n),0)$ are independent of $a$ and can be computed once and for all at execution.
|
|
The integral in\-~(\ref{Upsilon}) is computed using Gauss-Legendre quadratures, but without splitting into splines.
|
|
To maintain a high precision, we set the order of the integration to \refname{param:anyeq_J}{{\tt J}}$\times$\refname{param:anyeq_N}{{\tt N}}.
|
|
\bigskip
|
|
|
|
\subpoint
|
|
Finally,
|
|
\begin{equation}
|
|
(\mathds 1_{l',n}\odot\mathfrak v)_{l,i}=
|
|
\frac{\tau_{l'+1}-\tau_{l'}}{32\pi}w_n\cos({\textstyle\frac{\pi x_n}2})(1+\vartheta_{l'}(x_n))^2\vartheta_{l'}(x_n)\Upsilon(\vartheta(x_n),k_{l,i})
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\left<\mathds 1_{l',n}\mathfrak v\right>=
|
|
\frac{\tau_{l'+1}-\tau_{l'}}{32\pi}w_n\cos({\textstyle\frac{\pi x_n}2})(1+\vartheta_{l'}(x_n))^2\vartheta_{l'}(x_n)\Upsilon(\vartheta(x_n),0)
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subsubsubsection{Evaluating $\hat l_3$}
|
|
The only term in\-~(\ref{anyeq}) that does not involve just convolutions (whose computation was described above) is $\hat l_3$ (\ref{l3}).
|
|
To evaluate it, we first change variables, using a generalization of bipolar coordinates (see lemma\-~\ref{lemma:bipolar2}):
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\hat l_3(|k|)
|
|
=
|
|
\frac1{(2\pi)^5\rho^2| k|^2}\int_0^\infty dt\int_{|| k|-t|}^{| k|+t}ds\int_0^\infty dt'\int_{|| k|-t'|}^{| k|+t'}ds'\int_0^{2\pi}d\theta\ sts't'U(s)U(t)U(s')U(t')
|
|
\cdot\\\hfill\cdot
|
|
S(\mathfrak R(s,t,s',t',\theta,| k|))
|
|
\end{largearray}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\mathfrak R(s,t,s',t',\theta,| k|)
|
|
:=
|
|
\frac1{\sqrt 2| k|}
|
|
\Big(
|
|
| k|^2(s^2+t^2+(s')^2+(t')^2)-| k|^4-(s^2-t^2)((s')^2-(t')^2)
|
|
\\[0.5cm]\hfill
|
|
-
|
|
\sqrt{(4| k|^2s^2-(| k|^2+s^2-t^2)^2)(4| k|^2(s')^2-(| k|^2+(s')^2-(t')^2)^2)}\cos\theta
|
|
\Big)^{\frac12}
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
We change variables as in\-~(\ref{convvar}):
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\hat l_3(|k|)
|
|
=
|
|
\frac1{(2\pi)^5\rho^2| k|^2}
|
|
\int_{-1}^1 d\tau\ \frac{2(1-\tau)}{(1+\tau)^3}U({\textstyle\frac{1-\tau}{1+\tau}})
|
|
\int_{\alpha_-(| k|,\tau)}^{\alpha_+(| k|,\tau)} d\sigma\ \frac{2(1-\sigma)}{(1+\sigma)^3}U({\textstyle\frac{1-\sigma}{1+\sigma}})
|
|
\int_{-1}^1 d\tau'\ \frac{2(1-\tau')}{(1+\tau')^3}
|
|
\cdot\\[0.5cm]\hfill\cdot
|
|
U({\textstyle\frac{1-\tau'}{1+\tau'}})
|
|
\int_{\alpha_-(| k|,\tau')}^{\alpha_+(| k|,\tau')} d\sigma'\ \frac{2(1-\sigma')}{(1+\sigma')^3}U({\textstyle\frac{1-\sigma'}{1+\sigma'}})
|
|
\int_0^{2\pi}d\theta\ S(\mathfrak R({\textstyle \frac{1-\sigma}{1+\sigma},\frac{1-\tau}{1+\tau},\frac{1-\sigma'}{1+\sigma'},\frac{1-\tau'}{1+\tau'},\theta,| k|}))
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
We expand $U$ and $ S$ into Chebyshev polynomials as in\-~(\ref{a_chebyshev_spline}), and split the integrals into splines:
|
|
\begin{equation}
|
|
\hat l_3(|k_{l,j}|)
|
|
\approx
|
|
(\mathbb U^{\otimes4}\otimes\mathbb S)_{l,j}:=
|
|
\sum_{\lambda_1,\cdots,\lambda_5=0}^{J-1}
|
|
\sum_{n_1,\cdots,n_5=1}^\infty
|
|
\bar A_{\lambda_1,n_1;\cdots;\lambda_5,n_5}^{(\nu)}(|k_{l,j}|)
|
|
\prod_{i=1}^4
|
|
\left(\mathfrak F_{\lambda_i,n_i}^{(\nu)}(\mathbb U)\right)
|
|
\mathfrak F_{\lambda_5,n_5}^{(\nu)}(\mathbb S)
|
|
\label{otimes}
|
|
\end{equation}
|
|
with $\mathbb U_{l,j}:=U(k_{l,j})$, $\mathbb S_{l,j}:=S(k_{l,j})$ and
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\bar A_{\lambda_1,n_1;\cdots;\lambda_5,n_5}^{(\nu)}(| k|)
|
|
=
|
|
\frac1{(2\pi)^5\rho^2| k|^2}
|
|
\int_{\tau_{\lambda_1}}^{\tau_{\lambda_1+1}} d\tau\ \frac{2(1-\tau)}{(1+\tau)^{3-\nu}}T_{n_1}({\textstyle\frac{2\tau-(\tau_{\lambda_1}+\tau_{\lambda_1+1})}{\tau_{\lambda_1+1}-\tau_{\lambda_1}}})
|
|
\cdot\\[0.5cm]\hskip1cm\cdot
|
|
\mathds 1_{\alpha_-(| k|,\tau)<\tau_{\lambda_2+1}}\mathds 1_{\alpha_+(| k|,\tau)>\tau_{\lambda_2}}
|
|
\int_{\max(\tau_{\lambda_2},\alpha_-(| k|,\tau))}^{\min(\tau_{\lambda_2+1},\alpha_+(| k|,\tau))} d\sigma\ \frac{2(1-\sigma)}{(1+\sigma)^{3-\nu}}T_{n_2}({\textstyle\frac{2\sigma-(\tau_{\lambda_2}+\tau_{\lambda_2+1})}{\tau_{\lambda_2+1}-\tau_{\lambda_2}}})
|
|
\cdot\\[0.5cm]\hskip1cm\cdot
|
|
\int_{\tau_{\lambda_3}}^{\tau_{\lambda_3+1}} d\tau'\ \frac{2(1-\tau')}{(1+\tau')^{3-\nu}}T_{n_3}({\textstyle\frac{2\tau'-(\tau_{\lambda_3}+\tau_{\lambda_3+1})}{\tau_{\lambda_3+1}-\tau_{\lambda_3}}})
|
|
\cdot\\[0.5cm]\hskip1cm\cdot
|
|
\mathds 1_{\alpha_-(| k|,\tau')<\tau_{\lambda_4+1}}\mathds 1_{\alpha_+(| k|,\tau')>\tau_{\lambda_4}}
|
|
\int_{\max(\tau_{\lambda_4},\alpha_-(| k|,\tau'))}^{\min(\tau_{\lambda_4+1},\alpha_+(| k|,\tau'))} d\sigma'\ \frac{2(1-\sigma')}{(1+\sigma')^{3-\nu}}T_{n_4}({\textstyle\frac{2\sigma'-(\tau_{\lambda_4}+\tau_{\lambda_4+1})}{\tau_{\lambda_4+1}-\tau_{\lambda_4}}})
|
|
\cdot\\[0.5cm]\hskip1cm\cdot
|
|
\int_0^{2\pi}d\theta\
|
|
\frac{2^\nu}{(2+\mathfrak R)^\nu}
|
|
T_{n_5}({\textstyle\frac{2\frac{1-\mathfrak R}{1+\mathfrak R}-(\tau_{\lambda_5}+\tau_{\lambda_5+1})}{\tau_{\lambda_5+1}-\tau_{\lambda_5}}})
|
|
\mathds 1_{\tau_{\lambda_5}<\frac{1-\mathfrak R}{1+\mathfrak R}<\tau_{\lambda_5+1}}
|
|
\end{largearray}
|
|
\end{equation}
|
|
in which $\mathfrak R$ is a shorthand for $\mathfrak R({\textstyle \frac{1-\sigma}{1+\sigma},\frac{1-\tau}{1+\tau},\frac{1-\sigma'}{1+\sigma'},\frac{1-\tau'}{1+\tau'},\theta,| k|})$.
|
|
Note that $\bar A$ is independent of $U$, and can be computed once and for all at the beginning of execution.
|
|
Since the tensor $\bar A$ is quite large (it contains $(NJ)^5$ entries), and its computation can be rather long it can be inconvenient to compute $\bar A$ at every execution.
|
|
Instead, one can use the \refname{command:anyeq_save_Abar}{{\tt save\_Abar}} method to compute $\bar A$ and save it to a file, which can then be recalled via the {\tt -s} option on the command line.
|
|
\bigskip
|
|
|
|
\subsubsubsection{Main algorithm to compute $U$}
|
|
We are now ready to detail how $U\equiv\rho\hat u$ is computed.
|
|
All of the observables we are interested in are approximated from the values $\mathbb U_{l,j}:=U(k_{l,j})$ (\ref{klj}).
|
|
\bigskip
|
|
|
|
\point
|
|
The equation for $(\mathbb U_{l,j})_{l\in\{0,\cdots,J-1\},j\in\{1,\cdots,N\}}$ is obtained by approximating\-~(\ref{anyeq_hatu}) according to the prescriptions detailed above:
|
|
\begin{equation}
|
|
\mathbb U_{l,j}:=
|
|
\frac{1+\mathbb Y_{l,j}}{2(\mathbb X_{l,j}+1)}\ \Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
\label{bbU}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\mathbb X_{l,j}:=\frac1{\mathbb L_{l,j}}\left(\mathbb K_{l,j}-\mathbb L_{l,j}+\frac{k_{l,j}^2}{2\rho}+\mathbb I_{l,j}\right)
|
|
,\quad
|
|
\mathbb Y_{l,j}:=\frac1{\mathbb L_{l,j}}\left(\mathbb S_{l,j}-\mathbb L_{l,j}+\frac12\mathbb J_{l,j}+\mathbb G_{l,j}\right)
|
|
\end{equation}
|
|
\begin{equation}
|
|
\mathbb S_{l,j}:=
|
|
\mathfrak v_{l,j}-\frac1{\rho}(\mathbb U\odot\mathfrak v)_{l,j}
|
|
,\quad
|
|
\mathfrak v_{l,j}:=\hat v(k_{l,j})
|
|
,\quad
|
|
\mathbb E:=\hat v(0)-\frac1{\rho}\left<\mathbb U\mathfrak v\right>
|
|
\end{equation}
|
|
\begin{equation}
|
|
\mathbb I_{l,j}:=\frac1{\rho}\gamma_{L,2}(\beta_{L,2}(\mathbb U\odot(\mathbb U\mathbb S))_{l,j}+(1-\beta_{L,2})\mathbb E(\mathbb U\odot\mathbb U)_{l,j})
|
|
\end{equation}
|
|
\begin{equation}
|
|
\mathbb K_{l,j}:=\gamma_K(\beta_K\mathbb S_{l,j}+(1-\beta_K)\mathbb E)
|
|
,\quad
|
|
\mathbb L_{l,j}:=\gamma_{L,1}(\beta_{L,1}\mathbb S_{l,j}+(1-\beta_{L,1})\mathbb E)
|
|
\end{equation}
|
|
\begin{equation}
|
|
\mathbb J_{l,j}:=\gamma_{L,3}(1-\beta_{L,3})\frac{\mathbb E}{\rho^2}(\mathbb U\odot\mathbb U)^2_{l,j}
|
|
+\gamma_{L,3}\beta_{L,3}(\mathbb U^{\otimes4}\otimes\mathbb S)_{l,j}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\mathbb G_{l,j}:=
|
|
\frac2{\rho}\gamma_K\alpha_K(\mathbb U\odot((\beta_K\mathbb S+(1-\beta_K)\mathbb E)\mathbb U))_{l,j}
|
|
\\[0.5cm]\hfill
|
|
-
|
|
\frac1{\rho}\alpha_{L,1}(\mathbb U\odot((\beta_{L,1}\mathbb S+(1-\beta_{L,1})\mathbb E)\mathbb U^2))_{l,j}
|
|
+
|
|
\frac2{\rho}\alpha_{L,2}(\mathbb U\odot(\mathbb I\mathbb U))_{l,j}
|
|
-
|
|
\frac1{2\rho}\alpha_{L,3}(\mathbb U\odot \mathbb J)_{l,j}
|
|
\end{largearray}
|
|
\end{equation}
|
|
(see\-~(\ref{odot}), (\ref{otimes}) and\-~(\ref{avg}) for the definitions of $\odot$, $\otimes$ and $\left<\cdot\right>$).
|
|
\bigskip
|
|
|
|
\point
|
|
We rewrite\-~(\ref{bbU}) as a root finding problem:
|
|
\begin{equation}
|
|
\Xi_{l,j}(\mathbb U):=
|
|
\mathbb U_{l,j}-
|
|
\frac{1+\mathbb Y_{l,j}}{2(\mathbb X_{l,j}+1)}\ \Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
=0
|
|
\label{root_anyeq}
|
|
\end{equation}
|
|
which we solve using the Newton algorithm, that is, we define a sequence of $\mathbb U$'s:
|
|
\begin{equation}
|
|
\mathbb U^{(n+1)}=\mathbb U^{(n)}-(D\Xi(\mathbb U^{(n)}))^{-1}\Xi(\mathbb U^{(n)})
|
|
\end{equation}
|
|
where $D\Xi$ is the Jacobian of $\Xi$:
|
|
\begin{equation}
|
|
(D\Xi)_{l,j;l',i}:=\frac{\partial\Xi_{l,j}}{\partial\mathbb U_{l',i}}
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
We initialize the algorithm with the solution of the \refname{eq:easyeq_medeq}{Medium equation}, which is computed using the \refname{sec:easyeq}{{\tt easyeq}} method.
|
|
However, \refname{sec:easyeq}{{\tt easyeq}} only computes $\hat u$ at the momenta given by the Gauss-Legendre quadratures\-~(\ref{easyeq_ki}).
|
|
To obtain a value for $\hat u$ at $k_{l,j}$, we use a linear interpolation (code is provided for a polynomial interpolation, which has not performed as well).
|
|
The parameters \refname{param:anyeq_tolerance}{{\tt tolerance}}, \refname{param:anyeq_maxiter}{{\tt maxiter}} are passed on to \refname{sec:easyeq}{{\tt easyeq}} as is, and the order of the Gauss-Legendre quadratures is set to \refname{param:anyeq_J}{{\tt J}}$\times$\refname{param:anyeq_N}{{\tt N}}.
|
|
\bigskip
|
|
|
|
\point
|
|
We are left with computing the Jacobian of $\Xi$:
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\partial_{l',i}\Xi_{l,j}\equiv
|
|
\frac{\partial\Xi_{l,j}}{\partial\mathbb U_{l',i}}=
|
|
\delta_{l,l'}\delta_{i,j}
|
|
+\frac1{2(\mathbb X_{l,j}+1)}\left(
|
|
\frac{(1+\mathbb Y_{l,j})\partial_{l',i}\mathbb X_{l,j}}{\mathbb X_{l,j}+1}
|
|
-
|
|
\partial_{l',i}\mathbb Y_{l,j}
|
|
\right)
|
|
\Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
\\[0.5cm]\hfill
|
|
+\frac{(1+\mathbb Y_{l,j})}{2(\mathbb X_{l,j}+1)^3}\left(
|
|
\frac{2(1+\mathbb Y_{l,j})}{\mathbb X_{l,j}+1}\partial_{l',i}\mathbb X_{l,j}
|
|
-\partial_{l',i}\mathbb Y_{l,j}
|
|
\right)\ \partial\Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
\end{largearray}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\partial_{l',i}\mathbb X_{l,j}=
|
|
\frac1{\mathbb L_{l,j}}\left(\partial_{l',i}\mathbb K_{l,j}-\partial_{l',i}\mathbb L_{l,j}+\partial_{l',i}\mathbb I_{l,j}\right)
|
|
-
|
|
\frac{\partial_{l',i}\mathbb L_{l,j}}{\mathbb L_{l,j}}\mathbb X_{l,j}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_{l',i}\mathbb Y_{l,j}=
|
|
\frac1{\mathbb L_{l,j}}\left(\partial_{l',i}\mathbb S_{l,j}-\partial_{l',i}\mathbb L_{l,j}+\frac12\partial_{l',i}\mathbb J_{l,j}+\partial_{l',i}\mathbb G_{l,j}\right)
|
|
-
|
|
\frac{\partial_{l',i}\mathbb L_{l,j}}{\mathbb L_{l,j}}\mathbb Y_{l,j}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_{l',i}\mathbb S_{l,j}=
|
|
-\frac1{\rho}(\mathds 1_{l',i}\odot\mathfrak v)_{l,j}
|
|
,\quad
|
|
\partial_{l',i}\mathbb E=-\frac1{\rho}\left<\mathds 1_{l',i}\mathfrak v\right>
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_{l',i}\mathbb K_{l,j}:=\gamma_K(\beta_K\partial_{l',i}\mathbb S_{l,j}+(1-\beta_K)\partial_{l',i}\mathbb E)
|
|
,\quad
|
|
\partial_{l',i}\mathbb L_{l,j}:=\gamma_{L,1}(\beta_{L,1}\partial_{l',i}\mathbb S_{l,j}+(1-\beta_{L,1})\partial_{l',i}\mathbb E)
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\partial_{l',i}\mathbb I_{l,j}=
|
|
\frac1{\rho}\gamma_{L,2}\left(\beta_{L,2}(
|
|
\mathds 1_{l',i}\odot(\mathbb U\mathbb S)
|
|
+
|
|
\mathbb U\odot(\mathds 1_{l',i}\mathbb S+\mathbb U\partial_{l',i}\mathbb S)
|
|
)_{l,j}
|
|
\right.\\[0.5cm]\hfill\left.
|
|
+
|
|
(1-\beta_{L,2})(
|
|
\partial_{l',i}\mathbb E(\mathbb U\odot\mathbb U)_{l,j}
|
|
+
|
|
2\mathbb E(\mathbb U\odot\mathds 1_{l',i})_{l,j}
|
|
)\right)
|
|
\end{largearray}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\partial_{l',i}\mathbb J_{l,j}=
|
|
\frac1{\rho^2}\gamma_{L,3}(1-\beta_{L,3})\left(
|
|
\partial_{l',i}\mathbb E(\mathbb U\odot\mathbb U)^2_{l,j}
|
|
+
|
|
4\mathbb E(\mathbb U\odot\mathbb U)_{l,j}(\mathbb U\odot\mathds 1_{l',i})_{l,j}
|
|
\right)
|
|
\\[0.5cm]\hfill
|
|
+\gamma_{L,3}\beta_{L,3}(
|
|
4\mathds 1_{l',i}\otimes\mathbb U^{\otimes3}\otimes\mathbb S
|
|
+
|
|
\mathbb U^{\otimes4}\otimes\partial_{l',i}\mathbb S
|
|
)_{l,j}
|
|
\end{largearray}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\partial_{l',i}\mathbb G_{l,j}=
|
|
\frac2{\rho}\gamma_K\alpha_K\beta_K\left(
|
|
\mathds 1_{l',i}\odot(\mathbb S\mathbb U)
|
|
+
|
|
\mathbb U\odot(\mathbb S\mathds 1_{l',i}
|
|
+\partial_{l',i}\mathbb S\mathbb U)
|
|
\right)_{l,j}
|
|
\\[0.5cm]\hfill
|
|
+\frac2{\rho}\gamma_K\alpha_K(1-\beta_K)\left(
|
|
\partial_{l',i}\mathbb E(\mathbb U\odot\mathbb U)_{l,j}
|
|
+2\mathbb E(\mathds 1_{l',i}\odot\mathbb U)_{l,j}
|
|
\right)
|
|
\\[0.5cm]\indent
|
|
-
|
|
\frac1{\rho}\alpha_{L,1}\beta_{L,1}\left(
|
|
\mathds 1_{l',i}\odot(\mathbb S\mathbb U^2)
|
|
+
|
|
\mathbb U\odot(
|
|
\partial_{l',i}\mathbb S\mathbb U^2
|
|
+
|
|
2\mathbb S\mathbb U\mathds 1_{l',i}
|
|
)
|
|
\right)_{l,j}
|
|
\\[0.5cm]\hfill
|
|
-
|
|
\frac1{\rho}\alpha_{L,1}(1-\beta_{L,1})\left(
|
|
\mathbb E\mathds 1_{l',i}\odot\mathbb U^2
|
|
+
|
|
\mathbb U\odot(
|
|
\partial_{l',i}\mathbb E\mathbb U^2
|
|
+
|
|
2\mathbb E\mathbb U\mathds 1_{l',i}
|
|
)
|
|
\right)_{l,j}
|
|
\\[0.5cm]\hfill
|
|
+
|
|
\frac2{\rho}\alpha_{L,2}(
|
|
(\mathds 1_{l',i}\odot(\mathbb I\mathbb U))_{l,j}
|
|
+
|
|
\mathbb U\odot(
|
|
\partial_{l',i}\mathbb I\mathbb U
|
|
+
|
|
\mathbb I\mathds 1_{l',i})
|
|
)_{l,j}
|
|
-
|
|
\frac1{2\rho}\alpha_{L,3}(
|
|
\mathds 1_{l',i}\odot \mathbb J
|
|
+
|
|
\mathbb U\odot\partial_{l',i}\mathbb J
|
|
)_{l,j}
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
We iterate the Newton algorithm until the Newton relative error $\epsilon$ becomes smaller than the \refname{param:easyeq_tolerance}{{\tt tolerance}} parameter.
