667 lines
26 KiB
TeX
667 lines
26 KiB
TeX
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\date{}


\author{\alertb{Giovanni Gallavotti${}^1$ and


Ian Jauslin${}^2$}}




\title{\alertr{\bf A Theorem on Ellipses, an Integrable System and a Theorem


of Boltzmann}


}




\begin{document}


\maketitle


\kern8mm


\centerline{${}^1$ INFNRoma1 \& Universit\`a ``La Sapienza'', email: giovanni.gallavotti@roma1.infn.it}


\centerline{${}^2$ Department of Physics, Princeton University, email: ijauslin@princeton.edu}






%\kern1cm


\begin{abstract}


%


We study a mechanical system that was considered by Boltzmann in 1868 in the


context of the derivation of the canonical and microcanonical ensembles. This


system was introduced as an example of ergodic dynamics, which was central to


Boltzmann's derivation. It consists of a single particle in two dimensions,


which is subjected to a gravitational attraction to a fixed center. In


addition, an infinite plane is fixed at some finite distance from the center,


which acts as a hard wall on which the particle collides elastically. Finally,


an extra centrifugal force is added. We will show that, in the absence of this


extra centrifugal force, there are two independent integrals of motion.


Therefore the extra centrifugal force is necessary for Boltzmann's claim of


ergodicity to hold.


%


\end{abstract}


\*


%


\0Keywords: {\small Ergodicity, Chaotic hypothesis, Gibbs


distributions, Boltzmann, Integrable systems}


\*


%\end{document}




In 1868, \cite{Bo868a} laid the foundations for our modern understanding of the


behavior of manyparticle systems by introducing the ``microcanonical


ensemble'' (for more details on this history, see \cite{Ga016}). The principal


idea behind this ensemble is that one can achieve a good understanding of


manyparticle systems by focusing not on the dynamics of each individual


particle, but on the statistical properties of the whole. More precisely, the


state of the system becomes a random variable, chosen according to a


probability distribution on phase space, which came to be called the


``microcanonical ensemble''. An important assumption that was made implicitly


by Boltzmann is that the dynamics of the system be ergodic. In this case,


timeaverages of the dynamics can be rewritten as averages over phase space,


and the qualitative properties of the dynamics can be formulated as statistical


properties of the microcanonical ensemble.




To support this assumption, Boltzmann presented a mechanical system that


very same year (\cite{Bo868b}) as an example of an ergodic system. This


system consists of a particle in two dimensions that is attracted to a


fixed center via a gravitational potential $\frac{\alpha}{2r}$. In


addition, he added an extra centrifugal potential $\frac g{2r^2}$. As was


known since at least the times of Kepler, this system is subjected to a


central force, and is therefore integrable. In order to break the


integrability, Boltzmann added an extra ingredient: a rigid infinite planar


wall, located a finite distance away from the center (see figure


\ref{trajectory}). Whenever the particle hits the wall, it undergoes an elastic


collision and is reflected back. Boltzmann's argument was, roughly, that in


the absence of the wall, the dynamics is quasiperiodic, so the particle


should intersect the plane of the wall at points which should fill up a


segment of the wall densely as the dynamics evolves, and concluded that the


region of phase space in which the energy is constant must also be filled


densely. As we will show, this is not the whole story; following a


conjectured integrability for $g=0$, \cite[p.150]{Ga013b},


and first tests


%(by (GG) proposing a relation with KAM theory for $g>0$ and by (IJ) proposing chaotic motions at large $g$)


in


\cite[p.225228]{Ga016}, we have found that, in the absence of the


centrifugal term $g=0$, the dynamics (which has two degrees of freedom) still admits


two constants of motion even in presence of the hard wall. This


suggests that, if a suitable KAM analysis could be carried out, the system


would not be ergodic for small values of $g$.




\begin{figure}[ht]


\hfil\includegraphics[width=6cm]{trajectory.pdf}




\caption{A trajectory. The large dot is the attraction center $O$, and the line


is the hard wall $\LL$. In between collisions, the trajectories are ellipses.


