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\begin{document}
\title{Entropy, Lyapunov exponents and mean free path for billiards}
\author{N. Chernov
\\ Department of Mathematics\\
University of Alabama at Birmingham\\
Birmingham, AL 35294, USA\\
E-mail: chernov@vorteb.math.uab.edu
}
\date{\today}
\maketitle
\begin{abstract}
We review known results and derive some new ones about
the mean free path, Kolmogorov-Sinai entropy and Lyapunov
exponents for billiard type dynamical systems. We focus on
exact and asymptotic formulas for these quantities. The
dynamical systems covered in the paper include periodic
Lorentz gas, stadium and its modifications, and the gas
of hard balls, for which we prove rigorously the classical
Boltzmann formula for the intercollision time. Some open
questions and numerical observations are discussed.
\end{abstract}
{\em Keywords}: Billiards, hard balls, Lorentz gas, entropy,
mean free path, Lyapunov exponents.
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\section{Basic facts about billiards}
\label{secBFB}
\setcounter{equation}{0}
Hamiltonian systems with elastic collisions or specular
reflections are often called billiards. In particular,
these include gases of hard balls, Lorentz gases, and stadia.
Below we recall basic definitions and facts about billiards.
Let $Q$ be a compact closed connected domain in $\IR^d$
or on a $d$-torus $\IT^d=\IR^d/\ZZ^d$. Let the boundary
$\partial Q$ be a finite union of smooth (of class $C^3$)
compact manifolds of codimension one, $\partial Q=\Gamma_1\cup\cdots
\cup\Gamma_r$, $r\geq 1$. We call $Q$ a billiard table
and $\partial Q$ its wall. Let the set
$$
\Gamma^{\ast}=\cup_{i\neq j}(\Gamma_i\cap\Gamma_j)
$$
be a finite union of smooth compact submanifolds of
codimension $\geq 2$. This set includes all the corner
points of the wall $\partial Q$. We will call it the
singular part of $\partial Q$. The points $q\in (\partial Q)
\setminus\Gamma^\ast$ are said to be regular.
The billiard dynamical system in $Q$ is generated by
the free motion of a pointlike particle at unit speed
in the table $Q$ with specular reflections at the
wall $\partial Q$. The reflection rule ``the angle of
incidence equals the angle of reflection'' is specified
by the equation\footnote{Here and on $(u\cdot v)$ means
the scalar product of vectors $u$ and $v$.}
\be
v_+=v_--2\, (n(q)\cdot v_-)\, n(q)
\label{rr}
\ee
where $v_-$ and $v_+$ are the incoming and outgoing
velocity vectors, and $n(q)$ is the inward unit normal
vector to the wall $\partial Q$ at the point of reflection $q$.
The vector $n(q)$ is well defined at all regular points
of $\partial Q$. If the particle hits the singular set
$\Gamma^\ast$ (a corner point of the wall), its further
trajectory is not defined.
The phase space of the billiard system is
$M=Q\times S^{d-1}$, where $S^{d-1}$ is the unit
sphere of velocity vectors. We denote phase points
by $x=(q,v)$, where $q$ is the configuration point
(the position of the billiard particle in $Q$), and
$v$ is its velocity vector. We denote by $\Pi_Q$
and $\Pi_V$ the natural projections of $M$ onto
$Q$ and $S^{d-1}$, respectively. The billiard dynamics
generates a flow, $\Phi^t$, on $M$. This flow
preserves the normalized Liouville measure $d\mu= c_{\mu}\,
dq\, dv$, where $dq$ and $dv$ are the Lebesgue measures
on $Q$ and $S^{d-1}$, respectively, and
$$
c_{\mu}=\left (|Q|\cdot |S^{d-1}|\right )^{-1}
$$
is the normalizing factor. Here $|Q|$ is the volume
of the domain $Q$ and
$$
|S^{d-1}|=\frac{2\pi^{d/2}}{\Gamma(d/2)}
$$
is the $(d-1)$-dimensional volume of the unit sphere
in $\IR^d$. Here $\Gamma(x)$ is the gamma function,
$\Gamma(n+1)=n!$, $\Gamma(x+1)=x\Gamma(x)$, and
$\Gamma(1/2)=\sqrt{\pi}$.
The billiard flow $\Phi^t$ has a natural cross-section
associated to the wall of the billiard table. Let
$$
\Omega = \left\{ (q,v)\in M:\, q\in\partial Q\ \
{\rm and}\ \ (v\cdot n(q))\geq 0\right\}
$$
The first return map $T:\Omega\to\Omega$ takes a point
$x\in\Omega$ to the point on the trajectory of $x$
immediately after its first reflection in $\partial Q$.
This is called the billiard ball map.
The map $T$ preserves the probability measure
$d\nu=c_{\nu}\, (v\cdot n(q))\, dq\, dv$, where
$dq$ is now the Lebesgue measure on $\partial Q$,
and $c_{\nu}$ is the normalizing factor. It is
a simple calculation, cf. \cite{Ch91}, that
$$
c_{\nu}=\left (|\partial Q|\cdot |B^{d-1}|\right )^{-1}
$$
Here $|\partial Q|$ is the $(d-1)$-dimensional volume
of the wall $\partial Q$, and $|B^{d-1}|=|S^{d-2}|
/(d-1)$ is the volume of the unit ball in $\IR^{d-1}$.