|
|
The Newton error is defined as
|
|
\begin{equation}
|
|
\epsilon=\frac{\|\mathbb U^{(n+1)}-\mathbb U^{(n)}\|_2}{\|\mathbb U^{(n)}\|_2}
|
|
\end{equation}
|
|
where $\|\cdot\|_2$ is the $l_2$ norm.
|
|
The energy thus obtained is
|
|
\begin{equation}
|
|
e=\frac\rho2\mathds E
|
|
\end{equation}
|
|
the Fourier transform $\hat u$ of the solution is
|
|
\begin{equation}
|
|
\hat u(k_{l,j})\approx\frac{\mathbb U_{l,j}}\rho
|
|
\end{equation}
|
|
where $k_{l,j}$ was defined in\-~(\ref{klj}), and the solution $u$ in real space is obtained by inverting the Fourier transform, following the prescription of\-~(\ref{anyeq_integration}):
|
|
\begin{equation}
|
|
u(|x|)=\int \frac{dk}{(2\pi)^3}\ e^{-ikx}\hat u(|k|)
|
|
\approx
|
|
\sum_{l=0}^{J-1}
|
|
\frac{\tau_{l+1}-\tau_l}{16\pi}
|
|
\sum_{j=1}^Nw_j
|
|
\cos({\textstyle\frac{\pi x_j}2})(1+\vartheta_l(x_j))^2\vartheta_l(x_j)\mathbb U_{l,j}\sin(\vartheta_l(x_j)|x|)
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subsubsubsection{Condensate fraction}
|
|
To compute the condensate fraction, we solve the modified {\tt anyeq} (see\-~\cite{CJL21}):
|
|
\begin{equation}
|
|
(-\Delta+2\mu)u_\mu
|
|
=
|
|
(1-u_\mu)v-2\rho K+\rho^2 L
|
|
.
|
|
\end{equation}
|
|
where $K$ and $L$ are defined as in\-~(\ref{anyeq_K})-(\ref{anyeq_e}) in which $u$ is replaced with $u_\mu$.
|
|
The uncondensed fraction is then
|
|
\begin{equation}
|
|
\eta=\partial_\mu e|_{\mu=0}
|
|
=-\frac\rho2\int dx\ v(x)\partial_\mu u_\mu(x)|_{\mu=0}
|
|
.
|
|
\end{equation}
|
|
To compute the energy in the presence of the parameter $\mu$, we proceed in the same way as for $\mu=0$, the only difference being that $k^2$ should formally be replaced by $k^2+2\mu$.
|
|
In other words, we consider $\mathbb U_{j,l}=u_\mu(|k_{j,l}|)$ and define $\Xi(\mathbb U,\mu)$ in the same way as in\-~(\ref{root_anyeq}), except that $\mathbb X_i$ should be replaced by
|
|
\begin{equation}
|
|
\mathbb X_{l,j}=\frac1{\mathbb L_{l,j}}\left(\mathbb K_{l,j}-\mathbb L_{l,j}+\frac{k_{l,j}^2+2\mu}{2\rho}+\mathbb I_{l,j}\right)
|
|
.
|
|
\end{equation}
|
|
We then solve
|
|
\begin{equation}
|
|
\Xi(\mathbb U_\mu,\mu)=0
|
|
\end{equation}
|
|
By differentiating this identity with respect to $\mu$, we find $\partial_\mu u_\mu$:
|
|
\begin{equation}
|
|
\partial_\mu \mathbb U|_{\mu=0}=-(D\Xi)^{-1}\partial_\mu\Xi|_{\mu=0}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\partial_\mu\Xi|_{\mu=0}=
|
|
\frac{(1+\mathbb Y_{l,j})\partial_\mu\mathbb X_{l,j}}{2(\mathbb X_{l,j}+1)^2}
|
|
\Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
+\frac{(1+\mathbb Y_{l,j})^2}{(\mathbb X_{l,j}+1)^4}\partial_\mu\mathbb X_{l,j}
|
|
\partial\Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
,\quad
|
|
\partial_\mu\mathbb X_{l,j}=\frac1{\rho\mathbb L_{l,j}}
|
|
.
|
|
\end{equation}
|
|
We then approximate
|
|
\begin{equation}
|
|
\eta\approx
|
|
-\frac12\left<\mathfrak v\partial_\mu\mathbb U\right>
|
|
\end{equation}
|
|
(see\-~(\ref{avg})).
|
|
\bigskip
|
|
|
|
\subsubsubsection{Correlation function (spherical average)}
|
|
The two-point correlation function is
|
|
\begin{equation}
|
|
c_2(x):=
|
|
2\rho\frac{\delta e}{\delta v(x)}
|
|
\end{equation}
|
|
and its spherical average is
|
|
\begin{equation}
|
|
C_2(|x|):=\frac1{4\pi|x|^2}\int dy\ \delta(|x|-|y|)c_2(y)
|
|
.
|
|
\end{equation}
|
|
In Fourier space,
|
|
\begin{equation}
|
|
c_2(x)=
|
|
2\rho
|
|
\int dk\ e^{ikx}\frac{\delta e}{\delta\hat v(k)}
|
|
\end{equation}
|
|
so
|
|
\begin{equation}
|
|
C_2(|x|)=
|
|
2\rho\int dk\ \left(\frac1{4\pi|x|^2}\int dy\ \delta(|x|-|y|)e^{iky}\right)\frac{\delta e}{\delta\hat v(k)}
|
|
=2\rho\int dk\ \frac{\sin(|k||x|)}{|k||x|}\frac{\delta e}{\delta\hat v(k)}
|
|
.
|
|
\label{C2fourier}
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
We can compute this quantity by considering a modified {\tt anyeq} in Fourier space, by formally replacing $\hat v$ with
|
|
\begin{equation}
|
|
\hat v+\lambda g(|k|)
|
|
,\quad
|
|
g(|k|):=\frac{\sin(|k||x|)}{|k||x|}
|
|
.
|
|
\label{2pt_addv}
|
|
\end{equation}
|
|
Indeed, if $e_\lambda$ denotes the energy of this modified equation,
|
|
\begin{equation}
|
|
\partial_\lambda e_\lambda|_{\lambda=0}
|
|
=\int dk\ \frac{\delta e}{\delta\hat v(k)}\partial_\lambda({\textstyle \hat v(k)+\lambda g(|k|)})
|
|
=\int dk\ g(|k|)\frac{\delta e}{\delta\hat v(k)}
|
|
.
|
|
\end{equation}
|
|
So, denoting the solution of the modified equation by $u_\lambda$,
|
|
\begin{equation}
|
|
C_2(x)=2\rho\partial_\lambda e_\lambda|_{\lambda=0}
|
|
=\rho^2g(0)-\rho^2\int\frac{dk}{(2\pi)^3}\ (g(k)\hat u(k)+\hat v(k)\partial_\lambda\hat u_\lambda(k)|_{\lambda=0})
|
|
.
|
|
\end{equation}
|
|
We compute $\partial_\lambda u_\lambda|_{\lambda=0}$ in the same way as the uncondensed fraction: we define $\Xi(\mathbb U,\lambda)$ by formally adding $\lambda g(|k|)$ to $\hat v$, solve $\Xi(\mathbb U,\lambda)=0$, and differentiate:
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb U|_{\lambda=0}=-(D\Xi)^{-1}\partial_\lambda\Xi|_{\lambda=0}
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
We compute $\partial_\lambda\Xi|_{\lambda=0}$:
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\partial_\lambda\Xi_{l,j}=
|
|
\delta_{l,l'}\delta_{i,j}
|
|
+\frac1{2(\mathbb X_{l,j}+1)}\left(
|
|
\frac{(1+\mathbb Y_{l,j})\partial_\lambda\mathbb X_{l,j}}{\mathbb X_{l,j}+1}
|
|
-
|
|
\partial_\lambda\mathbb Y_{l,j}
|
|
\right)
|
|
\Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
\\[0.5cm]\hfill
|
|
+\frac{(1+\mathbb Y_{l,j})}{2(\mathbb X_{l,j}+1)^3}\left(
|
|
\frac{2(1+\mathbb Y_{l,j})}{\mathbb X_{l,j}+1}\partial_\lambda\mathbb X_{l,j}
|
|
-\partial_\lambda\mathbb Y_{l,j}
|
|
\right)\ \partial\Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
\end{largearray}
|
|
\label{2pt_dXi}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb S_{l,j}=
|
|
\mathfrak g_{l,j}
|
|
-\frac1{\rho}(\mathbb U\odot\mathfrak g)_{l,j}
|
|
,\quad
|
|
\partial_\lambda\mathbb E=g(0)-\frac1{\rho}\left<\mathbb U\mathfrak g\right>
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb X_{l,j}=
|
|
\frac1{\mathbb L_{l,j}}\left(\partial_\lambda\mathbb K_{l,j}-\partial_\lambda\mathbb L_{l,j}+\partial_\lambda\mathbb I_{l,j}\right)
|
|
-
|
|
\frac{\partial_\lambda\mathbb L_{l,j}}{\mathbb L_{l,j}}\mathbb X_{l,j}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb Y_{l,j}=
|
|
\frac1{\mathbb L_{l,j}}\left(\partial_\lambda\mathbb S_{l,j}-\partial_\lambda\mathbb L_{l,j}+\frac12\partial_\lambda\mathbb J_{l,j}+\partial_\lambda\mathbb G_{l,j}\right)
|
|
-
|
|
\frac{\partial_\lambda\mathbb L_{l,j}}{\mathbb L_{l,j}}\mathbb Y_{l,j}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb K_{l,j}:=\gamma_K(\beta_K\partial_\lambda\mathbb S_{l,j}+(1-\beta_K)\partial_\lambda\mathbb E)
|
|
,\quad
|
|
\partial_\lambda\mathbb L_{l,j}:=\gamma_{L,1}(\beta_{L,1}\partial_\lambda\mathbb S_{l,j}+(1-\beta_{L,1})\partial_\lambda\mathbb E)
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb I_{l,j}=
|
|
\frac1{\rho}\gamma_{L,2}\left(
|
|
\beta_{L,2}(\mathbb U\odot(\mathbb U\partial_\lambda\mathbb S))_{l,j}
|
|
+
|
|
(1-\beta_{L,2})\partial_\lambda\mathbb E(\mathbb U\odot\mathbb U)_{l,j}
|
|
\right)
|
|
\end{equation}
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb J_{l,j}=
|
|
\frac1{\rho^2}\gamma_{L,3}(1-\beta_{L,3})\partial_\lambda\mathbb E(\mathbb U\odot\mathbb U)^2_{l,j}
|
|
+\gamma_{L,3}\beta_{L,3}(\mathbb U^{\otimes4}\otimes\partial_\lambda\mathbb S)_{l,j}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\partial_\lambda\mathbb G_{l,j}=
|
|
\frac2{\rho}\gamma_K\alpha_K\beta_K\left(\mathbb U\odot\left(\partial_\lambda\mathbb S\mathbb U\right)\right)_{l,j}
|
|
+\frac2{\rho}\gamma_K\alpha_K(1-\beta_K)\partial_\lambda\mathbb E(\mathbb U\odot\mathbb U)_{l,j}
|
|
-
|
|
\frac1{\rho}\alpha_{L,1}\beta_{L,1}\left(\mathbb U\odot\partial_\lambda\mathbb S\mathbb U^2\right)_{l,j}
|
|
\\[0.5cm]\hfill
|
|
-
|
|
\frac1{\rho}\alpha_{L,1}(1-\beta_{L,1})\partial_\lambda\mathbb E\left(\mathbb U\odot(\mathbb U^2)\right)_{l,j}
|
|
+
|
|
\frac2{\rho}\alpha_{L,2}\mathbb U\odot(\partial_\lambda\mathbb I\mathbb U)_{l,j}
|
|
-
|
|
\frac1{2\rho}\alpha_{L,3}(\mathbb U\odot\partial_\lambda\mathbb J)_{l,j}
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
To evaluate $(\mathbb U\odot\mathfrak g)$ and $\left<\mathbb U\mathfrak g\right>$, we proceed as in\-~(\ref{adotv}) and\-~(\ref{<av>}).
|
|
To do so, we replace $\hat v$ with $g$ in the computation of $\Upsilon$.
|
|
\bigskip
|
|
|
|
\point
|
|
In order to invert the Fourier transform in\-~(\ref{C2fourier}) numerically, we will use a Hann window (see appendix\-~\ref{appendix:hann})
|
|
\begin{equation}
|
|
H_L(k):=\mathds 1_{|k|<\frac L2}\cos^2({\textstyle\frac{\pi|k|}{L}})
|
|
.
|
|
\label{hann}
|
|
\end{equation}
|
|
The parameter $L$ is set using \refname{param:anyeq_window_L}{{\tt window\_L}}.
|
|
The computation is changed only in that $g$ is changed to $H_L(k)\frac{\sin(|k||x|)}{|k||x|}$.
|
|
\bigskip
|
|
|
|
\point
|
|
To compute the maximum of $C_2$, we use a modified Newton algorithm.
|
|
The initial guess for the maximum is $|x_0|=\rho^{-\frac13}$\refname{param:anyeq_x0}{{\tt x0}}.
|
|
The modified Newton algorithm is an iteration:
|
|
\begin{equation}
|
|
x_{n+1}=x_n+\frac{\partial C_2(|x_n|)}{|\partial^2C_2(|x_n|)|}
|
|
\label{anyeq_newton_2pt}
|
|
\end{equation}
|
|
in which the derivatives are approximated using finite differences:
|
|
\begin{equation}
|
|
\partial C_2(x)\approx \frac{C_2(|x|+dx)-C_2(|x|)}{dx}
|
|
,\quad
|
|
\partial^2 C_2(x)\approx \frac{C_2(|x|+dx)+C_2(|x|-dx)-2C_2(|x|)}{dx^2}
|
|
.
|
|
\label{anyeq_dx_2pt}
|
|
\end{equation}
|
|
This is a modification of the usual Newton iteration $x_n+\partial C_2/\partial^2C_2$ which is designed to follow the direction of the gradient, and thus to move toward a local maximum.
|
|
In addition, if $|\partial C_2|/|\partial^2 C_2|$ is larger than \refname{param:anyeq_maxstep}{{\tt maxstep}}, then the step is replaced with $\pm$\refname{param:anyeq_maxstep}{{\tt maxstep}}.
|
|
This prevents the algorithm from stepping over a maximum and land on another, further away.
|
|
This is useful if one has a good idea of where the global maximum is, and does not want to get trapped in a smaller local maximum.
|
|
\bigskip
|
|
|
|
\indent
|
|
The algorithm is run for a maximum of \refname{param:anyeq_maxiter}{{\tt maxiter}} iterations, or until $|x_{n+1}-x_n|$ is smaller than \refname{param:anyeq_tolerance}{{\tt tolerance}}.
|
|
If the maximal number of iterations is reached, or if the solution found is not a local maximum, then the algorithm fails, and returns $+\infty$.
|
|
The point thus computed is therefore a local maximum, but it is not guaranteed to be the global maximum.
|
|
|
|
\subsubsubsection{Fourier transform of two-point correlation (spherical average)}
|
|
The Fourier transform of the two-point correlation function is
|
|
\begin{equation}
|
|
\hat c_2(q):=
|
|
2\rho\frac{\delta e}{\delta v(q)}
|
|
\end{equation}
|
|
and its spherical average is
|
|
\begin{equation}
|
|
\hat C_2(|q|):=\frac1{4\pi|q|^2}\int dk\ \delta(|q|-|k|)c_2(k)
|
|
=
|
|
\frac\rho{2\pi|q|^2}\int dk\ \delta(|q|-|k|)\frac{\delta e}{\delta\hat v(k)}
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
To compute $\frac{\delta e}{\delta\hat v(q)}$, one idea would be to proceed in the same way as for the two-point correlation function, by replacing $\hat v$ with
|
|
\begin{equation}
|
|
\hat v+\lambda g(|k|)
|
|
,\quad
|
|
g(|k|):=\frac1{4\pi|q|^2}\delta(|q|-|k|)
|
|
\end{equation}
|
|
where $\delta$ is the Dirac-delta function distribution (compare this with\-~(\ref{2pt_addv})).
|
|
However, the $\delta$ function causes all sorts of problems with the Chebyshev polynomial exansion and the quadratures.
|
|
\bigskip
|
|
|
|
\point
|
|
Instead, we approximate $\hat C_2$ by convolving it with a normalized Gaussian: let
|
|
\begin{equation}
|
|
\Gamma_L(|k|):=\left(\frac{L}{2\pi}\right)^{\frac32}e^{-\frac L2k^2}
|
|
\label{anyeq_gaussian}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\hat{\mathfrak C}_2(|q|)
|
|
:=
|
|
\int dp\ \hat C_2(|q-p|)\Gamma_L(|p|)
|
|
=\int dk\ \int dp\ \frac\rho{2\pi|q-p|^2}\delta(|q-p|-|k|)\frac{\delta e}{\delta\hat v(k)}\Gamma_L(|p|)
|
|
\end{equation}
|
|
which by lemma\-~\ref{lemma:bipolar} is
|
|
\begin{equation}
|
|
\hat{\mathfrak C}_2(|q|)
|
|
=\int dk\
|
|
\frac\rho{|q|}
|
|
\frac{\delta e}{\delta\hat v(k)}
|
|
\int_0^\infty dt\ \int_{||q|-t|}^{|q|+t}ds\ s\frac{\delta(t-|k|)}t\Gamma_L(s)
|
|
\end{equation}
|
|
that is
|
|
\begin{equation}
|
|
\hat{\mathfrak C}_2(|q|)
|
|
=\int dk\
|
|
\frac{\delta e}{\delta\hat v(k)}
|
|
\frac{\rho}{|q||k|}
|
|
\int_{||q|-|k||}^{|q|+|k|}ds\ s\Gamma_L(s)
|
|
\end{equation}
|
|
which is the directional derivative of $e$ with respect to $\hat v$ in the direction of $2\rho g$ with
|
|
\begin{equation}
|
|
g(|k|):=\frac1{2|q||k|}\int_{||q|-|k||}^{|q|+|k|}ds\ s\Gamma_L(s)
|
|
=
|
|
\frac1{2|k|rL}(\Gamma_L(|k|-r)-\Gamma_L(|k|+r))
|
|
.
|
|
\label{anyeq_2pt_fourier_g}
|
|
\end{equation}
|
|
Note that
|
|
\begin{equation}
|
|
g(0):=\Gamma_L(|q|)
|
|
.
|
|
\end{equation}
|
|
To compute this derivative, we replace $\hat v$ with
|
|
\begin{equation}
|
|
\hat v+\lambda g(|k|)
|
|
\end{equation}
|
|
so, denoting the solution of the modified equation by $u_\lambda$, for $q\neq 0$,
|
|
\begin{equation}
|
|
\hat{\mathfrak C}_2(|q|)=2\rho\partial_\lambda e_\lambda|_{\lambda=0}
|
|
=\rho^2\left(
|
|
-\int\frac{dk}{(2\pi)^3}\ g(|k|)\hat u(|k|)
|
|
-\int\frac{dk}{(2\pi)^3}\ \hat v(|k|)\partial_\lambda\hat u_\lambda(|k|)|_{\lambda=0}
|
|
\right)
|
|
.
|
|
\end{equation}
|
|
To compute $\partial_\lambda\hat u_\lambda|_{\lambda=0}$, we differentiate $\Xi(\mathbb U,\lambda)=0$:
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb U|_{\lambda=0}=-(D\Xi)^{-1}\partial_\lambda\Xi|_{\lambda=0}
|
|
.
|
|
\end{equation}
|
|
The computation of $\partial_\lambda\Xi|_{\lambda=0}$ is identical to\-~(\ref{2pt_dXi}), but taking
|
|
\begin{equation}
|
|
\mathfrak g_{l,j}=g(|k_{l,j}|)
|
|
.
|
|
\end{equation}
|
|
with $g$ defined in\-~(\ref{anyeq_2pt_fourier_g}).
|
|
\bigskip
|
|
|
|
\point
|
|
To compute the maximum of $\hat C_2$, we proceed as for $C_2$, see\-~(\ref{anyeq_newton_2pt})-(\ref{anyeq_dx_2pt}).
|
|
The only difference is that the algorithm is initialized with $|k_0|=\rho^{\frac13}$\refname{param:anyeq_k0}{{\tt k0}}.
