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\documentclass[10pt]{article}
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\usepackage{color}
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\usepackage{graphicx}
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\usepackage{amsfonts}
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\usepackage[hidelinks]{hyperref}
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\usepackage{natbib}
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\def\Eq#1{\label{#1}}
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% colors
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\definecolor{iblue}{RGB}{65,105,225}
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\definecolor{ired}{RGB}{220,20,60}
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\def\alertv#1{{\color{green}#1}}
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\def\alertn#1{{\color{black}#1}}
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%% symbols
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\let\a=\alpha \let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon
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\let\z=\zeta \let\h=\eta \let\th=\vartheta \let\k=\kappa \let\l=\lambda
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\let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho
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\let\s=\sigma \let\t=\tau \let\f=\varphi \let\ph=\varphi\let\ch=\chi
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\let\ch=\chi \let\ps=\psi \let\y=\upsilon \let\o=\omega \let\si=\varsigma
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\let\G=\Gamma \let\D=\Delta \let\Th=\Theta \let\L=\Lambda \let\X=\Xi
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\let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi
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\let\O=\Omega \let\Y=\Upsilon
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\def\V#1{{\bf#1}}\def\lhs{{\it l.h.s.}\ }\def\rhs{{\it r.h.s.}\ }
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\def\*{\vskip 3mm}
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\def\0{\noindent}
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\def\be{\begin{equation}}
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\def\ee{\end{equation}}
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\def\bea{\begin{eqnarray}}
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\def\eea{\end{eqnarray}}
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%\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
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\def\nn{\nonumber}
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\def\xx{{\V x}}\def\pp{{\V p}}\def\kk{{\V k}}
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\def\AA{{\mathcal A}}
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\def\BB{{\cal B}}\def\CC{{\cal C}}\def\DD{{\mathcal D}}
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\def\EE{{\cal E}}\def\HH{{\cal H}}\def\KK{{\cal K}}\def\LL{{\mathcal L}}
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\def\NN{{\cal N}}\def\FF{{\cal F}}\def\PP{{\mathcal P}}
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\def\QQ{{\mathcal Q}}\def\RR{{\cal R}}\def\TT{{\mathcal T}}
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\let\dpr=\partial\let\fra=\frac
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\def\ie{{\it i.e.}\ }
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\def\eg{{\it e.g.}\ }
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\def\equ#1{(\ref{#1})}
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\def\lis#1{\overline{#1}}
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\def\defi{{\buildrel def\over=}}
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\def\Media#1{{\Big\langle\,#1\,\Big\rangle}}
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\def\media#1{{\Blangle\,#1\,\Brangle}}
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\def\bra#1{{\Blangle#1\Bvert}}\def\ket#1{{\Bvert#1\Brangle}}
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\def\braket#1#2{\Blangle#1\Bvert#2\Brangle}
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\def\otto{\,{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}\,}
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\def\wt#1{\widetilde{#1}}
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\def\tende#1{\,\vtop{\ialign{##\crcr\rightarrowfill\crcr
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\noalign{\kern-1pt\nointerlineskip} \hskip3.pt${\scriptstyle
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#1}$\hskip3.pt\crcr}}\,}
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\def\ie{{\it i.e.\ }}\def\etc{{\it etc.\ }}
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\def\FF{{\mathcal F}}
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\def\tto{\Rightarrow}
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\def\ap{{\it a priori}}
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\def\Ba {{\mbox{\boldmath$ \alpha$}}}
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\def\Bb {{\mbox{\boldmath$ \beta$}}}
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\def\Bg {{\mbox{\boldmath$ \gamma$}}}
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\def\Bd {{\mbox{\boldmath$ \delta$}}}
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\def\Be {{\mbox{\boldmath$ \varepsilon$}}}
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\def\Bee {{\mbox{\boldmath$ \epsilon$}}}
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\def\Bz {{\mbox{\boldmath$\zeta$}}}
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\def\Bh {{\mbox{\boldmath$ \eta$}}}
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\def\Bthh {{\mbox{\boldmath$ \theta$}}}
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\def\Bth {{\mbox{\boldmath$ \vartheta$}}}
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\def\Bi {{\mbox{\boldmath$ \iota$}}}
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\def\Bk {{\mbox{\boldmath$ \kappa$}}}
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\def\Bl {{\mbox{\boldmath$ \lambda$}}}
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\def\Bm {{\mbox{\boldmath$ \mu$}}}
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\def\Bn {{\mbox{\boldmath$ \nu$}}}
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\def\Bx {{\mbox{\boldmath$ \xi$}}}
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\def\Bom {{\mbox{\boldmath$ \omega$}}}
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\def\Bp {{\mbox{\boldmath$ \pi$}}}
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\def\Br {{\mbox{\boldmath$ \rho$}}}
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\def\Bro {{\mbox{\boldmath$ \varrho$}}}
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\def\Bs {{\mbox{\boldmath$ \sigma$}}}
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\def\Bsi {{\mbox{\boldmath$ \varsigma$}}}
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\def\Bt {{\mbox{\boldmath$ \tau$}}}
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\def\Bu {{\mbox{\boldmath$ \upsilon$}}}