The ellipses are drawn in full, but the part that is not covered by the


particle is dashed.


}


\label{trajectory}


\end{figure}






\setcounter{section}{0}


\section{Definition of the model and main result}


\label{sec1}


\iniz






Let us now specify the model formally, and state our main result more


precisely. We fix the gravitational center to the origin of the $x,y$plane and


let $\LL$ be the line $y=h$. The Hamiltonian for the system in between


collisions is


%


\be H=\frac{p_x^2+p_y^2}2 \frac{\a}{2r}+\frac{g}{2r^2}


\Eq{e1.1}\ee


%


where $\a>0,g\ge0,r=\sqrt{x^2+y^2}$ and the particle moves following


Hamilton's equations as long as it stays away from the obstacle $\LL$. When


an encounter with $\LL$ occurs the particle is reflected elastically and


continues on.




\cite{Bo868b}, considered this system on the hypersurface $A={\V


p}^2\frac\a r+\frac{g}{r^2}$. The intersection of this hypersurface with


$y=h$ is the region $\FF_A$ enclosed within the curves


%


\be \pm\sqrt{(A\frac{g}{x^2+h^2} +\frac{\a}{\sqrt{x^2+h^2}})},\qquad


x_{min}<x<x_{max}\Eq{e1.2}\ee


%


with $x_{min}$ and $x_{\max}$ the roots of


$A=\frac{g}{x^2+h^2} \frac{\a}{\sqrt{x^2+h^2}}$. He argued that


all motions (with few exceptions) would cover densely the surfaces of


constant $A<0$ if $\a,g>0$.


\bigskip




From now on, unless it is explicitly stated otherwise, we will assume that


$g=0$.


\bigskip




In this case, the motion between collisions takes place at constant


energy $\frac12A$ and constant angular momentum $a$, and traces out an


ellipse. One of the foci of the ellipse is located at the origin, and we


will denote the angle that the aphelion of the ellipse makes with the


$x$axis by $\theta_0$. Thus, the ellipse is entirely determined by the triplet


$(A,a,\theta_0)$. When a collision occurs, $A$ remains unchanged, but $a$ and


$\theta_0$ change discontinuously to values $(a',\theta_0')=\FF(a,\theta_0)$, and thus


the Kepler ellipse of the trajectory changes. In addition, the semimajor


axis $a_M$ of the ellipse is also fixed to $a_M=\frac\a{2A}$ (Kepler's


law): so the successive ellipses have the same semimajor axis, while the


eccentricity varies because at each collision the angular momentum changes:


$e^2=1+ \frac{4 A a^2}{\a^2}$. Thus, the motion will take place on arcs of


various ellipses $\EE$, which all share the same focus and the same semimajor


axis, but whose angle and eccentricity changes at each collision.




Our main result is that the (canonical) map $(a',\f'_0)=\FF(a,\theta_0)$, which


maps the angular momentum and angle of the aphelion before a collision to their


values after the collision, admits a constant of motion. This follows from the


following geometric lemma about ellipses.


\bigskip




\0{\bf Lemma 1:} {\it Given an ellipse $\mathcal E$ with a focus at $O$ that


intersects $\LL$ at a point $P$. Let $Q$ denote the orthogonal projection of


$O$ onto $\LL$ (see figure \ref{fig1}). The distance $R_0$ between $Q$ and


the center of $\mathcal E$ depends solely on the semimajor axis $a_M$, the


distance $r$ from $O$ to $P$, and $\cos(2\lambda)$ where $\lambda$ is the angle


between the tangent of the ellipse at $P$ and $\LL$ (to define the direction of


the tangent, we parametrize the ellipse in the counterclockwise direction):


\be


R_0=\sqrt{\frac14 r^2+\frac14(2a_Mr)^2 +\frac12 r (2a_Mr)\cos(2\l)}


.