|
|
|
|
\subsubsubsection{Correlation function of uncondensed particles (spherical average)}
|
|
To compute the correlation function among uncondensed particles, denoted by $\gamma_2(|\xi|)$, we solve the modified {\tt anyeq} (see\-~\cite{Ja23}):
|
|
\begin{equation}
|
|
-\Delta u_\mu
|
|
=
|
|
(1-u_\mu)v-2\rho K+\rho^2 L
|
|
-\frac{\rho\mu}{2\pi|\xi|^2} u(|\xi|)\delta(|\xi|-|x|)
|
|
\end{equation}
|
|
where $K$ and $L$ are defined as in\-~(\ref{anyeq_K})-(\ref{anyeq_e}) in which $u$ is replaced with $u_\mu$.
|
|
In Fourier space,
|
|
\begin{equation}
|
|
k^2 \hat u_\mu
|
|
=
|
|
\hat S-2\rho\hat K+\rho^2\hat L
|
|
-2\rho\mu u(|\xi|)\frac{\sin(|k||\xi|)}{|k||\xi|}
|
|
.
|
|
\end{equation}
|
|
The uncondensed correlation function is then
|
|
\begin{equation}
|
|
\gamma_2(|\xi|)=\partial_\mu e|_{\mu=0}
|
|
=-\frac\rho2\int dx\ v(x)\partial_\mu u_\mu(x)|_{\mu=0}
|
|
.
|
|
\end{equation}
|
|
To compute the energy in the presence of the parameter $\mu$, we proceed in the same way as for $\mu=0$, the only difference being that the term
|
|
\begin{equation}
|
|
\mu g(|k|):=
|
|
-\mu 2\rho u(|\xi|)\frac{\sin(|k||\xi|)}{|k||\xi|}
|
|
\end{equation}
|
|
should formally be added to the right side of\-~(\ref{Omega}).
|
|
In other words, we consider $\mathbb U_{j,l}=u_\mu(|k_{j,l}|)$ and define $\Xi(\mathbb U,\mu)$ in the same way as in\-~(\ref{root_anyeq}), except that $\mathbb Y_{l,j}$ should be replaced by
|
|
\begin{equation}
|
|
\mathbb Y_{l,j}=
|
|
\frac1{\mathbb L_{l,j}}\left(\mathbb S_{l,j}-\mathbb L_{l,j}+\frac12\mathbb J_{l,j}+\mathbb G_{l,j}+\mu g(k_{l,j})\right)
|
|
.
|
|
\end{equation}
|
|
We then solve
|
|
\begin{equation}
|
|
\Xi(\mathbb U_\mu,\mu)=0
|
|
\end{equation}
|
|
By differentiating this identity with respect to $\mu$, we find $\partial_\mu u_\mu$:
|
|
\begin{equation}
|
|
\partial_\mu \mathbb U|_{\mu=0}=-(D\Xi)^{-1}\partial_\mu\Xi|_{\mu=0}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\partial_\mu\Xi|_{\mu=0}=
|
|
-\frac{\partial_\mu\mathbb Y_{l,j}}{2(\mathbb X_{l,j}+1)}
|
|
\Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
-\frac{(1+\mathbb Y_{l,j})\partial_\mu\mathbb Y_{l,j}}{2(\mathbb X_{l,j}+1)^3}
|
|
\partial\Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
,\quad
|
|
\partial_\mu\mathbb Y_{l,j}=\frac{g(k_{l,j})}{\mathbb L_{l,j}}
|
|
.
|
|
\end{equation}
|
|
We then approximate
|
|
\begin{equation}
|
|
\gamma_2(\xi)\approx
|
|
-\frac12\left<\mathfrak v\partial_\mu\mathbb U\right>
|
|
\end{equation}
|
|
(see\-~(\ref{avg})).
|
|
\bigskip
|
|
|
|
\indent
|
|
In order to avoid numerical oscillations due to the $\sin$ function, we will use a Hann window (see appendix\-~\ref{appendix:hann})
|
|
\begin{equation}
|
|
H_L(k):=\mathds 1_{|k|<\frac L2}\cos^2({\textstyle\frac{\pi|k|}{L}})
|
|
.
|
|
\label{hann}
|
|
\end{equation}
|
|
The parameter $L$ is set using \refname{param:anyeq_window_L}{{\tt window\_L}}.
|
|
The computation is changed only in that $g$ is changed to $H_L(k)g(k)$.
|
|
|
|
\subsubsubsection{Momentum distribution}
|
|
To compute the momentum distribution (see\-~\cite{CHe21}), we add a parameter $\lambda$ to {\tt anyeq}:
|
|
\begin{equation}
|
|
-\Delta u_\lambda(|x|)
|
|
=
|
|
(1-u_\lambda(|x|))v(|x|)-2\rho K(|x|)+\rho^2 L(|x|)
|
|
-2\lambda \hat u_0(q)\cos(q\cdot x)
|
|
\end{equation}
|
|
($\hat u_0\equiv\hat u_\lambda|_{\lambda=0}$).
|
|
The momentum distribution is then
|
|
\begin{equation}
|
|
\mathcal M(q)=\partial_\lambda e|_{\lambda=0}
|
|
=-\frac\rho2\int\frac{dk}{(2\pi)^3}\ \hat v(k)\partial_\lambda\hat u_\lambda(k)|_{\lambda=0}
|
|
.
|
|
\end{equation}
|
|
Note that the Fourier transform of $2\lambda\hat u_0(q)\cos(q\cdot x)$ is
|
|
\begin{equation}
|
|
-(2\pi)^3\lambda\hat u_0(q)(\delta(q+k)+\delta(q-k))
|
|
.
|
|
\label{fouriermomentum}
|
|
\end{equation}
|
|
The presence of delta functions does not play well with the Chebyshev polynomial expansion and the quadratures.
|
|
\bigskip
|
|
|
|
\point
|
|
We will consider two different ways of getting around this.
|
|
\bigskip
|
|
|
|
\subpoint
|
|
One idea is to compute a regularization of $\mathcal M(q)$ by convolving it with a peaked spherically symmetric function.
|
|
Let $\Gamma_L$ denote the Gaussian with variance $1/\sqrt L$:
|
|
\begin{equation}
|
|
\Gamma_L(|k|):=\left(\frac{L}{2\pi}\right)^{\frac32}e^{-\frac L2k^2}
|
|
.
|
|
\label{anyeq_gaussian_momt}
|
|
\end{equation}
|
|
In fact, we will scale $L$ with $k$, and set $L$ to
|
|
\begin{equation}
|
|
L=\sqrt{\mathrm{\refname{param:anyeq_momentum_window_L}{{\tt window\_L}}}}/k^2
|
|
.
|
|
\end{equation}
|
|
To compute
|
|
\begin{equation}
|
|
\mathfrak M(q):=\mathcal M\ast\Gamma_L(q)
|
|
\end{equation}
|
|
we solve the equation
|
|
\begin{equation}
|
|
-\Delta u_\lambda(|x|)
|
|
=
|
|
(1-u_\lambda(|x|))v(|x|)-2\rho K(|x|)+\rho^2 L(|x|)
|
|
-2\lambda\int dk\ \hat u_0(k)\cos(k\cdot x)\Gamma_L(q-k)
|
|
.
|
|
\end{equation}
|
|
Note that the Fourier transform of
|
|
\begin{equation}
|
|
-2\lambda\int dk\ \hat u_0(k)\cos(k\cdot x)\Gamma_L(q-k)
|
|
\end{equation}
|
|
is
|
|
\begin{equation}
|
|
-(2\pi)^3\lambda\hat u_0(q)(\Gamma_L(k+q)+\Gamma_L(k-q))
|
|
.
|
|
\end{equation}
|
|
Since the ground state is unique, $\mathcal M$ is spherically symmetric.
|
|
The term $\Gamma_L(k\pm q)$ is not, so we take its spherical average (which will not change the final result): by lemma\-~\ref{lemma:bipolar},
|
|
\begin{equation}
|
|
-\frac1{4\pi r^2}\int dq\ \delta(|q|-r)(2\pi)^3\lambda\hat u_0(q)(\Gamma_L(k+q)+\Gamma_L(k-q))
|
|
=
|
|
-\frac{(2\pi)^3}{|k|r}\lambda\hat u_0(r)\int_{||k|-r|}^{|k|+r} ds\ s\Gamma_L(s)
|
|
.
|
|
\end{equation}
|
|
In this setup, the approximation of the delta function is thus
|
|
\begin{equation}
|
|
\tilde\delta(|k|,r):=
|
|
\frac1{2|k|r}\int_{||k|-r|}^{|k|+r} ds\ s\Gamma_L(s)
|
|
=
|
|
\frac{1}{2|k|rL}(\Gamma_L(|k|-r)-\Gamma_L(|k|+r))
|
|
.
|
|
\end{equation}
|
|
To choose this method, set \refname{param:anyeq_window_L}{{\tt window\_L}} to a finite value.
|
|
\bigskip
|
|
|
|
\subpoint
|
|
Another approach is to contruct a discrete analog of the delta-functions in\-~(\ref{fouriermomentum}).
|
|
The starting point we take is, for $q\neq0$,
|
|
\begin{equation}
|
|
\int dk\ f(|k|)\delta(k-q)
|
|
\equiv\frac1{4\pi|q|^2}\int dk\ f(|k|)\delta(|k|-|q|)
|
|
=f(|q|)
|
|
\end{equation}
|
|
so, when approximating the integral according to\-~(\ref{anyeq_integration}), we find
|
|
\begin{equation}
|
|
\frac{\pi}{8|q|^2}\sum_{l=0}^{J-1}
|
|
(\tau_{l+1}-\tau_l)
|
|
\sum_{j=1}^Nw_j
|
|
\cos({\textstyle\frac{\pi x_j}2})(1+\vartheta_l(x_j))^2\vartheta_l^2(x_j)f(\vartheta_l(x_j))\tilde\delta(\vartheta_l(x_j),|q|)
|
|
=
|
|
f(|q|)
|
|
\end{equation}
|
|
where $\tilde\delta$ is the approximation of the delta-function.
|
|
Since
|
|
\begin{equation}
|
|
\vartheta_l(x_j)\equiv k_{l,j}
|
|
\end{equation}
|
|
(see\-~(\ref{chebyshevvars})), we define the approximation of the delta function as
|
|
\begin{equation}
|
|
\tilde\delta(k_{l,j},k_{l',i})
|
|
:=
|
|
\delta_{l,l'}\delta_{j,i}
|
|
\frac{8}{\pi}
|
|
\left(
|
|
(\tau_{l+1}-\tau_l)
|
|
w_j
|
|
\cos({\textstyle\frac{\pi x_j}2})
|
|
(1+k_{l,j})^2
|
|
\right)^{-1}
|
|
.
|
|
\end{equation}
|
|
To choose this method, set \refname{param:anyeq_window_L}{{\tt window\_L}} to {\tt Inf}.
|
|
{\color{red}This method seems to yield some fairly poor results!}
|
|
\bigskip
|
|
|
|
\point
|
|
To compute the momentum distribution at $q$, we define $\Xi(\mathbb U,\lambda)$ by formally adding
|
|
\begin{equation}
|
|
-2(2\pi)^3\lambda \hat u_0(|q|)\tilde\delta(k_{l,j},|q|)
|
|
\end{equation}
|
|
to $\mathbb G_{l,j}$, which corresponds to replacing $\mathbb Y$ with
|
|
\begin{equation}
|
|
\mathbb Y_{l,j}=
|
|
\frac1{\mathbb L_{l,j}}\left(\mathbb S_{l,j}-\mathbb L_{l,j}+\frac12\mathbb J_{l,j}+\mathbb G_{l,j}
|
|
-2(2\pi)^3\lambda \hat u(|q|)\tilde\delta(k_{l,j},|q|)
|
|
\right)
|
|
.
|
|
\end{equation}
|
|
Then we solve $\Xi(\mathbb U,\lambda)=0$, and differentiate:
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb U|_{\lambda=0}=-(D\Xi)^{-1}\partial_\lambda\Xi|_{\lambda=0}
|
|
.
|
|
\end{equation}
|
|
Finally,
|
|
\begin{equation}
|
|
\partial_\lambda\Xi_{l,j}|_{\lambda=0}
|
|
=
|
|
-\partial_\lambda\mathbb Y_{l,j}|_{\lambda=0}\left(
|
|
\frac1{2(\mathbb X_{l,j}+1)}
|
|
\Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
+\frac{(1+\mathbb Y_{l,j})}{2(\mathbb X_{l,j}+1)^3}
|
|
\partial\Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right)
|
|
\right)
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\partial_\lambda\mathbb Y_{l,j}|_{\lambda=0}=
|
|
-\frac{2(2\pi)^3}{\mathbb L_{l,j}} \hat u(|q|)\tilde\delta(k_{l,j},|q|)
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subsubsubsection{Compressibility}
|
|
The compressibility is defined as
|
|
\begin{equation}
|
|
\chi:=\frac1{\rho^2\partial_\rho^2(\rho e)}
|
|
=\frac1{\partial_{\log\rho}^2(\rho e)-\partial_{\log\rho}(\rho e)}
|
|
.
|
|
\end{equation}
|
|
We approximate these derivatives by finite differences:
|
|
\begin{equation}
|
|
\partial_{\log\rho}^2 f(\rho)
|
|
\approx
|
|
\frac{f(\rho_{j+1})+f(\rho_{j-1})-2f(\rho_j)}{(\log\rho_{j+2}-\log\rho_{j+1})(\log\rho_{j+1}-\log\rho_j)}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\partial_{\log\rho} f(\rho)
|
|
\approx
|
|
\frac12\left(
|
|
\frac{f(\rho_{j+1})-f(\rho_j)}{\log\rho_{j+1}-\log\rho_{j}}
|
|
+
|
|
\frac{f(\rho_{j})-f(\rho_{j-1})}{\log\rho_{j}-\log\rho_{j-1}}
|
|
\right)
|
|
.
|
|
\end{equation}
|
|
|
|
\subsection{{\tt simpleq-Kv}}\label{sec:simpleq-Kv}
|
|
\indent
|
|
The method is used to compute observables for the simple equation
|
|
\begin{equation}
|
|
-\Delta u
|
|
=
|
|
v(1-u)-4eu+2e\rho u\ast u
|
|
,\quad
|
|
e=\frac\rho2\int dx\ (1-u(|x|))v(|x|)
|
|
.
|
|
\label{simpleq}
|
|
\end{equation}
|
|
One can show\-~\cite[Theorem\-~1.6]{CJL21} that the condensate fraction is
|
|
\begin{equation}
|
|
\eta=\frac{\rho\int v(x)\mathfrak Ku(x)\ dx}{1-\rho\int v(x)\mathfrak K(2u(x)-\rho u\ast u(x))\ dx}
|
|
\label{simpleq-Kv_eta}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}\label{bg6}
|
|
\mathfrak K = (-\Delta + v + 4e(1 - \rho u\ast))^{-1}
|
|
.
|
|
\label{Kv}
|
|
\end{equation}
|
|
Similarly, the two-point correlation function is\-~\cite[(45)]{CHe21}
|
|
\begin{equation}
|
|
C_2=
|
|
\rho^2(1-u)+
|
|
\rho^2\frac{\mathfrak Kv(1-u)-2\rho u\ast \mathfrak Kv+\rho^2u\ast u\ast \mathfrak Kv}{1-\rho\int dx\ v(x)\mathfrak K(2u(x)-\rho u\ast u(x))}
|
|
.
|
|
\label{simpleq-Kv_C2}
|
|
\end{equation}
|
|
Thus, using the fact that $\mathfrak K$ is self-adjoint, we can compute these observables of the simple equation directly from the knowledge of $\mathfrak Kv$.
|
|
\bigskip
|
|
|
|
\subsubsection{Usage}
|
|
\indent
|
|
The computation uses the same approximation scheme as \refname{sec:anyeq}{{\tt anyeq}}, as well as using the solution of \refname{sec:anyeq}{{\tt anyeq}}.
|
|
As such, it takes a similar list of parameters:
|
|
\refname{param:anyeq_rho}{{\tt rho}},
|
|
\refname{param:anyeq_tolerance}{{\tt tolerance}},
|
|
\refname{param:anyeq_maxiter}{{\tt maxiter}},
|
|
\refname{param:anyeq_P}{{\tt P}},
|
|
\refname{param:anyeq_N}{{\tt N}},
|
|
\refname{param:anyeq_J}{{\tt J}},
|
|
\refname{param:anyeq_minlrho_init}{{\tt minlrho\_init}},
|
|
\refname{param:anyeq_nlrho_init}{{\tt nlrho\_init}}.
|
|
\bigskip
|
|
|
|
The available {\tt commands} are the following.
|
|
\bigskip
|
|
|
|
\begin{itemize}
|
|
\item\makelink{command:simpleq-Kv_Kv}{}
|
|
{\tt Kv}: compute $\mathfrak Kv$ as a function of $|x|$.\par
|
|
\underline{Extra parameters}:
|
|
\begin{itemize}
|
|
\item\makelink{param:simpleq-Kv_xmin}{}
|
|
{\tt xmin} ({\tt Float64}, default: 0): minimum of the range of $|x|$ to be printed.
|
|
|
|
\item\makelink{param:simpleq-Kv_xmax}{}
|
|
{\tt xmax} ({\tt Float64}, default: 100): maximum of the range of $|x|$ to be printed.
|
|
|
|
\item\makelink{param:simpleq-Kv_nx}{}
|
|
{\tt nx} ({\tt Int64}, default: 100): number of points to print (linearly spaced).
|
|
|
|
\end{itemize}
|
|
\underline{Output} (one line for each value of $|x|$): [$|x|$] [$\mathfrak K v$]
|
|
|
|
\item\makelink{command:simpleq-Kv_condensate-fraction}{}
|
|
{\tt condensate-fraction}: compute the uncondensed fraction as a function of $\rho$\par
|
|
\underline{Extra parameters}:
|
|
\begin{itemize}
|
|
\item\makelink{param:simpleq-Kv_minlrho}{}
|
|
{\tt minlrho} ({\tt Float64}, default: $10^{-6}$): minimal value for $\log_{10}\rho$.
|
|
|
|
\item\makelink{param:simpleq-Kv_maxlrho}{}
|
|
{\tt maxlrho} ({\tt Float64}, default: $10^{2}$): maximal value for $\log_{10}\rho$.
|
|
|
|
\item\makelink{param:simpleq-Kv_nlrho}{}
|
|
{\tt nlrho} ({\tt Int64}, default: $100$): number of values for $\rho$ (spaced logarithmically).
|
|
|
|
\item\makelink{param:simpleq-Kv_minrho}{}
|
|
{\tt minrho} ({\tt Float64}, default: $10^{-6}$): minimal value for $\rho$.
|
|
|
|
\item\makelink{param:simpleq-Kv_maxrho}{}
|
|
{\tt maxrho} ({\tt Float64}, default: $10^{2}$): maximal value for $\rho$.
|
|
|
|
\item\makelink{param:simpleq-Kv_nrho}{}
|
|
{\tt nrho} ({\tt Int64}, default: $0$): number of values for $\rho$ (spaced linearly). If {\tt nrho} is $\neq0$, then the linear spacing will be used, and \refname{param:simpleq-Kv_minlrho}{{\tt minlrho}}, \refname{param:simpleq-Kv_maxlrho}{{\tt maxlrho}}, \refname{param:simpleq-Kv_nlrho}{{\tt nlrho}} will be ignored. Otherwise, the logarithmic spacing will be used and \refname{param:simpleq-Kv_minrho}{{\tt minrho}}, \refname{param:simpleq-Kv_maxrho}{{\tt maxrho}} will be ignored.
|
|
|
|
\item\makelink{param:simpleq-Kv_rhos}{}
|
|
{\tt rhos} ({\tt Array\{Float64\}}): list of values for $\rho$, specified as a `{\tt,}' separated list.
|
|
This parameter takes precedence over \refname{param:simpleq-Kv_minlrho}{{\tt minlrho}}, \refname{param:simpleq-Kv_maxlrho}{{\tt maxlrho}}, \refname{param:simpleq-Kv_nlrho}{{\tt nlrho}}, \refname{param:simpleq-Kv_minrho}{{\tt minrho}}, \refname{param:simpleq-Kv_maxrho}{{\tt maxrho}}, \refname{param:simpleq-Kv_nrho}{{\tt nrho}}.
|
|
\end{itemize}
|
|
\underline{Output} (one line for each value of $\rho$): [$\rho$] [$\eta$]
|
|
|
|
\item\makelink{command:simpleq-Kv_2pt}{}
|
|
{\tt 2pt}: compute the two-point correlation as a function of $|x|$.\par
|
|
\underline{Extra parameters}: same as \refname{command:simpleq-Kv_Kv}{{\tt Kv}}.\par
|
|
\underline{Output} (one line for each value of $|x|$): [$|x|$] [$C_2$]
|
|
\end{itemize}
|
|
|
|
\subsubsection{Description}
|
|
\indent
|
|
In Fourier space\-~(\ref{anyeq_fourier}),
|
|
\begin{equation}
|
|
\hat{\mathfrak K}^{-1}f
|
|
:=
|
|
\int dx\ e^{ikx}
|
|
(-\Delta+4e(1-\rho u\ast)+v)f
|
|
=(k^2+4e(1-\rho\hat u(|k|))+\hat v\hat\ast)\hat f
|
|
\end{equation}
|
|
where $\hat\ast$ is defined in\-~(\ref{anyeq_hatast}).