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\def\Bf {{\mbox{\boldmath$ \phi$}}}
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\def\Bff {{\mbox{\boldmath$ \varphi$}}}
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\def\Bch {{\mbox{\boldmath$ \chi$}}}
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\def\Bps {{\mbox{\boldmath$ \psi$}}}
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\def\Bo {{\mbox{\boldmath$ \omega$}}}
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\def\Bome {{\mbox{\boldmath$ \varomega$}}}
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\def\BG {{\mbox{\boldmath$ \Gamma$}}}
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\def\BD {{\mbox{\boldmath$ \Delta$}}}
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\def\BTh {{\mbox{\boldmath$ \Theta$}}}
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\def\BL {{\mbox{\boldmath$ \Lambda$}}}
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\def\BX {{\mbox{\boldmath$ \Xi$}}}
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\def\BP {{\mbox{\boldmath$ \Pi$}}}
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\def\BS {{\mbox{\boldmath$ \Sigma$}}}
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\def\BU {{\mbox{\boldmath$ \Upsilon$}}}
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\def\BF {{\mbox{\boldmath$ \Phi$}}}
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\def\BPs {{\mbox{\boldmath$ \Psi$}}}
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\def\BO {{\mbox{\boldmath$ \Omega$}}}
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\def\BDpr {{\mbox{\boldmath$ \partial$}}}
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\def\Bstl {{\mbox{\boldmath$ *$}}}
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\def\Brangle {{\mbox{\boldmath$ \rangle$}}}
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\def\Blangle {{\mbox{\boldmath$ \langle$}}}
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\def\Bvert{{\mbox{\boldmath$|$}}}
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\def\Bell {{\mbox{\boldmath$\ell$}}}
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\let\up\uparrow
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\let\down\downarrow
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\def\({\left(}
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\def\){\right)}
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\let\mc\mathcal
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\let\mrm\mathrm
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\def\iniz{\setcounter{equation}{0}}
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\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
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\def\inizA{\setcounter{equation}{0}
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\renewcommand{\theequation}{\Alph{section}.\arabic{equation}}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newdimen\xshift \newdimen\xwidth \newdimen\yshift \newdimen\ywidth%
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\def\ins#1#2#3{\vbox to0pt{\kern-#2pt\hbox{\kern#1pt #3}\vss}\nointerlineskip}
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\def\eqfig#1#2#3#4#5{
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\par\xwidth=#1pt \xshift=\hsize \advance\xshift
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by-\xwidth \divide\xshift by 2
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\yshift=#2pt \divide\yshift by 2%
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{\hglue\xshift \vbox to #2pt{\vfil
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#3 \special{psfile=#4.eps}
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}\hfill\raise\yshift\hbox{#5}}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\def\Eqfig#1#2#3#4#5#6{
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\par\xwidth=#1pt \xshift=\hsize \advance\xshift
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by-\xwidth \divide\xshift by 2
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\yshift=#2pt \divide\yshift by 2%
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{\hglue\xshift \vbox to #2pt{\vfil
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#3 \special{psfile=#4.eps}\kern200pt\special{psfile=#5.eps}
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}\hfill\raise\yshift\hbox{#6}}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\def\eqalign#1{\null\,\vcenter{\openup\jot
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\ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
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\crcr#1\crcr}}\,}
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\def\qedsymbol{$\square$}
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\def\qed{\penalty10000\hfill\penalty10000\qedsymbol}
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\date{}
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\author{\alertb{Giovanni Gallavotti${}^1$ and
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Ian Jauslin${}^2$}}
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\title{\alertr{\bf A Theorem on Ellipses, an Integrable System and a Theorem
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of Boltzmann}
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}
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\begin{document}
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\maketitle
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\kern-8mm
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\centerline{${}^1$ INFN-Roma1 \& Universit\`a ``La Sapienza'', email: giovanni.gallavotti@roma1.infn.it}
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\centerline{${}^2$ Department of Physics, Princeton University, email: ijauslin@princeton.edu}
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%\kern-1cm
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\begin{abstract}
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%
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We study a mechanical system that was considered by Boltzmann in 1868 in the
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context of the derivation of the canonical and microcanonical ensembles. This
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system was introduced as an example of ergodic dynamics, which was central to
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Boltzmann's derivation. It consists of a single particle in two dimensions,
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which is subjected to a gravitational attraction to a fixed center. In
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addition, an infinite plane is fixed at some finite distance from the center,
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which acts as a hard wall on which the particle collides elastically. Finally,
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an extra centrifugal force is added. We will show that, in the absence of this
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extra centrifugal force, there are two independent integrals of motion.