\Eq{e1.3}\ee


}


\*




\begin{figure}[ht]


\hfil\includegraphics[width=100pt]{fig1.pdf}




\caption{The attractive center is $O$, hence it is the focus of the


ellipse in absence of centrifugal force $g=0$. $Q$ is the projection of


$O$ on the line $\LL$ and $P$ is a collision point. The arrow


represents the velocity of the particle after the collision.}


\label{fig1}


\end{figure}




\underline{Proof}: We switch to polar coordinates


$p=(r\cos\f,r\sin\f)$.




Let $O'$ denote the other focus of the ellipse, and $C$ denote its center. The


first step is to compute the vector $\protect\overrightarrow{O'P}$, which in polar


coordinates is


\be


\protect\overrightarrow{O'P}=((2a_Mr)\cos\f',(2a_Mr)\sin\f')\Eq{e1.4}\ee


%


Let $\psi:=\pi+\f\lambda$ denote the angle between the tangent of the


ellipse at $P$ and the vector $\protect\overrightarrow{PO}$ (see figure \ref{ellipse}),


and $\psi':=\pi+\f'\lambda$ denote the angle between the tangent of the


ellipse at $P$ and the vector $\protect\overrightarrow{PO'}$.




\begin{figure}[ht]


\hfil\includegraphics[width=8cm]{ellipse.pdf}




\caption{An ellipse with foci $O$ and $O'$ and center $C$. The thick line is


$\LL$, which intersects the ellipse at $P$, and $Q$ is the projection of $O$


onto $\LL$. The dashed line is the tangent at $P$. $\lambda$ is the angle


between $\LL$ and the tangent, $\f$ is the polar coordinate, $\f'$ is the angle


between $\LL$ and $\protect\overrightarrow{O'P}$. $\psi$ is the angle between the


tangent and $\protect\overrightarrow{PO}$, which is equal to the angle between the


tangent and $\protect\overrightarrow{PO'}$. $R_0$ is the distance between $Q$ and $C$.}


\label{ellipse}


\end{figure}




By the focustofocus


reflection property of ellipses, we have $\psi'=\pi\psi$. Thus


$\f'=2\lambda\pi\f$ and we find;






\begin{figure}[ht]


\hfil\includegraphics[width=8cm]{fig2.pdf}




\caption{Two ellipses, before and after a collision. The collision line $\LL$


is the line at $y=1$, $P$ is the collision point; $Q$ is the projection of $O$


onto $\LL$; the two ellipses $\EE$ and $\EE'$ have a common focus $O$, and


$O,O'$ are the foci of $\EE$, whereas $O,O''$ are the foci of $\EE'$; $C$ and


$C''$ are the centers of $\EE$ and $\EE'$ respectively; the ellipses are drawn


completely although the trajectory is restricted to the parts above $y=h=1$.


The distance from $C''$ to $Q$ is the same as that from $C$ to $Q$. The upper


ellipse $\EE$ contains the trajectory that starts at the collision point $P$


following the other ellipse $\EE'$ which has undergone reflection.}


\label{fig2}


\end{figure}




\be


R_0^2=QC^2


=


\frac14\left(r^2+(2a_Mr)^2+2r(2a_Mr)\cos(2\lambda)\right)


.


\Eq{e1.5}\ee


See figures \ref{ellipse} and \ref{fig2}.\qed




\*


\0{\bf Theorem 1}: {\it The quantity


%


\be R= a^2+h\a e \sin\theta_0\Eq{e1.6}


\equiv\frac\alpha{2a_M}(h^2+a_M^2R_0^2)


\Eq{e1.7}\ee


%


where $e$ is the eccentricity $e=\sqrt{1+\frac{4 A a^2}{\a^2}}$, is a constant


of motion.}


\*




\underline{Proof}:


During a collision, the value of $\l$ changes from $\l$ to $\p\l$, while


$r$ and $a_M$ stay the same. By lemma 1, this implies that the distance $R_0$


between $Q$ and the center of the ellipse is preserved during a collision.