|
|
We follow the same approximation scheme as \refname{sec:anyeq}{{\tt anyeq}}:
|
|
\begin{equation}
|
|
\hat{\mathfrak K}^{-1}f(k_{l,j})
|
|
\approx(k_{l,j}^2+4e(1-\mathbb U_{l,j}))\mathfrak f_{l,j}
|
|
+
|
|
(\mathfrak v\odot\mathfrak f)_{l,j}
|
|
\end{equation}
|
|
with $\mathfrak f_{l,j}:=f(k_{l,j})$, $\mathbb U_{l,j}:=\rho u(|k_{l,j}|)$, $\odot$ is defined in\-~(\ref{odot}), and $k_{l,j}$ is defined in\-~(\ref{klj}).
|
|
Therefore, we approximate the operator $\hat{\mathfrak K}^{-1}$ by a matrix:
|
|
\begin{equation}
|
|
\hat{\mathfrak K}^{-1}f(k_{l,j})\approx
|
|
\sum_{l'=0}^{J-1}\sum_{i=1}^N M_{l,j;l',i}\mathfrak f_{l',i}
|
|
\end{equation}
|
|
with, by\-~(\ref{odot}) and\-~(\ref{frakF}),
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
M_{l,j;l',i}:=
|
|
\delta_{l',l}\delta_{j,i}(k_{l,j}^2+4e(1-\mathbb U_{l,j}))
|
|
\\\hfill
|
|
+
|
|
\frac1{4\pi^2|k_{l,j}|}\sum_{n,m=0}^P\sum_{l''=0}^{J-1}\mathfrak F_{l'',n}^{(\nu_v)}(\mathfrak v)
|
|
A_{l'',n;l',m}^{(\nu_v,\nu_f)}(|k_{l,j}|)
|
|
\frac{\frac{2-\delta_{m,0}}2w_i\cos(\frac{m\pi(1+x_i)}2)}{(1-\frac{\tau_{l'+1}-\tau_{l'}}2\sin(\frac{\pi x_i}2)+\frac{\tau_{l'+1}+\tau_{l'}}2)^{\nu_f}}
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
Defining $\mathds 1_{l',i}$ as the vector whose only non-vanishing component is that indexed by $l',i$ which is equal to 1, we can rewrite
|
|
\begin{equation}
|
|
M_{l,j;l',i}:=
|
|
\delta_{l',l}\delta_{j,i}(k_{l,j}^2+4e(1-\mathbb U_{l,j}))
|
|
+
|
|
(\mathfrak v\odot\mathds 1_{l',i})_{l,j}
|
|
.
|
|
\end{equation}
|
|
Thus
|
|
\begin{equation}
|
|
\mathfrak Kv\approx M^{-1}\mathfrak v
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\indent
|
|
To compute\-~(\ref{simpleq-Kv_eta}), we write
|
|
\begin{equation}
|
|
\eta=\frac{\rho\int\frac{dk}{(2\pi)^3} \hat u(k)\hat{\mathfrak K}\hat v(k)}{1-\rho\int \frac{dk}{(2\pi)^3)}(2\hat u(k)-\rho\hat u^2(k))\hat{\mathfrak K}\hat v(k)}
|
|
\end{equation}
|
|
which we approximate as
|
|
\begin{equation}
|
|
\eta\approx\frac{\left<\mathbb UM^{-1}\mathfrak v\right>}{1-\left<(2\mathbb U-\mathbb U^2)M^{-1}\mathfrak v\right>}
|
|
\end{equation}
|
|
where $\left<\cdot\right>$ is defined in\-~(\ref{avg}).
|
|
We can thus compute $\eta$ using the solution $\mathbb U$ computed by \refname{sec:anyeq}{{\tt anyeq}}.
|
|
\bigskip
|
|
|
|
\indent
|
|
To compute\-~(\ref{simpleq-Kv_C2}), we write
|
|
\begin{equation}
|
|
C_2=
|
|
\rho^2(1-u(x))+
|
|
\rho^2\frac{\mathfrak Kv(x)(1-u(x))+\int\frac{dk}{(2\pi)^3}(-2\rho\hat u\hat{\mathfrak K}\hat v+\rho^2\hat u^2\hat{\mathfrak K}\hat v)}{1-\rho\int\frac{dk}{(2\pi)^3}\ (2\hat u(k)-\rho \hat u^2(k))\hat{\mathfrak K}\hat v(k)}
|
|
\end{equation}
|
|
which we approximate as
|
|
\begin{equation}
|
|
C_2\approx
|
|
\rho^2-\rho\left<e^{-ik_{l,j}x}\mathbb U\right>+
|
|
\rho^2\frac{\left<M^{-1}\mathfrak v\right>(1-\rho^{-1}\left<e^{-ik_{l,j}x}\mathbb U\right>)+\left<(-2\mathbb U M^{-1}\mathfrak v+\mathbb UM^{-1}\mathfrak v)\right>}{1-\left<(2\mathbb U-\mathbb U^2)M^{-1}\mathfrak v\right>}
|
|
.
|
|
\end{equation}
|
|
We can thus compute $C_2$ using the solution $\mathbb U$ computed by \refname{sec:anyeq}{{\tt anyeq}}.
|
|
|
|
\subsection{\tt simpleq-hardcore}\label{sec:simpleq-hardcore}
|
|
\indent
|
|
This method is used to solve the Simple equation with a hardcore potential:
|
|
\begin{equation}
|
|
\left\{\begin{array}{>\displaystyle ll}
|
|
(-\Delta+4e) u(x)=2e\rho u\ast u(x)
|
|
&\mathrm{for\ }|x|>1\\
|
|
u(x)=1
|
|
&\mathrm{for\ }|x|\leqslant1
|
|
\end{array}\right.
|
|
\label{linearhc}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
e=-\frac{4\pi\rho\partial u|_{|x|\searrow 1}}{2(1-\frac83\pi\rho+\rho^2\int_{|x|<1}dx\ u\ast u(x))}
|
|
.
|
|
\label{hardcore_energy}
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
This equation is solved in $x$-space, and as such is very different from \refname{sec:easyeq}{{\tt easyeq}}, and significantly longer to run.
|
|
\bigskip
|
|
|
|
\subsubsection{Usage}
|
|
\indent
|
|
Unless otherwise noted, this method takes the following parameters (specified via the {\tt [-p params]} flag, as a `{\tt;}' separated list of entries).
|
|
\begin{itemize}
|
|
\item\makelink{param:simpleq-hardcore_rho}{}
|
|
{\tt rho} ({\tt Float64}, default: $10^{-6}$): density $\rho$.
|
|
|
|
\item\makelink{param:simpleq-hardcore_tolerance}{}
|
|
{\tt tolerance} ({\tt Float64}, default: $10^{-11}$): maximal size of final step in Newton iteration.
|
|
|
|
\item\makelink{param:simpleq-hardcore_maxiter}{}
|
|
{\tt maxiter} ({\tt Int64}, default: 21): maximal number of iterations of the Newton algorithm before giving up.
|
|
|
|
\item\makelink{param:simpleq-hardcore_P}{}
|
|
{\tt P} ({\tt Int64}, default: 11): order of all Chebyshev polynomial expansions (denoted by $P$ below).
|
|
|
|
\item\makelink{param:simpleq-hardcore_N}{}
|
|
{\tt N} ({\tt Int64}, default: 12): order of all Gauss quadratures (denoted by $N$ below).
|
|
|
|
\item\makelink{param:simpleq-hardcore_J}{}
|
|
{\tt J} ({\tt Int64}, default: 10): number of splines (denoted by $J$ below).
|
|
\end{itemize}
|
|
\bigskip
|
|
|
|
The available {\tt commands} are the following.
|
|
|
|
\begin{itemize}
|
|
\item\makelink{command:simpleq-hardcore_energy_rho}{}
|
|
{\tt energy\_rho}: compute the energy $e$ as a function of $\rho$.\par
|
|
\underline{Disabled parameters}: \refname{param:simpleq-hardcore_rho}{{\tt rho}}.\par
|
|
\underline{Extra parameters}:
|
|
\begin{itemize}
|
|
\item\makelink{param:simpleq-hardcore_minlrho}{}
|
|
{\tt minlrho} ({\tt Float64}, default: $10^{-6}$): minimal value for $\log_{10}\rho$.
|
|
|
|
\item\makelink{param:simpleq-hardcore_maxlrho}{}
|
|
{\tt maxlrho} ({\tt Float64}, default: $10^{2}$): maximal value for $\log_{10}\rho$.
|
|
|
|
\item\makelink{param:simpleq-hardcore_nlrho}{}
|
|
{\tt nlrho} ({\tt Int64}, default: $100$): number of values for $\rho$ (spaced logarithmically).
|
|
|
|
\item\makelink{param:simpleq-hardcore_minrho}{}
|
|
{\tt minrho} ({\tt Float64}, default: $10^{-6}$): minimal value for $\rho$.
|
|
|
|
\item\makelink{param:simpleq-hardcore_maxrho}{}
|
|
{\tt maxrho} ({\tt Float64}, default: $10^{2}$): maximal value for $\rho$.
|
|
|
|
\item\makelink{param:simpleq-hardcore_nrho}{}
|
|
{\tt nrho} ({\tt Int64}, default: $0$): number of values for $\rho$ (spaced linearly). If {\tt nrho} is $\neq0$, then the linear spacing will be used, and \refname{param:simpleq-hardcore_minlrho}{{\tt minlrho}}, \refname{param:simpleq-hardcore_maxlrho}{{\tt maxlrho}}, \refname{param:simpleq-hardcore_nlrho}{{\tt nlrho}} will be ignored. Otherwise, the logarithmic spacing will be used and \refname{param:simpleq-hardcore_minrho}{{\tt minrho}}, \refname{param:simpleq-hardcore_maxrho}{{\tt maxrho}} will be ignored.
|
|
|
|
\item\makelink{param:simpleq-hardcore_rhos}{}
|
|
{\tt rhos} ({\tt Array\{Float64\}}): list of values for $\rho$, specified as a `{\tt,}' separated list.
|
|
This parameter takes precedence over \refname{param:simpleq-hardcore_minlrho}{{\tt minlrho}}, \refname{param:simpleq-hardcore_maxlrho}{{\tt maxlrho}}, \refname{param:simpleq-hardcore_nlrho}{{\tt nlrho}}, \refname{param:simpleq-hardcore_minrho}{{\tt minrho}}, \refname{param:simpleq-hardcore_maxrho}{{\tt maxrho}}, \refname{param:simpleq-hardcore_nrho}{{\tt nrho}}.
|
|
\end{itemize}
|
|
\underline{Output} (one line for each value of $\rho$): [$\rho$] [$e$] [Newton error $\epsilon$].\par
|
|
\underline{Multithread support}: yes, different values of $\rho$ split up among workers.
|
|
|
|
\item\makelink{command:simpleq-hardcore_condensate_fraction_rho}{}
|
|
{\tt condensate\_fraction\_rho}: compute the uncondensed fraction $\eta$ as a function of $\rho$.\par
|
|
\underline{Disabled parameters}: same as \refname{command:simpleq-hardcore_energy_rho}{{\tt energy\_rho}}.\par
|
|
\underline{Extra parameters}: same as \refname{command:simpleq-hardcore_energy_rho}{{\tt energy\_rho}}.\par
|
|
\underline{Output} (one line for each value of $\rho$): [$\rho$] [$\eta$] [Newton error $\epsilon$].\par
|
|
\underline{Multithread support}: yes, different values of $\rho$ split up among workers.
|
|
|
|
\item\makelink{command:simpleq-hardcore_ux}{}
|
|
{\tt ux}: compute $u$ as a function of $|x|$.\par
|
|
\underline{Extra parameters}:
|
|
\begin{itemize}
|
|
\item\makelink{param:simpleq-hardcore_xmin}{}
|
|
{\tt xmin} ({\tt Float64}, default: 0): minimum of the range of $|x|$ to be printed.
|
|
|
|
\item\makelink{param:simpleq-hardcore_xmax}{}
|
|
{\tt xmax} ({\tt Float64}, default: 100): maximum of the range of $|x|$ to be printed.
|
|
|
|
\item\makelink{param:simpleq-hardcore_nx}{}
|
|
{\tt nx} ({\tt Int64}, default: 100): number of points to print (linearly spaced).
|
|
|
|
\end{itemize}
|
|
\underline{Output} (one line for each value of $x$): [$|x|$] [$u(|x|)$]
|
|
|
|
\end{itemize}
|
|
|
|
\subsubsection{Description}
|
|
\indent
|
|
In order to carry out the computation of the solution of\-~(\ref{linearhc}) and compute the condensate fraction at the same time, we will consider the equation with an added parameter $\mu>0$:
|
|
\begin{equation}
|
|
(-\Delta+4\epsilon)u=2e\rho u\ast u
|
|
,\quad
|
|
\epsilon:=e+\frac\mu2
|
|
\label{hardcore_simple}
|
|
\end{equation}
|
|
for $|x|>1$.
|
|
\bigskip
|
|
|
|
\subsubsubsection{Energy}
|
|
To compute the energy $e$ of this equation, with the extra parameter $\mu$, we consider the limit of the soft sphere potential $\lambda\mathds 1_{|x|<1}$ (see\-~(\ref{easyeq}) with $\beta_K=\beta_L=0$):
|
|
\begin{equation}
|
|
(-\Delta+2\mu+4e)u(x)=s_\lambda(x)+2e\rho u\ast u
|
|
,\quad
|
|
s_\lambda(x):=\lambda(1-u(x))\mathds 1_{|x|\leqslant 1}
|
|
,\quad
|
|
\frac{2e}\rho=\int dx\ s_\lambda(x)
|
|
.
|
|
\end{equation}
|
|
Furthermore, since $\partial u$ need not be continuous at $|x|=1$, by integrating $-\Delta u$ over a thin spherical shell of radius 1, we find that, for $|x|\leqslant 1$,
|
|
\begin{equation}
|
|
-\Delta u(x)=-\delta(|x|-1)\partial u|_{|x|\searrow1}
|
|
\end{equation}
|
|
so, formally,
|
|
\begin{equation}
|
|
s_\infty(x)=\mathds 1_{|x|\leqslant 1}(2e(2-\rho u\ast u(x))+2\mu)-\delta(|x|-1)\partial u|_{|x|\searrow1}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\frac{2e}\rho=\int dx\ s_\infty(x)
|
|
=
|
|
2e\left(\frac{8\pi}3-\rho\int_{|x|<1}u\ast u(x)\right)+\frac{8\pi}3\mu-4\pi\partial u|_{|x|\searrow1}
|
|
.
|
|
\end{equation}
|
|
Therefore,
|
|
\begin{equation}
|
|
e=\frac{2\pi\rho(\frac23\mu-\partial u|_{|x|\searrow1})}{1-\frac{8\pi}3\rho+\rho^2\int_{|x|<1}dx\ u\ast u(x)}
|
|
.
|
|
\label{emu}
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subsubsubsection{Integral equation}
|
|
We turn the differential equation in\-~(\ref{hardcore_simple}) into an integral equation.
|
|
Let
|
|
\begin{equation}
|
|
w(|x|):=|x|u(x)
|
|
\end{equation}
|
|
we have, for $r>1$,
|
|
\begin{equation}
|
|
(-\partial_r^2+4\epsilon)w(r)=2e\rho ru\ast u(r)
|
|
.
|
|
\end{equation}
|
|
Furthermore, the bounded solution of
|
|
\begin{equation}
|
|
(-\partial_r^2+\alpha^2)w(r)=F(r)
|
|
,\quad
|
|
w(1)=1
|
|
\end{equation}
|
|
is
|
|
\begin{equation}
|
|
w(r)=e^{-\alpha(r-1)}+\int_0^\infty ds\ \frac{F(s+1)}{2\alpha}\left(e^{-\alpha|(r-1)-s|}-e^{-\alpha((r-1)+s)}\right)
|
|
\label{solw}
|
|
\end{equation}
|
|
so, for $r>1$,
|
|
\begin{equation}
|
|
u(r)=
|
|
\frac 1re^{-2\sqrt\epsilon(r-1)}
|
|
+
|
|
\frac{\rho e}{2r\sqrt\epsilon}\int_0^\infty ds\ (s+1)(u\ast u(s+1))\left(e^{-2\sqrt\epsilon|(r-1)-s|}-e^{-2\sqrt\epsilon((r-1)+s)}\right)
|
|
.
|
|
\end{equation}
|
|
In order to compute the integral more easily, we split it:
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
u(r)=
|
|
\frac 1re^{-2\sqrt\epsilon(r-1)}
|
|
+
|
|
\frac{\rho e}{r\sqrt\epsilon}\int_0^{r-1} ds\ (s+1)(u\ast u(s+1))\sinh(2\sqrt\epsilon s)e^{-2\sqrt\epsilon(r-1)}
|
|
\\[0.5cm]\hfill
|
|
+
|
|
\frac{\rho e}{r\sqrt\epsilon}\sinh(2\sqrt\epsilon(r-1))\int_{r-1}^\infty ds\ (s+1)(u\ast u(s+1))e^{-2\sqrt\epsilon s}
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
We change variables in the last integral:
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
u(r)=
|
|
\frac 1re^{-2\sqrt\epsilon(r-1)}
|
|
+
|
|
\frac{\rho e}{r\sqrt\epsilon}\int_0^{r-1} ds\ (s+1)(u\ast u(s+1))\sinh(2\sqrt\epsilon s)e^{-2\sqrt\epsilon(r-1)}
|
|
\\[0.5cm]\hfill
|
|
+
|
|
\frac{\rho e}{2r\sqrt\epsilon}\left(1-e^{-4\sqrt\epsilon(r-1)}\right)\int_0^\infty d\sigma\ (\sigma+r)(u\ast u(\sigma+r))e^{-2\sqrt\epsilon\sigma}
|
|
.
|
|
\end{largearray}
|
|
\label{uinteq}
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subsubsubsection{The auto-convolution term} We split
|
|
\begin{equation}
|
|
u(r)=\mathds 1_{r>1}u_+(r-1)+\mathds 1_{r\leqslant 1}
|
|
.
|
|
\end{equation}
|
|
in terms of which
|
|
\begin{equation}
|
|
u\ast u=
|
|
\mathds 1_{r\leqslant 1}\ast\mathds 1_{r\leqslant 1}
|
|
+2\mathds 1_{r\leqslant 1}\ast(u_+(r-1)\mathds 1_{r>1})
|
|
+(\mathds 1_{r>1}u_+(r-1))\ast(u_+(r-1)\mathds 1_{r>1})
|
|
\label{u*u_dcmp}
|
|
\end{equation}
|
|
In bipolar coordinates (see lemma\-~\ref{lemma:bipolar}),
|
|
\begin{equation}
|
|
\mathds 1_{r\leqslant 1}\ast\mathds 1_{r\leqslant 1}(r)
|
|
=
|
|
\frac{2\pi}{r}\int_0^1 dt\ t\int_{|r-t|}^{r+t}ds\ s\mathds 1_{s\leqslant 1}
|
|
\label{1*1}
|
|
\end{equation}
|
|
and, if $r\geqslant 0$ and $t\geqslant 0$, then
|
|
\begin{equation}
|
|
\int_{|r-t|}^{r+t}ds\ s\mathds 1_{s\leqslant 1}
|
|
=
|
|
\mathds 1_{r-1\leqslant t\leqslant r+1}\int_{|r-t|}^{\min(1,r+t)}ds\ s
|
|
=
|
|
\mathds 1_{r-1\leqslant t\leqslant r+1}
|
|
\left(
|
|
\mathds 1_{t\leqslant 1-r}
|
|
2rt
|
|
+
|
|
\mathds 1_{t>1-r}
|
|
\frac{1-(r-t)^2}2
|
|
\right)
|
|
.
|
|
\label{int1}
|
|
\end{equation}
|
|
Therefore, if $r\geqslant 1$
|
|
\begin{equation}
|
|
\mathds 1_{r\leqslant 1}\ast\mathds 1_{r\leqslant 1}(r)
|
|
=
|
|
\mathds 1_{r\leqslant 2}
|
|
\frac{2\pi}{r}\int_{r-1}^1 dt\ t
|
|
\frac{1-(r-t)^2}2
|
|
=
|
|
\mathds 1_{r\leqslant 2}\frac\pi{12}(r-2)^2(r+4)
|
|
\label{u*u_1}
|
|
\end{equation}
|
|
and if $r<1$,
|
|
\begin{equation}
|
|
\mathds 1_{r\leqslant 1}\ast\mathds 1_{r\leqslant 1}(r)
|
|
=
|
|
\frac{2\pi}r\int_0^1dt\ t
|
|
\left(
|
|
\mathds 1_{t\leqslant 1-r}
|
|
2rt
|
|
+
|
|
\mathds 1_{t>1-r}
|
|
\frac{1-(r-t)^2}2
|
|
\right)
|
|
\end{equation}
|
|
so
|
|
\begin{equation}
|
|
\mathds 1_{r\leqslant 1}\ast\mathds 1_{r\leqslant 1}(r)
|
|
=\frac43\pi(1-r)^3
|
|
+\frac\pi{12}r(36-48r+17r^2)
|
|
=
|
|
\frac\pi{12}(r-2)^2(r+4)
|
|
.