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Therefore the extra centrifugal force is necessary for Boltzmann's claim of
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ergodicity to hold.
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%
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\end{abstract}
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\*
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%
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\0Keywords: {\small Ergodicity, Chaotic hypothesis, Gibbs
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distributions, Boltzmann, Integrable systems}
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\*
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%\end{document}
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In 1868, \cite{Bo868a} laid the foundations for our modern understanding of the
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behavior of many-particle systems by introducing the ``microcanonical
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ensemble'' (for more details on this history, see \cite{Ga016}). The principal
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idea behind this ensemble is that one can achieve a good understanding of
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many-particle systems by focusing not on the dynamics of each individual
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particle, but on the statistical properties of the whole. More precisely, the
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state of the system becomes a random variable, chosen according to a
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probability distribution on phase space, which came to be called the
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``microcanonical ensemble''. An important assumption that was made implicitly
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by Boltzmann is that the dynamics of the system be ergodic. In this case,
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time-averages of the dynamics can be rewritten as averages over phase space,
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and the qualitative properties of the dynamics can be formulated as statistical
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properties of the microcanonical ensemble.
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To support this assumption, Boltzmann presented a mechanical system that
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very same year (\cite{Bo868b}) as an example of an ergodic system. This
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system consists of a particle in two dimensions that is attracted to a
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fixed center via a gravitational potential $-\frac{\alpha}{2r}$. In
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addition, he added an extra centrifugal potential $\frac g{2r^2}$. As was
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known since at least the times of Kepler, this system is subjected to a
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central force, and is therefore integrable. In order to break the
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integrability, Boltzmann added an extra ingredient: a rigid infinite planar
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wall, located a finite distance away from the center (see figure
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\ref{trajectory}). Whenever the particle hits the wall, it undergoes an elastic
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collision and is reflected back. Boltzmann's argument was, roughly, that in
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the absence of the wall, the dynamics is quasi-periodic, so the particle
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should intersect the plane of the wall at points which should fill up a
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segment of the wall densely as the dynamics evolves, and concluded that the
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region of phase space in which the energy is constant must also be filled
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densely. As we will show, this is not the whole story; following a
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conjectured integrability for $g=0$, \cite[p.150]{Ga013b},
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and first tests
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%(by (GG) proposing a relation with KAM theory for $g>0$ and by (IJ) proposing chaotic motions at large $g$)
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in
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\cite[p.225--228]{Ga016}, we have found that, in the absence of the
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centrifugal term $g=0$, the dynamics (which has two degrees of freedom) still admits
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two constants of motion even in presence of the hard wall. This
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suggests that, if a suitable KAM analysis could be carried out, the system
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would not be ergodic for small values of $g$.
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\begin{figure}[ht]
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\hfil\includegraphics[width=6cm]{trajectory.pdf}
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\caption{A trajectory. The large dot is the attraction center $O$, and the line
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is the hard wall $\LL$. In between collisions, the trajectories are ellipses.
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The ellipses are drawn in full, but the part that is not covered by the
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particle is dashed.
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}
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\label{trajectory}
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\end{figure}
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\setcounter{section}{0}
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\section{Definition of the model and main result}
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\label{sec1}
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\iniz
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Let us now specify the model formally, and state our main result more
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precisely. We fix the gravitational center to the origin of the $x,y$-plane and
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let $\LL$ be the line $y=h$. The Hamiltonian for the system in between
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collisions is
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%
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\be H=\frac{p_x^2+p_y^2}2 -\frac{\a}{2r}+\frac{g}{2r^2}
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\Eq{e1.1}\ee
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%
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where $\a>0,g\ge0,r=\sqrt{x^2+y^2}$ and the particle moves following
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Hamilton's equations as long as it stays away from the obstacle $\LL$. When
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an encounter with $\LL$ occurs the particle is reflected elastically and
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continues on.