Furthermore, the position of the center $C$ of the ellipse is given by


$C=a_Me(\cos\theta_0,\sin\theta_0)$


so


\be


R_0^2=QC^2=a_M^2e^22a_Meh\sin\theta_0+h^2.\Eq{e1.8}


\ee


Furthermore, the angular momentum is equal to


$a^2=\frac12a_M\alpha(1e^2)$


so


\be


R_0^2+h^2+a_M^2


=


\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{


&r_{max}<{2}{a_M}; \ x_{max}=\sqrt{r_{max}^2h^2};\


R_0^2\in ((a_Mr)^2,a_M^2);\cr


&\frac{\a h^2}{2a_M}\,<\,R\,<\,


(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




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 actionangle 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


nondifferentiable 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 actionangle 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 semimajor 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 actionangle variables in between collisions, but when


a collision occurs, $\theta_0$ and $a$ will change.


\bigskip




The following conjecture states that there exists an actionangle 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=Rh\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}


\hfil\includegraphics[width=8cm]{actionangle.pdf}


\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.}


\label{action_angle}


\end{figure}




\0{\it Remark:} The change of variables over the variables $a,\theta_0$ to


$R,\g$ at fixed $L$ is {\it remarkably} essentially the same as the one


(\ap\ unrelated) to find actionangle variable for the auxiliary


Hamiltonian $R=R(a,\theta_0)$. This might remain true even when $R<h\a$:


interpretable as a kind of auxiliary pendulum motion. \*




At the time of publication, it has been


brought to our attention that G. Felder has proved that the orbits are all


either periodic or quasiperiodic, which would be implied from conjecture


1.




\section{Conclusion and outlook}


In this brief note, we have shown that the system considered by Boltzmann in


1868, in the case $g=0$, admits two independent constants of motion.


This indicates that it should be possible to compute action


angle variables for this system, which is not entirely trivial because of the


discontinuous nature of the collision process. If such a construction could be


brought to its conclusion, then it would show that the trajectories are either


periodic or quasiperiodic, a fact which is consistent with the numerical


simulations we have run.




This is not a contradiction of Boltzmann's claim that this model is ergodic,


since Boltzmann considered the model at $g\neq 0$. However, we expect that a


KAMtype argument can be set up for this model, to show that the system cannot


be ergodic, even if $g>0$, provided $g$ is sufficiently small. However it


may still have invariant regions of positive volume where the motion is ergodic.




\*






\0{\bf Acknowledgements}: The authors thank G. Felder for giving us the impetus


to write this note up in its current form, and to publish it. I.J. gratefully


acknowledges support from NSF grants 31128155 and 1802170.




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\newblock Studien \"uber das Gleichgewicht der lebendingen Kraft zwischen bewegten materiellen Punkten.


\newblock \emph{Wiener Berichte}, {\bf 58}, 517560, (4996), 1868.




\bibitem[Boltzmann(1868b)]{Bo868b}


L.~Boltzmann.


\newblock L{\"o}sung eines mechanischen problems.


\newblock \emph{Wiener Berichte}, {\bf 58}, (W.A.,\#6):\penalty0 10351044,


(97105), 1868.




\bibitem[Gallavotti(2014)]{Ga013b}


G.~Gallavotti.


\newblock \emph{Nonequilibrium and irreversibility}.


\newblock Theoretical and Mathematical Physics. SpringerVerlag, 2014.




\bibitem[Gallavotti(2016)]{Ga016}


G.~Gallavotti.


\newblock Ergodicity: a historical perspective. equilibrium and nonequilibrium.


\newblock \emph{European Physics Journal H}, {\bf 41}, 181259, 2016.


\newblock \doi{DOI: 10.1140/epjh/e2016700308}.




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