|
|
\end{equation}
|
|
Thus, (\ref{u*u_1}) holds for all $r$.
|
|
Furthermore,
|
|
\begin{equation}
|
|
2\mathds 1_{r\leqslant 1}\ast(u_+(r-1)\mathds 1_{r>1})
|
|
=
|
|
\frac{4\pi}{r}\int_1^\infty dt\ tu_+(t-1)\int_{|r-t|}^{r+t}ds\ s\mathds 1_{s\leqslant 1}
|
|
\end{equation}
|
|
so, if $r\geqslant 0$ then, by\-~(\ref{int1}),
|
|
\begin{equation}
|
|
2\mathds 1_{r\leqslant 1}\ast(u_+(r-1)\mathds 1_{r>1})(r)
|
|
=
|
|
\frac{2\pi}{r}\int_{\max(1,r-1)}^{r+1} dt\ tu_+(t-1)(1-(r-t)^2)
|
|
.
|
|
\label{u*u_cross}
|
|
\end{equation}
|
|
Finally, if $r\geqslant 0$ then
|
|
\begin{equation}
|
|
(\mathds 1_{r>1}u_+(r-1))\ast(u_+(r-1)\mathds 1_{r>1})(r)
|
|
=
|
|
\frac{2\pi}{r}\int_1^\infty dt\ tu_+(t-1)\int_{\max{(1,|r-t|)}}^{r+t}ds\ su_+(s-1)
|
|
.
|
|
\label{u*u_u}
|
|
\end{equation}
|
|
Thus, by~\-(\ref{u*u_1}), (\ref{u*u_cross}) and\-~(\ref{u*u_u}), for $r\geqslant -1$,
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
u\ast u(1+r)=
|
|
\mathds 1_{r\leqslant 1}\frac\pi{12}(r-1)^2(r+5)
|
|
+
|
|
\frac{2\pi}{r+1}\int_{\max(0,r-1)}^{r+1} dt\ (t+1)u_+(t)(1-(r-t)^2)
|
|
\\[0.5cm]\hfill
|
|
+
|
|
\frac{2\pi}{r+1}\int_0^\infty dt\ (t+1)u_+(t)\int_{\max{(0,|r-t|-1)}}^{r+t+1}ds\ (s+1)u_+(s)
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
We then compactify the integrals by changing variables to $\tau=\frac{1-t}{1+t}$ and $\sigma=\frac{1-s}{1+s}$:
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
u\ast u(1+r)=
|
|
\mathds 1_{r\leqslant 1}\frac\pi{12}(r-1)^2(r+5)
|
|
+
|
|
\frac{8\pi}{r+1}\int_{-\frac r{2+r}}^{\min(1,\frac{2-r}r)} d\tau\ \frac1{(1+\tau)^3}u_+({\textstyle\frac{1-\tau}{1+\tau}})(1-(r-{\textstyle\frac{1-\tau}{1+\tau}})^2)
|
|
\\[0.5cm]\hfill
|
|
+
|
|
\frac{32\pi}{r+1}\int_{-1}^1 d\tau\ \frac1{(1+\tau)^3}u_+({\textstyle\frac{1-\tau}{1+\tau}})\int_{\alpha_-(1+r,\tau)}^{\min(1,\beta_+(r,\tau))}d\sigma\ \frac1{(1+\sigma)^3}u_+({\textstyle\frac{1-\sigma}{1+\sigma}})
|
|
.
|
|
\end{largearray}
|
|
\label{u*u}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\alpha_-(1+r,\tau):=\frac{-r-\frac{1-\tau}{1+\tau}}{2+r+\frac{1-\tau}{1+\tau}}
|
|
,\quad
|
|
\beta_+(r,\tau):=\frac{2-|r-\frac{1-\tau}{1+\tau}|}{|r-\frac{1-\tau}{1+\tau}|}
|
|
\end{equation}
|
|
(note that $\alpha_-$ is the same as defined for fulleq).
|
|
Finally, note that, if $\beta_+<1$, that is, if $|r-\frac{1-\tau}{1+\tau}|>1$, then
|
|
\begin{equation}
|
|
\beta_+(r,\tau)=\alpha_+({\textstyle |r-\frac{1-\tau}{1+\tau}|-1+\frac{1-\tau}{1+\tau}},\tau)
|
|
\end{equation}
|
|
where
|
|
\begin{equation}
|
|
\alpha_+(r,\tau):=\frac{1-|r-\frac{1-\tau}{1+\tau}|}{1+|r-\frac{1-\tau}{1+\tau}|}
|
|
\end{equation}
|
|
is the same as is defined for \refname{sec:anyeq}{{\tt anyeq}} (\ref{anyeq_alpha}).
|
|
\bigskip
|
|
|
|
\subsubsubsection{Chebyshev polynomial expansion} We use the same interpolation as we used in \refname{sec:anyeq}{{\tt anyeq}}: (\ref{a_chebyshev_spline})
|
|
\begin{equation}
|
|
\frac1{(1+\tau)^{\nu_u}}u_+({\textstyle\frac{1-\tau}{1+\tau}})=\sum_{l=0}^{J-1}\mathds 1_{\tau_l\leqslant\tau<\tau_{l+1}}\sum_{n=0}^\infty \mathbf F_{l,n}^{(\nu_u)}(u_+) T_n({\textstyle\frac{2\tau-(\tau_l+\tau_{l+1})}{\tau_{l+1}-\tau_l}})
|
|
\label{a_chebyshev_spline}
|
|
\end{equation}
|
|
($J$ is set by the parameter \refname{param:simpleq-hardcore_J}{{\tt J}})
|
|
with
|
|
\begin{equation}
|
|
\mathbf F_{l,n}^{(\nu_u)}(u_+):=
|
|
\frac{2-\delta_{n,0}}{\pi}
|
|
\int_0^{\pi} d\theta\
|
|
\frac{\cos(n\theta)}{(1+\frac{\tau_{l+1}-\tau_l}2\cos(\theta)+\frac{\tau_{l+1}+\tau_l}2)^{\nu_u}}
|
|
u_+({\textstyle\frac{2-(\tau_{l+1}-\tau_l)\cos(\theta)-(\tau_{l+1}+\tau_\alpha)}{2+(\tau_{l+1}-\tau_l)\cos(\theta)+(\tau_{l+1}+\tau_l)}})
|
|
\label{bfF}
|
|
\end{equation}
|
|
and we take $\nu_u$ to be the decay exponent of $u$, which we will assume is $\nu_u=4$.
|
|
In particular, by\-~(\ref{u*u}),
|
|
\begin{equation}
|
|
u\ast u(1+r)=\mathds 1_{r\leqslant 1}B^{(0)}(r)+\sum_l\sum_n\mathbf F_{l,n}^{(\nu_u)}(u_+)B^{(1)}_{l,n}(r)+\sum_{l,l'}\sum_{n,m}\mathbf F_{l,n}^{(\nu_u)}(u_+)\mathbf F_{l',m}^{(\nu_u)}(u_+)B^{(2)}_{l,n;l',m}(r)
|
|
\label{u*u}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
B^{(0)}(r):=
|
|
\frac\pi{12}(r-1)^2(r+5)
|
|
\end{equation}
|
|
\begin{equation}
|
|
B^{(1)}_{l,n}(r):=
|
|
\mathds 1_{\tau_l<\frac{2-r}r}\mathds 1_{\tau_{l+1}>-\frac r{2+r}}
|
|
\frac{8\pi}{r+1}\int_{\max(\tau_l,-\frac r{2+r})}^{\min(\tau_{l+1},\frac{2-r}r)} d\tau\ \frac1{(1+\tau)^{3-\nu_u}}(1-(r-{\textstyle\frac{1-\tau}{1+\tau}})^2)T_n({\textstyle\frac{2\tau-(\tau_l+\tau_{l+1})}{\tau_{l+1}-\tau_l}})
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
B^{(2)}_{l,n;l',m}(r):=
|
|
\frac{32\pi}{r+1}\int_{\tau_l}^{\tau_{l+1}} d\tau\ \frac1{(1+\tau)^{3-\nu_u}}T_n({\textstyle\frac{2\tau-(\tau_l+\tau_{l+1})}{\tau_{l+1}-\tau_l}})
|
|
\mathds 1_{\tau_{l'}<\alpha_+(|r-\frac{1-\tau}{1+\tau}|-\frac{2\tau}{1+\tau},\tau)}\mathds 1_{\tau_{l'+1}>\alpha_-(1+r,\tau)}
|
|
\cdot\\[0.5cm]\hfill\cdot
|
|
\int_{\max(\tau_{l'},\alpha_-(1+r,\tau))}^{\min(\tau_{l'+1},\alpha_+(|r-\frac{1-\tau}{1+\tau}|-\frac{2\tau}{1+\tau},\tau))}d\sigma\ \frac1{(1+\sigma)^{3-\nu_u}}T_m({\textstyle\frac{2\sigma-(\tau_{l'}+\tau_{l'+1})}{\tau_{l'+1}-\tau_{l'}}})
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
Thus, by\-~(\ref{uinteq}), for $r>0$,
|
|
\begin{equation}
|
|
u_+(r,e,\epsilon)=
|
|
D^{(0)}(r,e,\epsilon)
|
|
+
|
|
\sum_l\sum_n\mathbf F_{l,n}^{(\nu_u)}(u_+)D^{(1)}_{l,n}(r,e,\epsilon)
|
|
+
|
|
\sum_{l,l'}\sum_{n,m}\mathbf F_{l,n}^{(\nu_u)}(u_+)\mathbf F_{l',m}^{(\nu_u)}(u_+)D^{(2)}_{l,n;l',m}(r,e,\epsilon)
|
|
\label{hardcore_u+}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
D^{(0)}(r,e,\epsilon):=
|
|
\frac 1{r+1}e^{-2\sqrt\epsilon r}
|
|
+
|
|
\frac{\rho e}{(r+1)\sqrt\epsilon}\int_0^{\min(1,r)} ds\ (s+1)B^{(0)}(s)\sinh(2\sqrt\epsilon s)e^{-2\sqrt\epsilon r}
|
|
\\[0.5cm]\hfill
|
|
+
|
|
\mathds 1_{r\leqslant 1}
|
|
\frac{\rho e}{2(r+1)\sqrt\epsilon}\left(1-e^{-4\sqrt\epsilon r}\right)\int_0^{1-r} d\sigma\ (\sigma+r+1)B^{(0)}(\sigma+r)e^{-2\sqrt\epsilon\sigma}
|
|
\end{largearray}
|
|
\label{D0}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
D^{(1)}_{l,n}(r,e,\epsilon):=
|
|
\frac{\rho e}{(r+1)\sqrt\epsilon}\int_0^r ds\ (s+1)B^{(1)}_{l,n}(s)\sinh(2\sqrt\epsilon s)e^{-2\sqrt\epsilon r}
|
|
\\[0.5cm]\hfill
|
|
+
|
|
\frac{\rho e}{2(r+1)\sqrt\epsilon}\left(1-e^{-4\sqrt\epsilon r}\right)\int_0^\infty d\sigma\ (\sigma+r+1)B_{l,n}^{(1)}(\sigma+r)e^{-2\sqrt\epsilon\sigma}
|
|
.
|
|
\end{largearray}
|
|
\label{D1}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
D^{(2)}_{l,n;l',m}(r,e,\epsilon):=
|
|
\frac{\rho e}{(r+1)\sqrt\epsilon}\int_0^r ds\ (s+1)B^{(2)}_{l,n;l',m}(s)\sinh(2\sqrt\epsilon s)e^{-2\sqrt\epsilon r}
|
|
\\[0.5cm]\hfill
|
|
+
|
|
\frac{\rho e}{2(r+1)\sqrt\epsilon}\left(1-e^{-4\sqrt\epsilon r}\right)\int_0^\infty d\sigma\ (\sigma+r+1)B_{l,n;l',m}^{(2)}(\sigma+r)e^{-2\sqrt\epsilon\sigma}
|
|
.
|
|
\end{largearray}
|
|
\label{D2}
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subsubsubsection{Energy} We now compute the approximation for the energy, using\-~(\ref{hardcore_energy}).
|
|
\bigskip
|
|
|
|
\point We start with $\partial u|_{|x|\searrow 1}$.
|
|
By\-~(\ref{uinteq}),
|
|
\begin{equation}
|
|
\partial u|_{|x|\searrow 1}
|
|
=
|
|
-(1+2\sqrt\epsilon)
|
|
+
|
|
2\rho e\int_0^\infty d\sigma\ (\sigma+1)(u\ast u(\sigma+1))e^{-2\sqrt\epsilon\sigma}
|
|
\end{equation}
|
|
\bigskip
|
|
which, by\-~(\ref{u*u}), becomes
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\partial u|_{|x|\searrow 1}
|
|
=
|
|
-(1+2\sqrt\epsilon)
|
|
+
|
|
\gamma^{(0)}(e,\epsilon)
|
|
+
|
|
\sum_l\sum_n\mathbf F_{l,n}^{(\nu_u)}(u_+)\gamma^{(1)}_{l,n}(e,\epsilon)
|
|
+\\\hfill+
|
|
\sum_{l,l'}\sum_{n,m}\mathbf F_{l,n}^{(\nu_u)}(u_+)\mathbf F_{l',m}^{(\nu_u)}(u_+)\gamma^{(2)}_{l,n;l',m}(e,\epsilon)
|
|
\end{largearray}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\gamma^{(0)}(e,\epsilon):=
|
|
2\rho e\int_0^1 d\sigma\ (\sigma+1)B^{(0)}(\sigma)e^{-2\sqrt\epsilon\sigma}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\gamma^{(1)}_{l,n}(e,\epsilon):=
|
|
2\rho e\int_0^\infty d\sigma\ (\sigma+1)B_{l,n}^{(1)}(\sigma)e^{-2\sqrt\epsilon\sigma}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\gamma^{(2)}_{l,n;l',m}(e,\epsilon):=
|
|
2\rho e\int_0^\infty d\sigma\ (\sigma+1)B_{l,n;l',m}^{(2)}(\sigma)e^{-2\sqrt\epsilon\sigma}
|
|
.
|
|
\end{equation}
|
|
|
|
\point Let us now turn to $\int_{|x|<1}dx\ u\ast u(x)$.
|
|
We have
|
|
\begin{equation}
|
|
\int_{|x|<1}dx\ u\ast u(x)
|
|
=
|
|
4\pi\int_0^1 dr\ r^2 u\ast u(r)
|
|
\end{equation}
|
|
so, by\-~(\ref{u*u}),
|
|
\begin{equation}
|
|
\int_{|x|<1}dx\ u\ast u(x)
|
|
=
|
|
\bar\gamma^{(0)}(r)
|
|
+
|
|
\sum_l\sum_n\mathbf F_{l,n}^{(\nu_u)}(u_+)\bar\gamma^{(1)}_{l,n}(r)
|
|
+
|
|
\sum_{l,l'}\sum_{n,m}\mathbf F_{l,n}^{(\nu_u)}(u_+)\mathbf F_{l',m}^{(\nu_u)}(u_+)\bar\gamma^{(2)}_{l,n;l',m}(r)
|
|
\label{intusu}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\bar\gamma^{(0)}:=
|
|
4\pi\int_0^1 d\sigma\ \sigma^2B^{(0)}(\sigma-1)
|
|
\end{equation}
|
|
\begin{equation}
|
|
\bar\gamma^{(1)}_{l,n}:=
|
|
4\pi\int_0^1 d\sigma\ \sigma^2B_{l,n}^{(1)}(\sigma-1)
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\bar\gamma^{(2)}_{l,n;l',m}:=
|
|
4\pi\int_0^1 d\sigma\ \sigma^2B_{l,n;l',m}^{(2)}(\sigma-1)
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point Thus,
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
e=2\pi\rho
|
|
\cdot\\\hfill\cdot
|
|
\frac
|
|
{
|
|
C(\epsilon)
|
|
-
|
|
\gamma^{(0)}(e,\epsilon)
|
|
-
|
|
\sum_l\sum_n\mathbf F_{l,n}^{(\nu_u)}(u_+)\gamma^{(1)}_{l,n}(e,\epsilon)
|
|
-
|
|
\sum_{l,l'}\sum_{n,m}\mathbf F_{l,n}^{(\nu_u)}(u_+)\mathbf F_{l',m}^{(\nu_u)}(u_+)\gamma^{(2)}_{l,n;l',m}(e,\epsilon)
|
|
}
|
|
{1-\frac83\pi\rho+\rho^2\left(
|
|
\bar\gamma^{(0)}
|
|
+
|
|
\sum_l\sum_n\mathbf F_{l,n}^{(\nu_u)}(u_+)\bar\gamma^{(1)}_{l,n}
|
|
+
|
|
\sum_{l,l'}\sum_{n,m}\mathbf F_{l,n}^{(\nu_u)}(u_+)\mathbf F_{l',m}^{(\nu_u)}(u_+)\bar\gamma^{(2)}_{l,n;l',m}
|
|
\right)}
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
\begin{equation}
|
|
C(\epsilon):=\frac23\mu+(1+2\sqrt\epsilon)
|
|
=\frac43(\epsilon-e)+1+2\sqrt\epsilon
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subsubsubsection{Newton algorithm}
|
|
In this paragraph, we set $\epsilon=e$, that is, $\mu=0$.
|
|
\bigskip
|
|
|
|
\point
|
|
As we did for \refname{sec:anyeq}{{\tt anyeq}}, we discretize the integral in\-~(\ref{bfF}) by using a Gauss-Legendre quadrature, and truncate the sum over Chebyshev polynomials to order \refname{param:simpleq-hardcore_P}{{\tt P}}.
|
|
We then reduce the computation to a finite system of equations, whose variables are
|
|
\begin{equation}
|
|
e
|
|
,\quad
|
|
\mathfrak u_{l,j}:=u_+(r_{l,j})
|
|
,\quad
|
|
r_{l,j}:=
|
|
\frac{2+(\tau_{l+1}-\tau_l)\sin(\frac{\pi x_j}2)-(\tau_{l+1}+\tau_\alpha)}{2-(\tau_{l+1}-\tau_l)\sin(\frac{\pi x_j}2)+(\tau_{l+1}+\tau_l)}
|
|
\end{equation}
|
|
where $x_j$ are the abcissa for Gauss-Legendre quadratures (see\-~(\ref{klj})).
|
|
In other words, we define a vector $\mathbb U$ in dimension \refname{param:simpleq-hardcore_N}{{\tt N}}$\times$\refname{param:simpleq-hardcore_J}{{\tt J}}$+1$, where the first \refname{param:simpleq-hardcore_N}{{\tt N}}$\times$\refname{param:simpleq-hardcore_J}{{\tt J}} terms are $\mathfrak u_{l,j}$ and the last component is $e$.
|
|
We then write\-~(\ref{hardcore_u+}) as, for $l\in\{0,\cdots,J-1\}$, $j\in\{1,\cdots,N\}$,
|
|
\begin{equation}
|
|
\Xi_{lN+j}(\mathbb U)=0
|
|
,\quad
|
|
\Xi_{NJ+1}(\mathbb U)=0
|
|
\end{equation}
|
|
(note that $\Xi_{lN+j}$ corresponds to the pair $(l,j)$)
|
|
with
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\Xi_{lN+j}(\mathbb U):=
|
|
-\mathfrak u_{l,j}
|
|
+
|
|
\mathbb D^{(0)}(r_{l,j},e)
|
|
+
|
|
\sum_{l'}\sum_n\mathfrak F_{l',n}^{(\nu_u)}(\mathfrak u)\mathbb D^{(1)}_{l',n}(r_{l,j},e)
|
|
+\\\hfill+
|
|
\sum_{l',l''}\sum_{n,m}\mathfrak F_{l',n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l'',m}^{(\nu_u)}(\mathfrak u)\mathbb D^{(2)}_{l',n;l'',m}(r_{l,j},e)
|
|
\end{largearray}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\Xi_{NJ+1}(\mathbb U):=-e
|
|
+\\\hfill+
|
|
2\pi\rho
|
|
\frac
|
|
{
|
|
C(e)
|
|
-
|
|
\mathfrak g^{(0)}(e)
|
|
-
|
|
\sum_l\sum_n\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\mathfrak g^{(1)}_{l,n}(e)
|
|
-
|
|
\sum_{l,l'}\sum_{n,m}\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l',m}^{(\nu_u)}(\mathfrak u)\mathfrak g^{(2)}_{l,n;l',m}(e)
|
|
}
|
|
{1-\frac83\pi\rho+\rho^2\left(
|
|
\bar{\mathfrak g}^{(0)}
|
|
+
|
|
\sum_l\sum_n\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\bar{\mathfrak g}^{(1)}_{l,n}
|
|
+
|
|
\sum_{l,l'}\sum_{n,m}\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l',m}^{(\nu_u)}(\mathfrak u)\bar{\mathfrak g}^{(2)}_{l,n;l',m}
|
|
\right)}
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
where $\mathfrak F$ is defined in\-~(\ref{frakF}), and $\mathbb D^{(i)}$, $\mathfrak g^{(i)}$, $\bar{\mathfrak g}^{(i)}$ are defined like $D^{(i)}$, $\gamma^{(i)}$ and $\bar\gamma^{(i)}$ except that the integrals over bounded intervals are approximated using Gauss-Legendre quadratures, and the integrals from $0$ to $\infty$ are approximated using Gauss-Laguerre quadratures.