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\cite{Bo868b}, considered this system on the hyper-surface $A={\V
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p}^2-\frac\a r+\frac{g}{r^2}$. The intersection of this hyper-surface with
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$y=h$ is the region $\FF_A$ enclosed within the curves
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%
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\be \pm\sqrt{(A-\frac{g}{x^2+h^2} +\frac{\a}{\sqrt{x^2+h^2}})},\qquad
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x_{min}<x<x_{max}\Eq{e1.2}\ee
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%
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with $x_{min}$ and $x_{\max}$ the roots of
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$A=\frac{g}{x^2+h^2} -\frac{\a}{\sqrt{x^2+h^2}}$. He argued that
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all motions (with few exceptions) would cover densely the surfaces of
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constant $A<0$ if $\a,g>0$.
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\bigskip
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From now on, unless it is explicitly stated otherwise, we will assume that
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$g=0$.
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\bigskip
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In this case, the motion between collisions takes place at constant
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energy $\frac12A$ and constant angular momentum $a$, and traces out an
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ellipse. One of the foci of the ellipse is located at the origin, and we
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will denote the angle that the aphelion of the ellipse makes with the
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$x$-axis by $\theta_0$. Thus, the ellipse is entirely determined by the triplet
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$(A,a,\theta_0)$. When a collision occurs, $A$ remains unchanged, but $a$ and
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$\theta_0$ change discontinuously to values $(a',\theta_0')=\FF(a,\theta_0)$, and thus
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the Kepler ellipse of the trajectory changes. In addition, the semi-major
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axis $a_M$ of the ellipse is also fixed to $a_M=-\frac\a{2A}$ (Kepler's
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law): so the successive ellipses have the same semi-major axis, while the
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eccentricity varies because at each collision the angular momentum changes:
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$e^2=1+ \frac{4 A a^2}{\a^2}$. Thus, the motion will take place on arcs of
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various ellipses $\EE$, which all share the same focus and the same semi-major
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axis, but whose angle and eccentricity changes at each collision.
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Our main result is that the (canonical) map $(a',\f'_0)=\FF(a,\theta_0)$, which
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maps the angular momentum and angle of the aphelion before a collision to their
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values after the collision, admits a constant of motion. This follows from the
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following geometric lemma about ellipses.
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\bigskip
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\0{\bf Lemma 1:} {\it Given an ellipse $\mathcal E$ with a focus at $O$ that
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intersects $\LL$ at a point $P$. Let $Q$ denote the orthogonal projection of
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$O$ onto $\LL$ (see figure \ref{fig1}). The distance $R_0$ between $Q$ and
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the center of $\mathcal E$ depends solely on the semi-major axis $a_M$, the
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distance $r$ from $O$ to $P$, and $\cos(2\lambda)$ where $\lambda$ is the angle
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between the tangent of the ellipse at $P$ and $\LL$ (to define the direction of
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the tangent, we parametrize the ellipse in the counter-clockwise direction):
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\be
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R_0=\sqrt{\frac14 r^2+\frac14(2a_M-r)^2 +\frac12 r (2a_M-r)\cos(2\l)}
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.
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\Eq{e1.3}\ee
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}
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\*
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\begin{figure}[ht]
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\hfil\includegraphics[width=100pt]{fig1.pdf}
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\caption{The attractive center is $O$, hence it is the focus of the
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ellipse in absence of centrifugal force $g=0$. $Q$ is the projection of
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$O$ on the line $\LL$ and $P$ is a collision point. The arrow
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|
represents the velocity of the particle after the collision.}
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\label{fig1}
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\end{figure}
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\underline{Proof}: We switch to polar coordinates
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$p=(r\cos\f,r\sin\f)$.
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Let $O'$ denote the other focus of the ellipse, and $C$ denote its center. The
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first step is to compute the vector $\protect\overrightarrow{O'P}$, which in polar
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coordinates is
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\be
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\protect\overrightarrow{O'P}=((2a_M-r)\cos\f',(2a_M-r)\sin\f')\Eq{e1.4}\ee
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|
%
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Let $\psi:=\pi+\f-\lambda$ denote the angle between the tangent of the
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ellipse at $P$ and the vector $\protect\overrightarrow{PO}$ (see figure \ref{ellipse}),
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and $\psi':=\pi+\f'-\lambda$ denote the angle between the tangent of the
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ellipse at $P$ and the vector $\protect\overrightarrow{PO'}$.
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\begin{figure}[ht]
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\hfil\includegraphics[width=8cm]{ellipse.pdf}
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\caption{An ellipse with foci $O$ and $O'$ and center $C$. The thick line is
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|
$\LL$, which intersects the ellipse at $P$, and $Q$ is the projection of $O$
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|
onto $\LL$. The dashed line is the tangent at $P$. $\lambda$ is the angle
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|
between $\LL$ and the tangent, $\f$ is the polar coordinate, $\f'$ is the angle
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|
between $\LL$ and $\protect\overrightarrow{O'P}$. $\psi$ is the angle between the
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|
tangent and $\protect\overrightarrow{PO}$, which is equal to the angle between the
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|
tangent and $\protect\overrightarrow{PO'}$. $R_0$ is the distance between $Q$ and $C$.}
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\label{ellipse}
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|
\end{figure}
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By the focus-to-focus
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reflection property of ellipses, we have $\psi'=\pi-\psi$. Thus
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$\f'=2\lambda-\pi-\f$ and we find;
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|
\begin{figure}[ht]
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|
\hfil\includegraphics[width=8cm]{fig2.pdf}
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\caption{Two ellipses, before and after a collision. The collision line $\LL$
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|
is the line at $y=1$, $P$ is the collision point; $Q$ is the projection of $O$
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|
onto $\LL$; the two ellipses $\EE$ and $\EE'$ have a common focus $O$, and
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|
$O,O'$ are the foci of $\EE$, whereas $O,O''$ are the foci of $\EE'$; $C$ and
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$C''$ are the centers of $\EE$ and $\EE'$ respectively; the ellipses are drawn
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|
completely although the trajectory is restricted to the parts above $y=h=1$.
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|
The distance from $C''$ to $Q$ is the same as that from $C$ to $Q$. The upper
|
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|
ellipse $\EE$ contains the trajectory that starts at the collision point $P$
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|
|
following the other ellipse $\EE'$ which has undergone reflection.}
|
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|
\label{fig2}
|
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|
\end{figure}
|
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|
\be
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|
R_0^2=|Q-C|^2
|
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|
=
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|
\frac14\left(r^2+(2a_M-r)^2+2r(2a_M-r)\cos(2\lambda)\right)
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.
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|
\Eq{e1.5}\ee
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See figures \ref{ellipse} and \ref{fig2}.\qed
|
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|
\*
|
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|
\0{\bf Theorem 1}: {\it The quantity
|
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|
%
|
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|
|
\be R= a^2+h\a e \sin\theta_0\Eq{e1.6}
|
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|
|
\equiv\frac\alpha{2a_M}(h^2+a_M^2-R_0^2)
|
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|
|
\Eq{e1.7}\ee
|
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|
|
%
|
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|
|
where $e$ is the eccentricity $e=\sqrt{1+\frac{4 A a^2}{\a^2}}$, is a constant
|
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|
|
of motion.}
|
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|
|
\*
|
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|
|
\underline{Proof}:
|
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|
|
During a collision, the value of $\l$ changes from $\l$ to $\p-\l$, while
|
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|
|
|
$r$ and $a_M$ stay the same. By lemma 1, this implies that the distance $R_0$
|
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|
|
between $Q$ and the center of the ellipse is preserved during a collision.
|
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|
|
Furthermore, the position of the center $C$ of the ellipse is given by
|
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|
|
$C=a_Me(\cos\theta_0,\sin\theta_0)$
|
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|
|
so
|
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|
|
\be
|
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|
|
R_0^2=|Q-C|^2=a_M^2e^2-2a_Meh\sin\theta_0+h^2.\Eq{e1.8}
|
|
|
|
|
\ee
|
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|
|
Furthermore, the angular momentum is equal to
|
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|
|
$a^2=\frac12a_M\alpha(1-e^2)$
|
|
|
|
|
so
|
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|
|
|
\be
|
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|
|
|
-R_0^2+h^2+a_M^2
|
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|
|
=
|
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|
|
|
\frac{2a_M}\alpha(a^2+e\alpha h\sin\theta_0)\Eq{e1.9}
|
|
|
|
|
\ee
|
|
|
|
|
is a conserved quantity. \qed
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
\0{\bf Remark:} Some useful inequalities are
|
|
|
|
|
%
|
|
|
|
|
\be
|
|
|
|
|
\eqalign{
|
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|
|
|
&r_{max}<{2}{a_M}; \ x_{max}=\sqrt{r_{max}^2-h^2};\
|
|
|
|
|
R_0^2\in ((a_M-r)^2,a_M^2);\cr
|
|
|
|
|
&\frac{\a h^2}{2a_M}\,<\,R\,<\,
|
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|
|
|
(1+\frac{a_M^2}{h^2}-\Big(\frac{a_M}{h}-
|
|
|
|
|
\frac{r}h\Big)^2)\frac{\a h^2}{2a_M}\cr}
|
|
|
|
|
\Eq{e1.10}\ee
|
|
|
|
|
%
|
|
|
|
|
hence in the plane $(x,\l)$ the rectangle $(-x_{max},x_{max})\times(0,\p)$
|
|
|
|
|
(recall that $x_{max}$ is the largest $x$ accessible at energy $\frac12A$)
|
|
|
|
|
is the surface of energy $\frac12A$ and the trajectories are the curves of
|
|
|
|
|
constant $R$ inside this rectangle.