|
|
Gauss-Legendre and Gauss-Laguerre quadratures and their errors are discussed in appendix\-~\ref{appGL}.
|
|
The orders of the quadratures are given by the variable \refname{param:simpleq-hardcore_N}{{\tt N}}.
|
|
\bigskip
|
|
|
|
\point
|
|
We then solve $\Xi=0$ using the Newton algorithm, that is, we define a sequence of $\mathbb U$'s:
|
|
\begin{equation}
|
|
\mathbb U^{(n+1)}=\mathbb U^{(n)}-(D\Xi(\mathbb U^{(n)}))^{-1}\Xi(\mathbb U^{(n)})
|
|
\end{equation}
|
|
where $D\Xi$ is the Jacobian of $\Xi$:
|
|
\begin{equation}
|
|
(D\Xi)_{\alpha,\beta}:=\frac{\partial\Xi_\alpha}{\partial\mathbb U_\beta}
|
|
.
|
|
\end{equation}
|
|
We initialize the algorithm with
|
|
\begin{equation}
|
|
\mathbb U^{(0)}_{lN+j}=\frac1{(1+r_{l,j}^2)^2}
|
|
,\quad
|
|
\mathbb U^{(0)}_{JN+1}=\pi\rho
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\point
|
|
We are left with computing the Jacobian of $\Xi$:
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\frac{\partial\Xi_{lN+j}(\mathbb U)}{\partial\mathfrak u_{l',i}}=
|
|
-\delta_{l,l'}\delta_{j,i}
|
|
+
|
|
\sum_{l''}\sum_n\mathfrak F_{l'',n}^{(\nu_u)}(\mathds 1_{l',i})\mathbb D^{(1)}_{l'',n}(r_{l,j},e)
|
|
+\\\hfill+
|
|
2\sum_{l'',l'''}\sum_{n,m}\mathfrak F_{l'',n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l''',m}^{(\nu_u)}(\mathds 1_{l',j})\mathbb D^{(2)}_{l'',n;l''',m}(r_{l,j},e)
|
|
\end{largearray}
|
|
\end{equation}
|
|
where $\mathds 1_{l',i}$ is a vector whose only non-vanishing entry is the $(l',i)$-th, which is 1,
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\frac{\partial\Xi_{JN+1}(\mathbb U)}{\partial\mathfrak u_{l',i}}
|
|
=\\\hfill=
|
|
2\pi\rho
|
|
\frac
|
|
{
|
|
-
|
|
\sum_l\sum_n\mathfrak F_{l,n}^{(\nu_u)}(\mathds 1_{l',i})\mathfrak g^{(1)}_{l,n}(e)
|
|
-
|
|
2\sum_{l,l''}\sum_{n,m}\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l'',m}^{(\nu_u)}(\mathds 1_{l',i})\mathfrak g^{(2)}_{l,n;l'',m}(e)
|
|
}
|
|
{1-\frac83\pi\rho+\rho^2\left(
|
|
\bar{\mathfrak g}^{(0)}
|
|
+
|
|
\sum_l\sum_n\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\bar{\mathfrak g}^{(1)}_{l,n}
|
|
+
|
|
\sum_{l,l''}\sum_{n,m}\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l'',m}^{(\nu_u)}(\mathfrak u)\bar{\mathfrak g}^{(2)}_{l,n;l'',m}
|
|
\right)}
|
|
\\[1cm]\hfill
|
|
-(\Xi_{NJ+1}(\mathbb U)+e)
|
|
\frac
|
|
{\rho^2\left(
|
|
\sum_l\sum_n\mathfrak F_{l,n}^{(\nu_u)}(\mathds 1_{l'.i})\bar{\mathfrak g}^{(1)}_{l,n}
|
|
+
|
|
2\sum_{l,l''}\sum_{n,m}\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l'',m}^{(\nu_u)}(\mathds 1_{l',i})\bar{\mathfrak g}^{(2)}_{l,n;l'',m}
|
|
\right)}{1-\frac83\pi\rho+\rho^2\left(
|
|
\bar{\mathfrak g}^{(0)}
|
|
+
|
|
\sum_l\sum_n\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\bar{\mathfrak g}^{(1)}_{l,n}
|
|
+
|
|
\sum_{l,l''}\sum_{n,m}\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l'',m}^{(\nu_u)}(\mathfrak u)\bar{\mathfrak g}^{(2)}_{l,n;l'',m}
|
|
\right)}
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\frac{\partial\Xi_{lN+j}(\mathbb U)}{\partial e}=
|
|
\partial_e\mathbb D^{(0)}(r_{l,j},e)
|
|
+
|
|
\sum_{l'}\sum_n\mathfrak F_{l',n}^{(\nu_u)}(\mathfrak u)\partial_e\mathbb D^{(1)}_{l',n}(r_{l,j},e)
|
|
+\\\hfill+
|
|
\sum_{l',l''}\sum_{n,m}\mathfrak F_{l',n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l'',m}^{(\nu_u)}(\mathfrak u)\partial_e\mathbb D^{(2)}_{l',n;l'',m}(r_{l,j},e)
|
|
\end{largearray}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\frac{\partial\Xi_{NJ+1}(\mathbb U)}{\partial e}
|
|
=
|
|
-1
|
|
+\\\hfill+
|
|
2\pi\rho
|
|
\frac
|
|
{
|
|
\partial_e C(e)
|
|
-
|
|
\partial_e\mathfrak g^{(0)}(e)
|
|
-
|
|
\sum_l\sum_n\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\partial_e\mathfrak g^{(1)}_{l,n}(e)
|
|
-
|
|
\sum_{l,l'}\sum_{n,m}\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l',m}^{(\nu_u)}(\mathfrak u)\partial_e\mathfrak g^{(2)}_{l,n;l',m}(e)
|
|
}
|
|
{1-\frac83\pi\rho+\rho^2\left(
|
|
\bar{\mathfrak g}^{(0)}
|
|
+
|
|
\sum_l\sum_n\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\bar{\mathfrak g}^{(1)}_{l,n}
|
|
+
|
|
\sum_{l,l'}\sum_{n,m}\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l',m}^{(\nu_u)}(\mathfrak u)\bar{\mathfrak g}^{(2)}_{l,n;l',m}
|
|
\right)}
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
To compute $\partial_e\mathbb D^{(i)}$ and $\partial_e\mathfrak g^{(i)}$, we use $\partial_e=\frac1{2\sqrt e}\frac\partial{\partial\sqrt e}$ and
|
|
\begin{equation}
|
|
\frac{\partial C(e)}{\partial\sqrt e}=2
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\frac{\partial D^{(0)}(r,e)}{\partial\sqrt e}
|
|
=
|
|
-\frac{2r}{r+1}e^{-2\sqrt er}
|
|
\\[0.5cm]\hfill
|
|
+
|
|
\frac{\rho}{r+1}\int_0^{\min(1,r)} ds\ (s+1)B^{(0)}(s)e^{-2\sqrt er}\left(
|
|
(1-2\sqrt er)\sinh(2\sqrt e s)
|
|
+2\sqrt es\cosh(2\sqrt e s)
|
|
\right)
|
|
\\[0.5cm]
|
|
+
|
|
\mathds 1_{r\leqslant 1}
|
|
\frac{\rho}{2(r+1)}\int_0^{1-r} d\sigma\ (\sigma+r+1)B^{(0)}(\sigma+r)
|
|
\cdot\\[0.5cm]\hfill\cdot
|
|
\left(
|
|
(1-2\sqrt e\sigma)\left(1-e^{-4\sqrt er}\right)e^{-2\sqrt e\sigma}
|
|
+
|
|
4\sqrt e re^{-2\sqrt e(2r+\sigma)}
|
|
\right)
|
|
\end{largearray}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\frac{\partial D_{l,n}^{(1)}(r,e)}{\partial\sqrt e}
|
|
=
|
|
\frac{\rho}{r+1}\int_0^{r} ds\ (s+1)B_{l,n}^{(1)}(s)e^{-2\sqrt er}\left(
|
|
(1-2\sqrt er)\sinh(2\sqrt e s)
|
|
+2\sqrt es\cosh(2\sqrt e s)
|
|
\right)
|
|
\\[0.5cm]
|
|
+
|
|
\frac{\rho}{2(r+1)}\int_0^\infty d\sigma\ (\sigma+r+1)B_{l,n}^{(1)}(\sigma+r)
|
|
\cdot\\[0.5cm]\hfill\cdot
|
|
\left(
|
|
(1-2\sqrt e\sigma)\left(1-e^{-4\sqrt er}\right)e^{-2\sqrt e\sigma}
|
|
+
|
|
4\sqrt e re^{-2\sqrt e(2r+\sigma)}
|
|
\right)
|
|
\end{largearray}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\frac{\partial D_{l,n;l',m}^{(2)}(r,e)}{\partial\sqrt e}
|
|
\\[0.5cm]
|
|
=
|
|
\frac{\rho}{r+1}\int_0^{r} ds\ (s+1)B_{l,n;l',m}^{(2)}(s)e^{-2\sqrt er}\left(
|
|
(1-2\sqrt er)\sinh(2\sqrt e s)
|
|
+2\sqrt es\cosh(2\sqrt e s)
|
|
\right)
|
|
\\[0.5cm]
|
|
+
|
|
\frac{\rho}{2(r+1)}\int_0^\infty d\sigma\ (\sigma+r+1)B_{l,n;l',m}^{(2)}(\sigma+r)
|
|
\cdot\\[0.5cm]\hfill\cdot
|
|
\left(
|
|
(1-2\sqrt e\sigma)\left(1-e^{-4\sqrt er}\right)e^{-2\sqrt e\sigma}
|
|
+
|
|
4\sqrt e re^{-2\sqrt e(2r+\sigma)}
|
|
\right)
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
Furthermore,
|
|
\begin{equation}
|
|
\frac{\partial\gamma^{(0)}(e)}{\partial\sqrt e}=
|
|
4\rho\sqrt e\int_0^1 d\sigma\ (\sigma+1)B^{(0)}(\sigma)(1-\sqrt e\sigma)e^{-2\sqrt e\sigma}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\frac{\partial\gamma^{(1)}_{l,n}(e)}{\partial\sqrt e}=
|
|
4\rho e\int_0^\infty d\sigma\ (\sigma+1)B_{l,n}^{(1)}(\sigma)(1-\sqrt e\sigma)e^{-2\sqrt e\sigma}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\frac{\partial\gamma^{(2)}_{l,n;l',m}(e)}{\partial\sqrt e}=
|
|
4\rho e\int_0^\infty d\sigma\ (\sigma+1)B_{l,n;l',m}^{(2)}(\sigma)(1-\sqrt e\sigma)e^{-2\sqrt e\sigma}
|
|
.
|
|
\end{equation}
|
|
Finally, to get from $D$ to $\mathbb D$ and $\gamma$ to $\mathfrak g$, we approximate the integrate using Gauss-Legendre and Gauss-Laguerre quadratures (see appendix\-~\ref{appGL}), as described above.
|
|
\bigskip
|
|
|
|
\point
|
|
We iterate the Newton algorithm until the Newton relative error $\epsilon$ becomes smaller than the \refname{param:easyeq_tolerance}{{\tt tolerance}} parameter.
|
|
The Newton error is defined as
|
|
\begin{equation}
|
|
\epsilon=\frac{\|\mathbb U^{(n+1)}-\mathbb U^{(n)}\|_2}{\|\mathbb U^{(n)}\|_2}
|
|
\end{equation}
|
|
where $\|\cdot\|_2$ is the $l_2$ norm.
|
|
The energy thus obtained is
|
|
\begin{equation}
|
|
e=\mathbb U_{JN+1}
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subsubsubsection{Condensate fraction}
|
|
To compute the condensate fraction, we use the parameter $\mu$ in\-~(\ref{hardcore_simple}).
|
|
The uncondensed fraction is
|
|
\begin{equation}
|
|
\eta=\partial_\mu e|_{\mu=0}
|
|
.
|
|
\end{equation}
|
|
To compute $\partial_\mu e$, we use
|
|
\begin{equation}
|
|
\Xi(\mathbb U)=0
|
|
\end{equation}
|
|
which we differentiate with respect to $\mu$:
|
|
\begin{equation}
|
|
\partial_\mu\mathbb U=-(D\Xi)^{-1}\partial_\mu\Xi
|
|
.
|
|
\end{equation}
|
|
We are left with computing $\partial_\mu\Xi$:
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\frac{\partial\Xi_{lN+j}(\mathbb U)}{\partial\mu}=
|
|
\partial_\mu\mathbb D^{(0)}(r_{l,j},e,\epsilon)
|
|
+
|
|
\sum_{l'}\sum_n\mathfrak F_{l',n}^{(\nu_u)}(\mathfrak u)\partial_\mu\mathbb D^{(1)}_{l',n}(r_{l,j},e,\epsilon)
|
|
+\\\hfill+
|
|
\sum_{l',l''}\sum_{n,m}\mathfrak F_{l',n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l'',m}^{(\nu_u)}(\mathfrak u)\partial_\mu\mathbb D^{(2)}_{l',n;l'',m}(r_{l,j},e,\epsilon)
|
|
\end{largearray}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\frac{\partial\Xi_{NJ+1}(\mathbb U)}{\partial \mu}
|
|
=\\\hfill=
|
|
2\pi\rho
|
|
\frac
|
|
{
|
|
\partial_\mu C(\epsilon)
|
|
-
|
|
\partial_\mu\mathfrak g^{(0)}(e,\epsilon)
|
|
-
|
|
\sum_l\sum_n\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\partial_\mu\mathfrak g^{(1)}_{l,n}(e,\epsilon)
|
|
-
|
|
\sum_{l,l'}\sum_{n,m}\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l',m}^{(\nu_u)}(\mathfrak u)\partial_\mu\mathfrak g^{(2)}_{l,n;l',m}(e,\epsilon)
|
|
}
|
|
{1-\frac83\pi\rho+\rho^2\left(
|
|
\bar{\mathfrak g}^{(0)}
|
|
+
|
|
\sum_l\sum_n\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\bar{\mathfrak g}^{(1)}_{l,n}
|
|
+
|
|
\sum_{l,l'}\sum_{n,m}\mathfrak F_{l,n}^{(\nu_u)}(\mathfrak u)\mathfrak F_{l',m}^{(\nu_u)}(\mathfrak u)\bar{\mathfrak g}^{(2)}_{l,n;l',m}
|
|
\right)}
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
We then use $\partial_\mu=\frac1{4\sqrt\epsilon}\partial_{\sqrt\epsilon}$ and
|
|
\begin{equation}
|
|
\left.\frac{\partial C(\epsilon)}{\partial\sqrt\epsilon}\right|_{\mu=0}
|
|
=
|
|
\frac83\sqrt\epsilon+2
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\left.\frac{\partial D^{(0)}(r)}{\partial\sqrt\epsilon}\right|_{\mu=0}=
|
|
-\frac{2r}{r+1}e^{-2\sqrt{e}r}
|
|
\\[0.5cm]\hfill
|
|
+
|
|
\frac{\rho}{(r+1)}\int_0^{\min(1,r)} ds\ (s+1)B^{(0)}(s)e^{-2\sqrt{e}r}((-1-2\sqrt er)\sinh(2\sqrt{e} s)+2\sqrt es\cosh(2\sqrt{e} s))
|
|
\\[0.5cm]
|
|
+
|
|
\mathds 1_{r\leqslant 1}
|
|
\frac{\rho}{2(r+1)}\int_0^{1-r} d\sigma\ (\sigma+r+1)B^{(0)}(\sigma+r)
|
|
\cdot\\[0.5cm]\hfill\cdot
|
|
\left((-1-2\sqrt e\sigma)\left(1-e^{-4\sqrt er}\right)e^{-2\sqrt{e}\sigma}+4\sqrt ere^{-2\sqrt{e}(2r+\sigma)}\right)
|
|
\end{largearray}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\left.\frac{\partial D_{l,n}^{(1)}(r)}{\partial\sqrt\epsilon}\right|_{\mu=0}
|
|
=
|
|
\frac{\rho}{r+1}\int_0^{r} ds\ (s+1)B_{l,n}^{(1)}(s)e^{-2\sqrt er}\left(
|
|
(-1-2\sqrt er)\sinh(2\sqrt e s)
|
|
+2\sqrt es\cosh(2\sqrt e s)
|
|
\right)
|
|
\\[0.5cm]
|
|
+
|
|
\frac{\rho}{2(r+1)}\int_0^\infty d\sigma\ (\sigma+r+1)B_{l,n}^{(1)}(\sigma+r)
|
|
\cdot\\[0.5cm]\hfill\cdot
|
|
\left(
|
|
(-1-2\sqrt e\sigma)\left(1-e^{-4\sqrt er}\right)e^{-2\sqrt e\sigma}
|
|
+
|
|
4\sqrt e re^{-2\sqrt e(2r+\sigma)}
|
|
\right)
|
|
\end{largearray}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\left.\frac{\partial D_{l,n;l',m}^{(2)}(r)}{\partial\sqrt\epsilon}\right|_{\mu=0}
|
|
\\[0.5cm]
|
|
=
|
|
\frac{\rho}{r+1}\int_0^{r} ds\ (s+1)B_{l,n;l',m}^{(2)}(s)e^{-2\sqrt er}\left(
|
|
(-1-2\sqrt er)\sinh(2\sqrt e s)
|
|
+2\sqrt es\cosh(2\sqrt e s)
|
|
\right)
|
|
\\[0.5cm]
|
|
+
|
|
\frac{\rho}{2(r+1)}\int_0^\infty d\sigma\ (\sigma+r+1)B_{l,n;l',m}^{(2)}(\sigma+r)
|
|
\cdot\\[0.5cm]\hfill\cdot
|
|
\left(
|
|
(-1-2\sqrt e\sigma)\left(1-e^{-4\sqrt er}\right)e^{-2\sqrt e\sigma}
|
|
+
|
|
4\sqrt e re^{-2\sqrt e(2r+\sigma)}
|
|
\right)
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\left.\frac{\partial \gamma^{(0)}}{\partial\sqrt\epsilon}\right|_{\mu=0}
|
|
=
|
|
-4\rho e\int_0^1 d\sigma\ \sigma(\sigma+1)B^{(0)}(\sigma)e^{-2\sqrt e\sigma}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\left.\frac{\partial\gamma^{(1)}_{l,n}}{\partial\sqrt\epsilon}\right|_{\mu=0}
|
|
=
|
|
-4\rho e\int_0^\infty d\sigma\ (\sigma+1)\sigma B_{l,n}^{(1)}(\sigma)e^{-2\sqrt e\sigma}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\left.\frac{\partial\gamma^{(2)}_{l,n;l',m}}{\partial\sqrt\epsilon}\right|_{\mu=0}
|
|
=
|
|
-4\rho e\int_0^\infty d\sigma\ (\sigma+1)\sigma B_{l,n;l',m}^{(2)}(\sigma)e^{-2\sqrt e\sigma}
|
|
.
|
|
\end{equation}
|
|
|
|
\subsection{\tt simpleq-iteration}\label{sec:simpleq-iteration}
|
|
\indent
|
|
This method is used to solve the Simple equation using the iteration described in\-~\cite{CJL20}.
|
|
The Simple equation is
|
|
\begin{equation}
|
|
-\Delta u
|
|
=
|
|
S-4e u+2e\rho u\ast u
|
|
\label{simpleq_iteration}
|
|
\end{equation}
|
|
\begin{equation}
|
|
S:=(1-u)v
|
|
,\quad
|
|
\rho:=\frac{2e}{\int dx\ (1-u(|x|))v(|x|)}
|
|
.
|
|
\label{simpleq_Se}
|
|
\end{equation}
|
|
for a soft potential $v$ at fixed energy $e>0$.
|
|
The iteration is defined as
|
|
\begin{equation}
|
|
-\Delta u_n=S_n-4eu_n+2e\rho_{n-1}u_{n-1}\ast u_{n-1}
|
|
,\quad
|
|
u_0=0
|
|
,\quad
|
|
\rho_{n-1}=
|
|
\frac{2e}{\int dx\ (1-u_{n-1}(|x|))v(|x|)}
|
|
.
|
|
\label{iteration}
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\subsubsection{Usage}
|
|
\indent
|
|
Unless otherwise noted, this method takes the following parameters (specified via the {\tt [-p params]} flag, as a `{\tt;}' separated list of entries).
|
|
\begin{itemize}
|
|
\item\makelink{param:simpleq-iteration_e}{}
|
|
{\tt e} ({\tt Float64}, default: $10^{-4}$): energy $e$.
|
|
|
|
\item\makelink{param:simpleq-iteration_maxiter}{}
|
|
{\tt maxiter} ({\tt Int64}, default: 21): maximal number of iterations.
|
|
|
|
\item\makelink{param:simpleq-iteration_order}{}
|
|
{\tt order} ({\tt Int64}, default: 100): order used for all Gauss quadratures (denoted by $N$ below).
|
|
|
|
\end{itemize}
|
|
\bigskip
|
|
|
|
The available {\tt commands} are the following.