|
|
|
|
|
|
|
|
|
|
\def\SEC{Conjectures on action angle variables}
|
|
|
|
|
\section{\SEC}
|
|
|
|
|
\label{sec2}
|
|
|
|
|
\iniz
|
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|
|
|
|
|
|
|
|
In the previous section, we exhibited a constant of motion, which, along
|
|
|
|
|
with the conservation of energy, brings the number of independent conserved
|
|
|
|
|
quantities to two. In a continuous Hamiltonian system, this would imply the
|
|
|
|
|
existence of action-angle variables, which are canonically conjugate to the
|
|
|
|
|
position and momentum of the particle, in terms of which the dynamics
|
|
|
|
|
reduces to a linear evolution on a torus. In this case, the collision
|
|
|
|
|
with the wall introduces some discreteness into the problem, and the
|
|
|
|
|
existence of the action angle variables is not guaranteed by standard
|
|
|
|
|
theorems. Indeed, in the presence of the collisions, we no longer have a
|
|
|
|
|
Hamiltonian system, but rather a discrete symplectic map (or a
|
|
|
|
|
non-differentiable Hamiltonian), which describes the change in the state of
|
|
|
|
|
the particle during a collision. In this section, we present some
|
|
|
|
|
conjectures pertaining to the existence of action angle variables for this
|
|
|
|
|
problem.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
The first step is to change to variables which are action-angle variables for
|
|
|
|
|
the motion in between collision. We choose the {\it Delaunay} variables, whose
|
|
|
|
|
angles are the argument of the aphelion $\theta_0$ defined above, the {\it mean
|
|
|
|
|
anomaly} $M$, and whose actions are the angular momentum $a$, and another
|
|
|
|
|
momentum usually denoted by $L$ and related to the semi-major axis $a_M$ and
|
|
|
|
|
to the energy $E=\frac12 A$:
|
|
|
|
|
\be L:=-\sqrt{\frac{\alpha}2a_M},\quad a_M:=-\frac\alpha{2A}
|
|
|
|
|
,\quad
|
|
|
|
|
A:=p^2+\frac{a^2}{r^2}-\frac\alpha{r}\equiv-\frac{\alpha^2}{4L^2}
|
|
|
|
|
\Eq{e2.1}\ee
|
|
|
|
|
It is well known that this change of variables is canonical. In between
|
|
|
|
|
collisions, the dynamics of the particle in the variables
|
|
|
|
|
$(M,\theta_0;L,a)$ is, simply,
|
|
|
|
|
\be
|
|
|
|
|
\dot M=\frac{\alpha^2}{4L^3}
|
|
|
|
|
,\quad
|
|
|
|
|
\dot\theta_0=0
|
|
|
|
|
,\quad
|
|
|
|
|
\dot L=0
|
|
|
|
|
,\quad
|
|
|
|
|
\dot a=0
|
|
|
|
|
.\Eq{e2.2}
|
|
|
|
|
\ee
|
|
|
|
|
These variables are thus action-angle variables in between collisions, but when
|
|
|
|
|
a collision occurs, $\theta_0$ and $a$ will change.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
The following conjecture states that there exists an action-angle variable
|
|
|
|
|
during the collisions.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
\0{\bf Conjecture 1:} {\it There exists a variable $\gamma$ and an integer
|
|
|
|
|
$k$ such that, every $k$ collisions, the change in $\gamma$ is
|
|
|
|
|
\begin{equation}
|
|
|
|
|
\gamma'=\gamma+\o(L,R)\Eq{e2.3}
|
|
|
|
|
\end{equation}
|
|
|
|
|
in which case $\gamma$ is an angle that rotates on a circle of radius depending
|
|
|
|
|
on $L,R$. The function $\o(L,R)$ has a non zero derivative with respect to
|
|
|
|
|
$R$ at constant $L$, \ie the motion on the energy surface is quasi periodic
|
|
|
|
|
and anisochronous.}
|
|
|
|
|
\*
|
|
|
|
|
|
|
|
|
|
We will now sketch a construction of this variable $\gamma$, which we obtain
|
|
|
|
|
using a generating function $F(L,R,M,\theta_0)$.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
First of all, by theorem 1, the angular momentum $a(\theta_0)$ is a solution of
|
|
|
|
|
\begin{equation}
|
|
|
|
|
a^2=R-h\a\sin\theta_0\sqrt{1-\fra{a^2}{L^2}}\Eq{e2.4}
|
|
|
|
|
\end{equation}
|
|
|
|
|
that is, if $\e=\pm$,
|
|
|
|
|
%
|
|
|
|
|
\be
|
|
|
|
|