|
|
|
|
\begin{itemize}
|
|
\item {\tt rho\_e}\makelink{command:simpleq-iteration_rho_e}{}:
|
|
compute the density $\rho$ as a function of $e$.\par
|
|
\underline{Disabled parameters}: \refname{param:simpleq-iteration_e}{{\tt e}}.\par
|
|
\underline{Extra parameters}:
|
|
\begin{itemize}
|
|
\item\makelink{param:simpleq-iteration_minle}{}
|
|
{\tt minle} ({\tt Float64}, default: $10^{-6}$): minimal value for $\log_{10}e$.
|
|
|
|
\item\makelink{param:simpleq-iteration_maxle}{}
|
|
{\tt maxle} ({\tt Float64}, default: $10^{2}$): maximal value for $\log_{10}e$.
|
|
|
|
\item\makelink{param:simpleq-iteration_nle}{}
|
|
{\tt nle} ({\tt Int64}, default: $100$): number of values for $e$ (spaced logarithmically).
|
|
|
|
\item\makelink{param:simpleq-iteration_es}{}
|
|
{\tt es} ({\tt Array\{Float64\}}, default: $(10^{{\tt minle}+\frac{{\tt maxle}-{\tt minle}}{{\tt nle}}n})_n$: list of values for $e$, specified as a `{\tt,}' separated list.
|
|
\end{itemize}
|
|
\underline{Output} (one line for each value of $e$): [$e$] [$\rho$].
|
|
|
|
\item\makelink{command:simpleq-iteration_ux}{}
|
|
{\tt ux}: compute $u$ as a function of $|x|$.\par
|
|
\underline{Extra parameters}:
|
|
\begin{itemize}
|
|
\item\makelink{param:simpleq-iteration_xmin}{}
|
|
{\tt xmin} ({\tt Float64}, default: 0): minimum of the range of $|x|$ to be printed.
|
|
|
|
\item\makelink{param:simpleq-iteration_xmax}{}
|
|
{\tt xmax} ({\tt Float64}, default: 100): maximum of the range of $|x|$ to be printed.
|
|
|
|
\item\makelink{param:simpleq-iteration_nx}{}
|
|
{\tt nx} ({\tt Int64}, default: 100): number of points to print (linearly spaced).
|
|
\end{itemize}
|
|
\underline{Output} (one line for each value of $x$): [$|x|$] [$u_1(|x|)$] [$u_2(|x|)$] [$u_3(|x|)$] $\cdots$
|
|
|
|
\end{itemize}
|
|
|
|
\subsubsection{Description}
|
|
\indent
|
|
In Fourier space
|
|
\begin{equation}
|
|
\hat u_n(|k|):=\int dx\ e^{ikx}u_n(|x|)
|
|
\end{equation}
|
|
(\ref{iteration}) becomes
|
|
\begin{equation}
|
|
(k^2+4e)\hat u_n(|k|)=\hat S_n(|k|)+2e\rho_{n-1}\hat u_{n-1}(k)^2
|
|
,\quad
|
|
\rho_n:=\frac{2e}{\hat S_n(0)}
|
|
\label{iteration_uk}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\hat S_n(|k|)
|
|
=\hat v(|k|)-\frac1{(2\pi)^3}\int dp\ \hat u_n(|p|)\hat v(|k-p|)
|
|
.
|
|
\end{equation}
|
|
We write $\hat S_n$ in bipolar coordinates (see lemma\-~\ref{lemma:bipolar}):
|
|
\begin{equation}
|
|
\hat S_n(|k|)=\hat v(|k|)-\frac1{(2\pi)^3}\int_0^\infty dt\ t\hat u_n(t)\Eta(|k|,t)
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\Eta(y,t):=\frac{2\pi}y\int_{|y-t|}^{y+t}ds\ s\hat v(s)
|
|
\end{equation}
|
|
(note that this agrees with the function\-~(\ref{easyeq_Eta}) defined for \refname{sec:easyeq}{{\tt easyeq}}).
|
|
We also change variables to
|
|
\begin{equation}
|
|
y=\frac1{t+1}
|
|
,\quad
|
|
t=\frac{1-y}y
|
|
\end{equation}
|
|
\begin{equation}
|
|
\hat S_n(|k|)=\hat v(|k|)-\frac1{(2\pi)^3}\int_0^1 dy\ \frac{(1-y)\hat u_n(\frac{1-y}y)\Eta(|k|,\frac{1-y}y)}{y^3}
|
|
.
|
|
\end{equation}
|
|
We approximate this integral using a Gauss-Legendre quadrature (see appendix\-~\ref{appGL}) and discretize Fourier space:
|
|
\begin{equation}
|
|
k_i:=\frac{1-x_i}{1+x_i}
|
|
,\quad
|
|
y_i:=\frac{x_i+1}2
|
|
\end{equation}
|
|
where $x_i$ are the Gauss-Legendre abscissa, and
|
|
\begin{equation}
|
|
\hat S_n(k_i)\approx\hat v(k_i)-\frac1{2(2\pi)^3}\sum_{j=1}^Nw_j\frac{(1-y_j)\hat u_n(\frac{1-y_j}{y_j})\Eta(k_i,\frac{1-y_j}{y_j})}{y_j^3}
|
|
\end{equation}
|
|
so if
|
|
\begin{equation}
|
|
\mathbb U_i(n):=\hat u_n(k_i)
|
|
,\quad
|
|
\mathbf v_i:=\hat v(k_i)
|
|
\end{equation}
|
|
we have
|
|
\begin{equation}
|
|
\sum_{j=1}^NA_{i,j}\mathbb U_j(n)=b_i^{(n)}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
A_{i,j}:=(k_i^2+4e)\delta_{i,j}+\frac{w_j(1-y_j)\Eta(k_i,\frac{1-y_j}{y_j})}{2(2\pi)^3y_j^3}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
b_i^{(n)}:=\mathbf v_i+2e\rho_{n-1}\mathbb U_i(n-1)^2
|
|
\end{equation}
|
|
in terms of which
|
|
\begin{equation}
|
|
\mathbb U=A^{-1}b^{(n)}
|
|
.
|
|
\end{equation}
|
|
Finally, we compute $\rho_n$ using the second of\-~(\ref{iteration_uk}):
|
|
\begin{equation}
|
|
\rho_n=\frac{2e}{\hat S_n(0)}
|
|
,\quad
|
|
S_n(0)\approx\hat v(0)
|
|
-\frac1{2(2\pi)^3}\sum_{j=1}^Nw_j\frac{(1-y_j)\hat u_n(\frac{1-y_j}{y_j})\Eta(0,\frac{1-y_j}{y_j})}{y_j^3}
|
|
.
|
|
\end{equation}
|
|
|
|
\section{Potentials}\label{sec:potentials}
|
|
\indent
|
|
In this section, we describe the potentials available in {\tt simplesolv}, and provide documentation to add custom potentials to {\tt simplesolv}.
|
|
\bigskip
|
|
|
|
\subsection{Built-in potentials}
|
|
\subsubsection{{\tt exp}}\label{sec:exp}
|
|
\indent
|
|
In $x$-space,
|
|
\begin{equation}
|
|
v(|x|)=ae^{-|x|}
|
|
.
|
|
\end{equation}
|
|
The constant $a$ is specified through the {\tt v\_a} parameter, and can be any real number.
|
|
Note that $v\geqslant 0$ if and only if $a\geqslant 0$.
|
|
This is the potential that is selected by default.
|
|
\bigskip
|
|
|
|
\point
|
|
In Fourier space,
|
|
\begin{equation}
|
|
\hat v(|k|)=\int dx\ e^{ikx}v(|x|)
|
|
=\frac{8\pi a}{(1+k^2)^2}
|
|
.
|
|
\end{equation}
|
|
In particular, $v$ is of positive type (that is, $\hat v\geqslant 0$) if and only if $a\geqslant 0$.
|
|
\bigskip
|
|
|
|
\point
|
|
The zero energy scattering solution, that is, the solution of
|
|
\begin{equation}
|
|
(-\Delta+v)\psi=0
|
|
,\quad
|
|
\lim_{|x|\to\infty}\psi=1
|
|
\end{equation}
|
|
is
|
|
\begin{equation}
|
|
\psi(r)=\frac{cI_0(2\sqrt a e^{-\frac r2})+2K_0(2\sqrt a e^{-\frac r2})}r
|
|
,\quad
|
|
c=-\frac{2K_0(2\sqrt a)}{I_0(2\sqrt a)}
|
|
\end{equation}
|
|
where $I_0$ and $K_0$ are the modified Bessel functions, and the square root is taken with a branch cut on the negative imaginary axis.
|
|
In other words, if $a<0$, $\sqrt a=i\sqrt{|a|}$ and\-~\dlmfcite{(10.27.6),(10.27.9),(10.27.10)}{1.1.3}
|
|
\begin{equation}
|
|
I_0(ix)=J_0(x)
|
|
,\quad
|
|
K_0(ix)=-\frac\pi2(Y_0(x)+iJ_0(x))
|
|
\end{equation}
|
|
where $J_0$ and $Y_0$ are the Bessel functions, so
|
|
\begin{equation}
|
|
\psi(r)=\pi\frac{c'J_0(2\sqrt{|a|} e^{-\frac r2})-Y_0(2\sqrt{|a|} e^{-\frac r2})}r
|
|
,\quad
|
|
c'=\frac{Y_0(2\sqrt{|a|})}{J_0(2\sqrt{|a|})}
|
|
.
|
|
\end{equation}
|
|
The scattering length is\-~\dlmfcite{(10.25.2),(10.31.2)}{1.1.3}
|
|
\begin{equation}
|
|
\mathfrak a_0=\lim_{r\to\infty}r(1-\psi(r))
|
|
=\log(a)+2\gamma+\frac{2K_0(2\sqrt a)}{I_0(2\sqrt a)}
|
|
\end{equation}
|
|
where $\gamma$ is the Euler constant, which, for $a<0$, is
|
|
\begin{equation}
|
|
\mathfrak a_0=\log|a|+2\gamma-\frac{\pi Y_0(2\sqrt{|a|})}{J_0(2\sqrt{|a|})}
|
|
.
|
|
\end{equation}
|
|
|
|
\subsubsection{{\tt tent}}\label{sec:tent}
|
|
\indent
|
|
In $x$-space,
|
|
\begin{equation}
|
|
v(|x|)=\mathds 1_{|x|<b}a\frac{2\pi}3\left(1-\frac{|x|}b\right)^2\left(\frac{|x|}b+2\right)
|
|
.
|
|
\end{equation}
|
|
The constants $a$ and $b$ are specified through the {\tt v\_a} and {\tt v\_b} parameters, and $a\in\mathbb R$, $b>0$.
|
|
Note that $v\geqslant 0$ if and only if $a\geqslant 0$.
|
|
Note that
|
|
\begin{equation}
|
|
v(|x|)=\frac a{b^3}\mathds 1_{|x|<\frac b2}\ast\mathds 1_{|x|<\frac b2}
|
|
\end{equation}
|
|
(this is more easily checked in Fourier space than explicitly computing the convolution).
|
|
\bigskip
|
|
|
|
\indent
|
|
In Fourier space,
|
|
\begin{equation}
|
|
\hat v(|k|)=\int dx\ e^{ikx}v(|x|)=a\frac{b^3}8\left(4\pi\frac{\sin(\frac{|k|b}2)-\frac{|k|b}2\cos(\frac{|k|b}2)}{\frac{(|k|b)^3}8}\right)^2
|
|
.
|
|
\end{equation}
|
|
Note that this is $\frac a{b^3}(\hat\mathds 1_{|x|<\frac b2})^2$.
|
|
In particular, $v$ is of positive type (that is, $\hat v\geqslant 0$) if and only if $a\geqslant 0$.
|
|
|
|
\subsubsection{{\tt expcry}}\label{sec:expcry}
|
|
\indent
|
|
In $x$-space
|
|
\begin{equation}
|
|
v(|x|)=e^{-|x|}-ae^{-b|x|}
|
|
.
|
|
\end{equation}
|
|
The constants $a$ and $b$ are specified through the {\tt v\_a} and {\tt v\_b} parameters, and $a\in\mathbb R$, $b>0$.
|
|
Note that $v\geqslant 0$ if and only if $a\leqslant 1$ and $b\geqslant 1$.
|
|
\bigskip
|
|
|
|
\indent
|
|
In Fourier space
|
|
\begin{equation}
|
|
\hat v(|k|)=\int dx\ e^{ikx}v(|x|)
|
|
=8\pi\left(\frac1{(1+k^2)^2}-\frac{ab}{(b^2+k^2)^2}\right)
|
|
\end{equation}
|
|
In particular, $\hat v$ is of positive type (that is, $\hat v\geqslant 0$) if and only if $ab\leqslant 1$, $a\leqslant b$ and $a\leqslant b^3$.
|
|
If $a\leqslant 1$, $b\geqslant 1$ and $ab>1$, then $\hat v$ has a unique minimum at
|
|
\begin{equation}
|
|
|k_*|=\sqrt{\frac{b^2-(ab)^{\frac13}}{(ab)^{\frac13}-1}}
|
|
,\quad
|
|
\hat v(|k_*|)=
|
|
-8\pi\frac{((ab)^{\frac13}-1)^3}{b^2-1}
|
|
.
|
|
\end{equation}
|
|
|
|
\subsubsection{{\tt npt}}\label{sec:npt}
|
|
\indent
|
|
In $x$-space
|
|
\begin{equation}
|
|
v(|x|)=x^2e^{-|x|}
|
|
.
|
|
\end{equation}
|
|
Note that $v\geqslant 0$.
|
|
\bigskip
|
|
|
|
\indent
|
|
In Fourier space
|
|
\begin{equation}
|
|
\hat v(|k|)=\int dx\ e^{ikx}v(|x|)
|
|
=96\pi\frac{1-k^2}{(1+k^2)^4}
|
|
.
|
|
\end{equation}
|
|
In particular, $v$ is not of positive type (that is, $\hat v$ is not $\geqslant 0$).
|
|
|
|
\subsubsection{{\tt alg}}\label{sec:alg}
|
|
\indent
|
|
In $x$-space
|
|
\begin{equation}
|
|
v(|x|)=\frac1{1+\frac14|x|^4}
|
|
.
|
|
\end{equation}
|
|
Note that $v\geqslant 0$.
|
|
\bigskip
|
|
|
|
\indent
|
|
In Fourier space
|
|
\begin{equation}
|
|
\hat v(|k|)=\int dx\ e^{ikx}v(|x|)
|
|
=4\pi^2 e^{-|k|}\frac{\sin(|k|)}{|k|}
|
|
.
|
|
\end{equation}
|
|
In particular, $v$ is not of positive type (that is, $\hat v$ is not $\geqslant 0$).
|
|
|
|
\subsubsection{{\tt algwell}}\label{sec:algwell}
|
|
\indent
|
|
In $x$-space
|
|
\begin{equation}
|
|
v(|x|)=\frac{1+a|x|^4}{(1+|x|^2)^4}
|
|
.
|
|
\end{equation}
|
|
The constant $a$ can be set using the parameter {\tt v\_a} and can be any real number.
|
|
Note that $v\geqslant 0$ if and only if $a\geqslant 0$.
|
|
If $a>8$, this potential has a local minimum at $|x_-|$ and a local maximum at $|x_+|$:
|
|
\begin{equation}
|
|
|x_\pm|=\sqrt{\frac12(1\pm\sqrt{1-8a^{-1}})}
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\indent
|
|
In Fourier space
|
|
\begin{equation}
|
|
\hat v(|k|)=\int dx\ e^{ikx}v(|x|)
|
|
=\frac{\pi^2}{24}e^{-|k|}(a(k^2-9|k|+15)+k^2+3|k|+3)
|
|
.
|
|
\end{equation}
|
|
In particular, $v$ is of positive type (that is, $\hat v$ is not $\geqslant 0$) if and only if
|
|
\begin{equation}
|
|
-\frac15\leqslant a\leqslant 3+\frac87\sqrt7\approx6.02
|
|
.
|
|
\end{equation}
|
|
|
|
\subsubsection{{\tt exact}}\label{sec:exact}
|
|
\indent
|
|
In $x$-space
|
|
\begin{equation}
|
|
v(|x|)=
|
|
\frac{12c( x^6b^6(2e-b^2) +b^4x^4(9e-7b^2) +4b^2x^2(3e-2b^2) +(5e+16b^2))}{(1+b^2x^2)^2(4+b^2x^2)^2((1+b^2x^2)^2-c)}
|
|
\end{equation}
|
|
The constants $a,b,c,e$ can be set using the parameters {\tt v\_a}, {\tt v\_b}, {\tt v\_c}, {\tt v\_e}, and $a,e\in\mathbb R$, $b\neq0$, $c>0$, $c\neq9$.
|
|
Note that $v\geqslant 0$ if and only if\-~\cite{CJL21}
|
|
\begin{equation}
|
|
b>0
|
|
,\quad
|
|
0<c<1
|
|
,\quad
|
|
\frac e{b^2}\geqslant\frac{-263+23\sqrt{161}}{48}\approx 0.601
|
|
.
|
|
\end{equation}
|
|
With this potential, the Simple equation has an exact solution:
|
|
\begin{equation}
|
|
u=\frac c{(1+b^2x^2)^2}
|
|
,\quad
|
|
\rho=\frac{b^3}{c\pi^2}
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\indent
|
|
In Fourier space $\hat v(|k|)=\int dx\ e^{ikx}v(x)$ is
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\hat v(|k|)=\frac{48 c\pi^2}{|k|}
|
|
\left(
|
|
\frac{18+3\sqrt c-(4-3\alpha)c-(1-2\alpha)c^{\frac32}}{4(3+\sqrt c)^2c^{\frac32}}
|
|
e^{-\sqrt{1-\sqrt c}\frac{|k|}b}
|
|
\right.\\[0.5cm]\hfill\left.
|
|
+
|
|
\frac{-18+3\sqrt c+(4-3\alpha)c-(1-2\alpha)c^{\frac32}}{4(3-\sqrt c)^2c^{\frac32}}
|
|
e^{-\sqrt{1+\sqrt c}\frac{|k|}b}
|
|
+\frac{1-\frac{|k|}b}{2c}
|
|
e^{-\frac{|k|}b}
|
|
-\frac{\alpha(3(9-c)\frac{|k|}b+8c)}{8(9-c)^2}
|
|
e^{-2\frac{|k|}b}
|
|
\right)
|
|
\end{largearray}
|
|
\end{equation}
|
|
with $\alpha:=\frac e{b^2}$.
|
|
|
|
\subsection{Programming custom potentials}
|
|
\indent
|
|
In this section, we provide documentation for programming custom potentials.
|
|
\bigskip
|
|
|
|
\indent
|
|
The potentials are implemented in the file `{\tt \$SIMPLESOLV/src/potentials.jl}', and consist of two functions, one specifying the potential in Fourier space (the ``potential function''), and the other returning an approximate value for the scattering length (the ``scatterin glength function'') (as is explained below, a precise value of the scattering length is not actually needed).
|
|
For instance, the potential \refname{sec:exp}{{\tt exp}} has two functions: {\tt v\_exp} and {\tt a0\_exp}.
|
|
The potential function should take the following arguments:
|
|
\begin{itemize}
|
|
\item {\tt k} ({\tt Float64}): the Fourier momentum
|
|
\item and any parameters that the potential depends on, such as $a$ in \refname{sec:exp}{{\tt exp}} (can be of any type, provided the appropriate changes are made to {\tt main.jl} as explained below)
|
|
\end{itemize}
|
|
and it must return a {\tt Float64}: the value of $\hat v$ at {\tt k}.
|
|
The scattering length function takes the same parameters as an input, and returns a {\tt Float64}: the approximate value for the scattering length.
|
|
\bigskip
|
|
|
|
\indent
|
|
In addition, the potential must be linked in `{\tt \$SIMPLESOLV/src/main.jl}'.
|
|
In that file, the potential is read from the command line option {\tt U}.
|
|
The relevant code is in lines 197-222.
|
|
To add a new potential, add
|
|
\begin{code}
|
|
elseif potential=="{\rm \{name of potential\}}"\par
|
|
\indent v=k->{\rm \{potential function\}}(k,v\_param\_a,v\_param\_b,{\rm...})\par
|
|
\indent a0={\rm \{scattering length function\}}(v\_param\_a,v\_param\_b,{\rm...})\par
|
|
\end{code}
|
|
where the number of {\tt v\_param} entries should be the number of parameters of the potential.
|
|
The parameters that are currently read from the parameters list are {\tt a}, {\tt b}, {\tt c} and {\tt e}.
|
|
To add a parameter, it must first be declared and initialized after line 35, and code to read it should be added after line 172:
|
|
\begin{code}
|
|
elseif lhs="v\_{\rm\{name of parameter\}}"\par
|
|
\indent v\_param\_{\rm\{name of parameter\}}=parse(Float64,rhs)
|
|
\end{code}
|
|
If the new parameter has a type other than {\tt Float64}, this should be changed in the {\tt parse} function, and in the initialization.
|
|
\bigskip
|
|
|
|
\indent
|
|
The approximation of the scattering length is only used to initialize the Newton algorithm for \refname{sec:easyeq}{{\tt easyeq}}, so it is not important that it be exact.
|
|
In fact, some of the built-in potentials set the scattering length to 1, when it has proved too difficult to compute it exactly.