a^2=R-\frac{h^2\alpha^2}{2L^2}\sin^2\theta_0+
|
|
|
|
|
\e\sqrt{\frac{h^4\alpha^2}{4L^4}\sin^4\theta_0+h^2\a^2\sin^2\theta_0-
|
|
|
|
|
\frac{R\alpha^2h^2}{L^2}\sin^2\theta_0}\Eq{e2.5}\ee
|
|
|
|
|
%
|
|
|
|
|
and $a=\h\sqrt{a^2}$, so that there may be four possibilities for the value of
|
|
|
|
|
$a$ denoted $a=a_{\e,\h}(\theta_0,R,L)$ with $\e=\pm,\h=\pm$. The choice of the
|
|
|
|
|
signs $\e=\pm1$, and $\h$ must be examined carefully.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
We then define the generating function
|
|
|
|
|
\be F(L,R,M,\theta_0)=LM+\int_0^{\theta_0} a(L,R,\ps)d\ps
|
|
|
|
|
%-\int_0^{2\pi} a(L,R,\ps)d\ps
|
|
|
|
|
\Eq{e2.6}\ee
|
|
|
|
|
%
|
|
|
|
|
which yields the following canonical transformation:
|
|
|
|
|
|
|
|
|
|
\be\eqalign{
|
|
|
|
|
\g=&\dpr_R\int^{\theta_0}_0 a_{\e,\h}(L,R,\ps)\,d\ps
|
|
|
|
|
%-\dpr_R\int^{2\pi}_0 a_{\e,\h}(L,R,\ps)\,d\ps
|
|
|
|
|
\cr
|
|
|
|
|
M'=&M+\dpr_L\int^{\theta_0}_0 a_{\e,\h}(L,R,\ps)\,d\ps
|
|
|
|
|
%-\dpr_L\int^{2\pi}_0 a_{\e,\h}(L,R,\ps)\,d\ps
|
|
|
|
|
\cr
|
|
|
|
|
}\Eq{e2.7}\ee
|
|
|
|
|
%
|
|
|
|
|
It is natural, if Boltzmann's system is integrable (at $g=0$), that the new
|
|
|
|
|
variables are its action angle variables and $M',\gamma$ rotate uniformly in spite of
|
|
|
|
|
the collisions.
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
However, in this case, the signs $\e$ and $\h$ may change from one collision to
|
|
|
|
|
the next, complicating the situation. A careful numerical study of the system
|
|
|
|
|
has led us to the following conjecture (see figure \ref{action_angle}).
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
%
|
|
|
|
|
\0{\bf Conjecture 2:} {\it If $R>h\a$ (which is the case in which the
|
|
|
|
|
circle, of radius $R_0$, of the centers encloses the focus $O$), when the
|
|
|
|
|
motion collides for the $n$-th time, the angular momentum is proportional
|
|
|
|
|
to $(-1)^n$, and, thus, $\epsilon=(-1)^n$. The sign $\eta$ is fixed to
|
|
|
|
|
$+$. The increment $\Delta_2\gamma$ in $\gamma$ between the $n$-th and
|
|
|
|
|
the $n+2$-th collision is independent of $n$. } \*
|
|
|
|
|
\bigskip
|
|
|
|
|
|
|
|
|
|
\begin{figure}
|
|
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\hfil\includegraphics[width=8cm]{action-angle.pdf}
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\caption{A plot of the increment in $\gamma$ between the $n$-th and the $n+2$-nd collision as a function of $n$. The blue `+' signs correspond to even $n$, and the red `$\times$' to odd $n$. The variation of $\Delta_2\gamma$ is as small as 1 part per million, thus supporting conjecture 2.}
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\label{action_angle}
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\end{figure}
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\0{\it Remark:} The change of variables over the variables $a,\theta_0$ to
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$R,\g$ at fixed $L$ is {\it remarkably} essentially the same as the one
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(\ap\ unrelated) to find action-angle variable for the auxiliary
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Hamiltonian $R=R(a,\theta_0)$. This might remain true even when $R<h\a$:
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interpretable as a kind of auxiliary pendulum motion. \*
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At the time of publication, it has been
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brought to our attention that G. Felder has proved that the orbits are all
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either periodic or quasi-periodic, which would be implied from conjecture
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1.