|
|
|
|
\appendix
|
|
|
|
\section{Chebyshev polynomial expansion}\label{appChebyshev}
|
|
\indent
|
|
In this appendix, we compute the error of Chebyshev polynomial expansions of regular functions.
|
|
Specifically, we will consider class-$s$ Gevrey functions (which is a generalization of the notion of analyticity: analytic functions are Gevrey functions with $s=1$).
|
|
A class-$s$ Gevrey function on $[-1,1]$ is a $\mathcal C^\infty$ function that satisfies, $\forall n\in\mathbb N$,
|
|
\begin{equation}
|
|
\mathop{\mathrm{sup}}_{x\in[-1,1]}\left|\frac{d^nf(x)}{dx^n}\right|\leqslant C_0C^n(n!)^s.
|
|
\end{equation}
|
|
Formally, the Chebyshev polynomial expansion of $f$ is
|
|
\begin{equation}
|
|
f(x)=\frac{c_0}2+\sum_{j=1}^\infty c_j T_j(x)
|
|
\label{eqcheby}
|
|
\end{equation}
|
|
where $T_j$ is the $j$-th Chebyshev polynomial:
|
|
\begin{equation}
|
|
T_j(x):=\cos(j\arccos(x))
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
c_j:=\frac2\pi\int_0^\pi d\theta\ f(\cos\theta)\cos(j\theta)
|
|
.
|
|
\end{equation}
|
|
|
|
\bigskip
|
|
|
|
\theo{Lemma}\label{lemmaChebyshev}
|
|
Let $f$ be a class-$s$ Gevrey function on $[-1,1]$ with $s\in\mathbb N\setminus\{0\}$.
|
|
There exist $b_0,b>0$, which are independent of $s$, such that the coefficients $c_j$ of the Chebyshev polynomial expansion are bounded as
|
|
\begin{equation}
|
|
c_j\leqslant b_0e^{-bj^{\frac1s}}
|
|
.
|
|
\label{boundchebyshevcoef}
|
|
\end{equation}
|
|
In particular, the Chebyshev polynomial expansion is absolutely convergent pointwise (and, therefore in any $L_p$ norm), and, for every $N\geqslant 1$,
|
|
\begin{equation}
|
|
\left|f(x)-\frac{c_0}2-\sum_{j=1}^N c_j T_j(x)\right|
|
|
\leqslant
|
|
\frac{b_0}{1-e^{-b}}(s-1)!(N^{\frac1s}+b^{-1})^{s-1}e^{-bN^{\frac1s}}
|
|
.
|
|
\label{boundchebyshev}
|
|
\end{equation}
|
|
\endtheo
|
|
\bigskip
|
|
|
|
\indent\underline{Proof}:
|
|
Note that~\-(\ref{eqcheby}) is nothing other than the Fourier cosine series expansion of $F(\theta):=f(\cos(\theta))$, which is an even, periodic, class-$s$ Gevrey function on $[-\pi,\pi]$, whose $j$-th Fourier coefficient for $j\in\mathbb Z$ is equal to $\frac12c_{|j|}$.
|
|
The bound\-~(\ref{boundchebyshevcoef}) follows from a well-known estimate of the decay of Fourier coefficients of class-$s$ Gevrey functions (see e.g.~\-\cite[Theorem~\-3.3]{Ta87}).
|
|
The bound\-~(\ref{boundchebyshev}) then follows from $|T_j(x)|\leqslant 1$ and lemma\-~\ref{lemma:ineqgevrey} below.
|
|
\qed
|
|
\bigskip
|
|
|
|
\theo{Lemma}\label{lemma:ineqgevrey}
|
|
Given $b>0$ and two integers $N,s>0$,
|
|
\nopagebreakaftereq
|
|
\begin{equation}
|
|
\sum_{j=N}^\infty e^{-bj^{\frac1s}}
|
|
\leqslant
|
|
(s-1)!\left(N^{\frac1s}+\frac1{1-e^{-b}}\right)^{s-1}\frac{e^{-bN^{\frac1s}}}{1-e^{-b}}
|
|
\end{equation}
|
|
\endtheo
|
|
\restorepagebreakaftereq
|
|
\bigskip
|
|
|
|
\indent\underline{Proof}:
|
|
If $\nu_{N,s}^s:=\lfloor N^{\frac1s}\rfloor^s$ denotes the largest integer that is $\leqslant N$ and has an integer $s$-th root, then
|
|
\begin{equation}
|
|
\sum_{j=N}^\infty e^{-bj^{\frac1s}}\leqslant
|
|
\sum_{j=\nu_{N,s}^s}^\infty e^{-bj^{\frac1s}}\leqslant
|
|
\sum_{k=\nu_{N,s}}^\infty(k^s-(k-1)^s)e^{-bk}\leqslant
|
|
s\sum_{k=\nu_{N,s}}^\infty k^{s-1}e^{-bk}
|
|
.
|
|
\end{equation}
|
|
We then estimate
|
|
\begin{equation}
|
|
\sum_{k=\nu_{N,s}}^\infty k^{s-1}e^{-bk}=\frac{d^{s-1}}{d(-b)^{s-1}}\sum_{k=\nu_{N,s}}^\infty e^{-bk}
|
|
\leqslant (s-1)!\left(\nu_{N,s}+\frac1{1-e^{-b}}\right)^{s-1}\frac{e^{-b\nu_{N,s}}}{1-e^{-b}}
|
|
\end{equation}
|
|
which concludes the proof of the lemma.
|
|
\qed
|
|
|
|
\section{Gauss quadratures}\label{appGL}
|
|
\indent
|
|
Gauss quadratures are approximation schemes to compute integrals of the form
|
|
\begin{equation}
|
|
\int_a^bdt\ \omega(t)f(t)
|
|
\end{equation}
|
|
where $\omega(x)\geqslant 0$ is one of several functions that Gauss quadratures can treat.
|
|
The possible choices of $\omega$ and $(a,b)$ are
|
|
\begin{itemize}
|
|
\item $(a,b)=(-1,1)$, $\omega(t)=1$: Gauss-Legendre quadratures.
|
|
\item $(a,b)=(-1,1)$, $\omega(t)=(1-t)^\alpha(1+t)^\beta$, $\alpha,\beta>-1$: Gauss-Jacobi quadratures.
|
|
\item $(a,b)=(-1,1)$, $\omega(t)=(1-t^2)^{-\frac12}$: Gauss-Chebyshev quadratures.
|
|
\item $(a,b)=(0,\infty)$, $\omega(t)=e^{-t}$: Gauss-Laguerre quadratures.
|
|
\item $(a,b)=(0,\infty)$, $\omega(t)=t^\alpha e^{-t}$, $\alpha>-1$: generalized Gauss-Laguerre quadratures.
|
|
\item $(a,b)=(0,\infty)$, $\omega(t)=e^{-t^2}$: Gauss-Hermite quadratures.
|
|
\end{itemize}
|
|
It is not our goal here to discuss Gauss quadratures in detail, or their relation to orthogonal polynomials.
|
|
Instead, we will compute the error made when approximating such an integral by a Gauss quadrature.
|
|
\bigskip
|
|
|
|
\indent
|
|
For each Gauss quadrature, the integral is approximated in the form
|
|
\begin{equation}
|
|
\int_a^b dt\ \omega(t) f(t)
|
|
\approx
|
|
\sum_{i=1}^Nw_if(r_i)
|
|
\end{equation}
|
|
where $w_i$ are called the {\it weights}, $r_i$ are the {\it abcissa}, and $N$ is the {\it order}.
|
|
The weights and abcissa depend on both $\omega$ and the order $N$.
|
|
The crucial property of Gauss quadratures is that they are {\it exact} when $f$ is a polynomial of order $\leqslant 2N-1$.
|
|
\bigskip
|
|
|
|
\indent
|
|
In this appendix, we compute the error of Gauss quadratures when used to integrate regular functions.
|
|
Specifically, we will consider class-$s$ Gevrey functions (which is a generalization of the notion of analyticity: analytic functions are Gevrey functions with $s=1$).
|
|
A class-$s$ Gevrey function on is a $\mathcal C^\infty$ function that satisfies, $\forall n\in\mathbb N$,
|
|
\begin{equation}
|
|
\mathop{\mathrm{sup}}_{x\in[-1,1]}\left|\frac{d^nf(x)}{dx^n}\right|\leqslant C_0C^n(n!)^s.
|
|
\end{equation}
|
|
|
|
\bigskip
|
|
|
|
\theo{Lemma}\label{lemmaGL}
|
|
Let $f$ be a class-$s$ Gevrey function with $s\in\mathbb N\setminus\{0\}$.
|
|
There exist $b_0,b>0$, which are independent of $s$, and $N_0>0$, which is independent of $s$ and $f$, such that, if $N\geqslant N_0$, then, denoting the Gauss weights and abcissa by $w_i$ and $r_i$,
|
|
\begin{equation}
|
|
\left|\int_a^bdt\ \omega(t)f(t)-\sum_{i=1}^Nw_if(r_i)\right|
|
|
\leqslant
|
|
b_0(s-1)!((2N-1)^{\frac1s}+b^{-1})^{s-1}e^{-b(2N-1)^{\frac1s}}\left(\int_a^b\omega(t)+\sum_{i=1}^Nw_i\right)
|
|
.
|
|
\end{equation}
|
|
In particular, if $f$ is analytic (i.e. $s=1$), then
|
|
\begin{equation}
|
|
\left|\int_a^bdt\ \omega(t)f(t)-\sum_{i=1}^Nw_if(r_i)\right|
|
|
\leqslant b_0e^{-b(2N-1)}\left(\int_a^b\omega(t)+\sum_{i=1}^Nw_i\right)
|
|
.
|
|
\end{equation}
|
|
\endtheo
|
|
\bigskip
|
|
|
|
\indent\underline{Proof}:
|
|
We expand $f$ into Chebyshev polynomials as in\-~(\ref{eqcheby}):
|
|
\begin{equation}
|
|
f(x)=\frac{c_0}2+\sum_{j=1}^{\infty}c_jT_j(x)
|
|
\end{equation}
|
|
Let
|
|
\begin{equation}
|
|
g(x):=f(x)-\frac{c_0}2-\sum_{j=1}^{2N-1} c_j T_j(x)
|
|
.
|
|
\end{equation}
|
|
Since order-$N$ Gauss quadratures are exact on polynomials of order $\leqslant 2N-1$, we have
|
|
\begin{equation}
|
|
\int_a^bdt\ \omega(t)f(t)-\sum_{i=1}^Nw_if(r_i)
|
|
=
|
|
\int_a^bdt\ \omega(t)g(t)-\sum_{i=1}^Nw_ig(r_i)
|
|
\end{equation}
|
|
and, by lemma\-~\ref{lemmaChebyshev},
|
|
\begin{equation}
|
|
|g(x)|\leqslant
|
|
(\mathrm{const}.)(s-1)!((2N-1)^{\frac1s}+b^{-1})^{s-1}e^{-b(2N-1)^{\frac1s}}
|
|
.
|
|
\label{boundchebyshev}
|
|
\end{equation}
|
|
\qed
|
|
|
|
\section{Bipolar coordinates}\label{appendix:bipolar}
|
|
\indent
|
|
Bipolar coordinates are very useful for computing convolutions of radial functions in three dimensions.
|
|
\bigskip
|
|
|
|
\theo{Lemma}\label{lemma:bipolar}
|
|
For $y\in\mathbb R^3$,
|
|
\nopagebreakaftereq
|
|
\begin{equation}
|
|
\int_{\mathbb R^3}dx\ f(|x|,|x-y|)
|
|
=
|
|
\frac{2\pi}{|y|}\int_0^\infty dt\int_{||y|-t|}^{|y|+t}ds\ stf(s,t)
|
|
\end{equation}
|
|
\endtheo
|
|
\restorepagebreakaftereq
|
|
|
|
\indent\underline{Proof}:
|
|
Without loss of generality, we assume that $y=(0,0,a)$ with $a>0$. We first change to cylindrical coordinates: $(\rho,\theta,x_3)$:
|
|
\begin{equation}
|
|
\int_{\mathbb R^3}dx\ f(|x|,|x-y|)
|
|
=
|
|
2\pi\int_0^\infty d\rho\int_{-\infty}^\infty dx_3\ \rho f(|(\rho,0,x_3)|,|(\rho,0,x_3-a)|)
|
|
.
|
|
\end{equation}
|
|
Next, we change variables to
|
|
\begin{equation}
|
|
s=|(\rho,0,x_3)|
|
|
,\quad
|
|
t=|(\rho,0,x_3-a)|
|
|
.
|
|
\end{equation}
|
|
The inverse of this transformation is
|
|
\begin{equation}
|
|
x_3=\pm\frac{a^2+s^2-t^2}{2a}
|
|
,\quad
|
|
\rho=\frac1{2a}\sqrt{4a^2s^2-(a^2+s^2-t^2)^2}
|
|
\label{inverse}
|
|
\end{equation}
|
|
and its Jacobian is
|
|
\begin{equation}
|
|
\frac{2ts}{\sqrt{-2a^4+2a^2(t^2+s^2)-(t^2-s^2)^2}}
|
|
=\frac{ts}{\rho a}
|
|
.
|
|
\label{jacobian}
|
|
\end{equation}
|
|
\qed
|
|
\bigskip
|
|
|
|
\indent
|
|
The following is a generalization of the previous lemma to functions of four variables.
|
|
\bigskip
|
|
|
|
\theo{Lemma}\label{lemma:bipolar2}
|
|
For $y\in\mathbb R^3$,
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\int_{\mathbb R^6}dxdx'\ f(|x|,|x-y|,|x'|,|x'-y|,|x-x'|)
|
|
\\[0.5cm]\hfill
|
|
=
|
|
\frac{2\pi}{|y|^2}\int_0^\infty dt\int_{||y|-t|}^{|y|+t}ds\int_0^\infty dt'\int_{||y|-t'|}^{|y|+t'}ds'\int_0^{2\pi}d\theta\ sts't'f(s,t,s',t',\xi(s,t,s',t',\theta,|y|))
|
|
\end{largearray}
|
|
\end{equation}
|
|
with
|
|
\nopagebreakaftereq
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\xi(s,t,s',t',\theta,|y|)
|
|
:=
|
|
\frac1{\sqrt 2|y|}
|
|
\Big(
|
|
|y|^2(s^2+t^2+(s')^2+(t')^2)-|y|^4-(s^2-t^2)((s')^2-(t')^2)
|
|
\\[0.5cm]\hfill
|
|
-
|
|
\sqrt{(4|y|^2s^2-(|y|^2+s^2-t^2)^2)(4|y|^2(s')^2-(|y|^2+(s')^2-(t')^2)^2)}\cos\theta
|
|
\Big)^{\frac12}
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
\endtheo
|
|
\restorepagebreakaftereq
|
|
|
|
\indent\underline{Proof}:
|
|
Without loss of generality, we assume that $y=(0,0,a)$ with $a>0$. We first change to cylindrical coordinates: $(\rho,\theta,x_3;\rho',\theta',x_3')$:
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
\int_{\mathbb R^6}dxdx'\ f(|x|,|x-y|,|x'|,|x'-y|,|x-x'|)
|
|
\\[0.5cm]\hfill
|
|
=
|
|
2\pi\int_0^\infty d\rho\int_{-\infty}^\infty dx_3
|
|
\int_0^\infty d\rho'\int_{-\infty}^\infty dx_3'\int_0^{2\pi}d\theta'
|
|
\ \rho\rho'
|
|
f(
|
|
s,t,s',t',
|
|
|(\rho-\rho'\cos\theta',\rho'\sin\theta',x_3-x_3')|
|
|
)
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
where
|
|
\begin{equation}
|
|
s:=|(\rho,0,x_3)|
|
|
,\quad
|
|
t:=|(\rho,0,x_3-a)|
|
|
,\quad
|
|
s':=|(\rho'\cos\theta',\rho'\sin\theta',x_3')|,
|
|
t':=|(\rho'\cos\theta',\rho'\sin\theta',x_3'-a)|
|
|
\end{equation}
|
|
Next, we change variables to $(s,t,s',t',\theta')$.
|
|
The Jacobian of this transformation is, by\-~(\ref{jacobian}),
|
|
\begin{equation}
|
|
\frac{tst's'}{\rho\rho' a^2}
|
|
.
|
|
\end{equation}
|
|
Furthermore, by\-~(\ref{inverse}),
|
|
\begin{equation}
|
|
x_3-x_3'=\frac{s^2-t^2-(s')^2+(t')^2}{2a}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\rho=\frac{\sqrt{4a^2s^2-(a^2+s^2-t^2)^2}}{2a}
|
|
,\quad
|
|
\rho'=\frac{\sqrt{4a^2(s')^2-(a^2+(s')^2-(t')^2)^2}}{2a}
|
|
\end{equation}
|
|
so
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
|(\rho-\rho'\cos\theta',\rho'\sin\theta',x_3-x_3')|
|
|
=
|
|
\frac1{2a}
|
|
\left(
|
|
4a^2s^2-(a^2+s^2-t^2)^2
|
|
+
|
|
4a^2(s')^2-(a^2+(s')^2-(t')^2)^2
|
|
\right.\\[0.5cm]\hfill\left.
|
|
-
|
|
2\sqrt{(4a^2s^2-(a^2+s^2-t^2)^2)(4a^2(s')^2-(a^2+(s')^2-(t')^2)^2)}\cos\theta'
|
|
+
|
|
(s^2-t^2-(s')^2+(t')^2)^2
|
|
\right)^{\frac12}
|
|
\end{largearray}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\begin{largearray}
|
|
|(\rho-\rho'\cos\theta',\rho'\sin\theta',x_3-x_3')|
|
|
=
|
|
\frac1{\sqrt 2a}
|
|
\Big(
|
|
a^2(s^2+t^2+(s')^2+(t')^2)-a^4-(s^2-t^2)((s')^2-(t')^2)
|
|
\\[0.5cm]\hfill
|
|
-
|
|
\sqrt{(4a^2s^2-(a^2+s^2-t^2)^2)(4a^2(s')^2-(a^2+(s')^2-(t')^2)^2)}\cos\theta'
|
|
\Big)^{\frac12}
|
|
.
|
|
\end{largearray}
|
|
\end{equation}
|
|
\qed
|
|
|
|
\section{Hann windowing}\label{appendix:hann}
|
|
\indent
|
|
In this appendix, we discuss the use of Hann windows to compute Fourier transforms.
|
|
Consider the Fourier transform
|
|
\begin{equation}
|
|
\hat f(k)=\int dx\ e^{ikx}f(x)
|
|
.
|
|
\end{equation}
|
|
Evaluating this integral numerically can be tricky, especially at high $|k|$, because of the rapid oscillations at large $|x|$.
|
|
A trick to palliate such a problem is to multiply $f$ by a {\it window function} $h_L$, which cuts off distances of order $L$.
|
|
We then compute, instead of $\hat f$,
|
|
\begin{equation}
|
|
\tilde f(k)=\int dx\ e^{ikx}h_L(x)f(x)
|
|
.
|
|
\end{equation}
|
|
We can then evaluate $\tilde f$ using standard numerical techniques, such as Gauss quadratures (see appendix\-~\ref{appGL}), without issues at large $|x|$.
|
|
However, in doing so, we will make an error in the Fourier transform.
|
|
To quantify this error, note that
|
|
\begin{equation}
|
|
\tilde f(k)
|
|
=\hat h_L\hat\ast\hat f(k)
|
|
\equiv\int\frac{dq}{(2\pi)^d}\ \hat h_L(q)\hat f(k-q)
|
|
\end{equation}
|
|
so if we choose $h_L$ in such a way that $\hat h_L$ is peaked around the origin, then $\tilde f$ will not differ too much from $\hat f$:
|
|
\begin{equation}
|
|
\hat f(k)-\tilde f(k)
|
|
=((2\pi)^d\delta(k)-\hat h_L)\hat\ast\hat f(k)
|
|
.
|
|
\end{equation}
|
|
\bigskip
|
|
|
|
\indent
|
|
The {\it Hann} window is defined as
|
|
\begin{equation}
|
|
h_L(x)=\cos^2({\textstyle\frac{\pi|x|}L})\mathds 1_{|x|<\frac L2}
|
|
\end{equation}
|
|
whose Fourier transform is, in $d=3$,
|
|
\begin{equation}
|
|
\hat h_L(k)=
|
|
\frac{4\pi^3 L^2}{|k|}\frac{((|k|L)^3-4|k|L\pi^2)\cos(\frac{|k|L}2)-2(3(|k|L)^2-4\pi^2)\sin(\frac{|k|L}2)}{((|k|L)^3-4|k|L\pi^2)^2}
|
|
\end{equation}
|
|
which decays at large $|k|L$ as
|
|
\begin{equation}
|
|
\hat h_L(k)\sim\frac{4\pi^3}{|k|^4L}\cos({\textstyle\frac{|k|L}2})
|
|
.
|
|
\end{equation}
|
|
|
|
|
|
|
|
\vfill
|
|
\eject
|
|
|
|
\begin{thebibliography}{WWW99}
|
|
\small
|
|
\IfFileExists{bibliography/bibliography.tex}{\input bibliography/bibliography.tex}{}
|
|
\end{thebibliography}
|
|
|
|
|
|
\end{document}
|