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\section{Conclusion and outlook}
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In this brief note, we have shown that the system considered by Boltzmann in
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1868, in the case $g=0$, admits two independent constants of motion.
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This indicates that it should be possible to compute action
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angle variables for this system, which is not entirely trivial because of the
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discontinuous nature of the collision process. If such a construction could be
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brought to its conclusion, then it would show that the trajectories are either
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periodic or quasi-periodic, a fact which is consistent with the numerical
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simulations we have run.
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This is not a contradiction of Boltzmann's claim that this model is ergodic,
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since Boltzmann considered the model at $g\neq 0$. However, we expect that a
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KAM-type argument can be set up for this model, to show that the system cannot
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be ergodic, even if $g>0$, provided $g$ is sufficiently small. However it
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may still have invariant regions of positive volume where the motion is ergodic.
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\*
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\0{\bf Acknowledgements}: The authors thank G. Felder for giving us the impetus
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to write this note up in its current form, and to publish it. I.J. gratefully
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acknowledges support from NSF grants 31128155 and 1802170.
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\bibliographystyle{plainnat}
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%\bibliographystyle{alpha}
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%\bibliographystyle{amsref}
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%\bibliographystyle{apsrmp}
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%\bibliographystyle{spmpsci}
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%\bibliographystyle{annotate}
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%\bibliography{0Bib}
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\begin{thebibliography}{3}
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\providecommand{\natexlab}[1]{#1}
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\providecommand{\url}[1]{\texttt{#1}}
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\expandafter\ifx\csname urlstyle\endcsname\relax
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\providecommand{\doi}[1]{doi: #1}\else
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\providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi
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\bibitem[Boltzmann(1868a)]{Bo868a}
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L.~Boltzmann.
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\newblock Studien \"uber das Gleichgewicht der lebendingen Kraft zwischen bewegten materiellen Punkten.
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\newblock \emph{Wiener Berichte}, {\bf 58}, 517--560, (49--96), 1868.
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\bibitem[Boltzmann(1868b)]{Bo868b}
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L.~Boltzmann.
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\newblock L{\"o}sung eines mechanischen problems.
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\newblock \emph{Wiener Berichte}, {\bf 58}, (W.A.,\#6):\penalty0 1035--1044,
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(97--105), 1868.
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\bibitem[Gallavotti(2014)]{Ga013b}
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G.~Gallavotti.
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\newblock \emph{Nonequilibrium and irreversibility}.
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\newblock Theoretical and Mathematical Physics. Springer-Verlag, 2014.
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\bibitem[Gallavotti(2016)]{Ga016}
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G.~Gallavotti.
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\newblock Ergodicity: a historical perspective. equilibrium and nonequilibrium.
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\newblock \emph{European Physics Journal H}, {\bf 41}, 181--259, 2016.
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\newblock \doi{DOI: 10.1140/epjh/e2016-70030-8}.
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\end{thebibliography}
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\*\*
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\end{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###################
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%plot "lambdak-0.25-3.4--0.3333" u 1:2 every 2::1:::2000
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% plot 'file' every {<point_incr>}
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% {:{<block_incr>}
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% {:{<start_point>}
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% {:{<start_block>}
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% {:{<end_point>}
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% {:<end_block>}}}}}
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%plot "grafk-0.326753-2--0.3333" u 3:4 every 2:1:0:0:5:0 w l
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%plot "gammak-0.3-1--0.2" u 1:3 every 2:1:0:0:2:0 w l
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% Syntax:
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% plot 'file' every {<point_incr>}
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% {:{<block_incr>}
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% {:{<start_point>}
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% {:{<start_block>}
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% {:{<end_point>}
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% {:<end_block>}}}}}
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%
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% every 3:1:2::1024:1
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% 2 significa che inizia dal #3
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