For every phase point $x=(q,v)\in M$ let
$\tau_+(x)=\min\{t>0:\, \Phi^{t+0}x\in \Omega\}$ and
$\tau_-(x)=\max\{t<0:\, \Phi^{t+0}x\in \Omega\}$ be the
first positive and negative moments of reflection,
respectively. We then have two canonical projections
of $M$ onto $\Omega$: $\tilde{\pi}_{\pm}x=\Phi^{\tau_{\pm}
(x)+0}x$. Note that $T^{\pm 1}x=\tilde{\pi}_{\pm}x$ for
$x\in\Omega$.
Any point $x=(q,v)\in M$ can be specified by
the point $\tilde{x}=(\tilde{q},v)=
\tilde{\pi}_-(x)\in\Omega$ and the positive number
$t=|\tau_-(x)|\geq 0$. In this way $(\tilde{q},v,t)$ make a
coordinate system in $M$, with $\tilde{q}\in\partial Q$,
$v\in S^{d-1}$, and $t>0$. In these coordinates
\be
d\mu = c_{\mu}\, (v\cdot n(\tilde{q}))\, d\tilde{q}\, dv\, dt
\label{change}
\ee
where $d\tilde{q}$ is the Lebesgue measure on $\partial Q$,
cf. \cite{Ch91}. Actually, it is Eq. (\ref{change}) on which
the invariance of the measure $\nu$ under $T$ is based,
see also \cite{Ch91}.
\section{Mean free path}
\label{secMFP}
\setcounter{equation}{0}
For every phase point $x\in M$ the free path is the distance
the billiard particle covers before it collides with
$\partial Q$. Since the speed of the particle is unit,
the free path is $\tau(x)=\tau_+(x)=\min\{t>0:\, \Phi^{t+0}x\in \Omega\}$.
As for the {\em mean} free path, there is a natural ambiguity in
this concept, since one can integrate $\tau(x)$ with respect to
either the Liouville measure $\mu$ or the ``boundary'' measure
$\nu$ arriving at two different values.
It is, however, more sensible and traditional to define the mean free path by
\be
\bar{\tau}=\int_\Omega\tau(x)\, d\nu(x)
\label{bartaud}
\ee
which is the definition we adopt here. The main reason
for this is the `dynamical' interpretation of $\bar{\tau}$:
it is the time average of the free paths along typical
trajectories. Precisely, the Birkhoff ergodic theorem
implies that for almost every $x\in\Omega$ we have
\be
\frac {\tau(x)+\tau(Tx)+\cdots +\tau(T^{n-1}x)}{n}
\to\hat{\tau}(x)\ \ \ \ {\rm as}\ \ n\to\infty
\label{hattau1}
\ee
Then the mean value of $\hat{\tau}(x)$ coincides with $\bar{\tau}$:
\be
\int_{\Omega}\hat{\tau}(x)\, d\nu(x) = \bar{\tau}
\label{hattau2}
\ee
If the billiard ball map $T$ is ergodic, then
we simply have $\hat{\tau}(x)=\bar{\tau}$ a.e.
There is a remarkably simple formula for the mean free path
in {\em any} billiard table $Q$ in terms of its geometric
parameters:
\be
\bar{\tau}=\frac{|Q|\cdot |S^{d-1}|}{|\partial Q|\cdot |B^{d-1}|}
\ \left (=\frac{c_{\nu}}{c_{\mu}}\right )
\label{bartau}
\ee
This follows from Eq. (\ref{change}):
$$
\int_{\Omega}\tau(x)\, d\nu(x) =
c_{\nu}\int_{\Omega}\tau(x)\, (v\cdot n(\tilde{q}))\, d\tilde{q}\, dv =
c_{\nu}\int_M(v\cdot n(\tilde{q}))\, d\tilde{q}\, dv\, dt = c_{\nu}/c_{\mu}
$$
In particular, for planar billiard tables $d=2$, and we have
\be
\bar{\tau}=\frac{\pi\, |Q|}{|\partial Q|}
\label{bartau2}
\ee
and for 3-D billiard tables we have
\be
\bar{\tau}=\frac{4\, |Q|}{|\partial Q|}
\label{bartau3}
\ee
We emphasize that (\ref{bartau})-(\ref{bartau3}) are
{\em exact} formulas.
The formulas (\ref{bartau})-(\ref{bartau3}) with the
definition (\ref{bartaud}) of $\bar{\tau}$ are
known in integral geometry and geometric probability,
see, e.g., Eq. (4-3-4) in \cite{Ma}. Eq. (\ref{bartau2}) is often
referred to as Santalo's formula, since it is given
in Santalo's book \cite{Sa}.
The formulas (\ref{bartau})-(\ref{bartau3}) with the
dynamical interpretation (\ref{hattau1})-(\ref{hattau2})
of $\bar{\tau}$ have been also discussed
and derived from (\ref{change}) at Moscow seminar on dynamical
systems directed by Sinai and Alekseev in the seventies.
Regrettably, it has never been considered to be worth publishing, so
I could not locate any reference to it until mid-eighties.
Wojtkowski proved the 2-D formula (\ref{bartau2}) based on Eq. (\ref{change})
in Ref. \cite{W88}. A proof of Eq. (\ref{bartau}) in any dimension
based on (\ref{change}) was given in \cite{Ch91}.
Another proof of (\ref{bartau}), based on Green's theorem
in vector analysis, was independently found by Golse
(private communication \cite{Go}). Unfortunately, the lack of references
to the above formulas leads to repeated attempts by various
researches to estimate $\bar{\tau}$ numerically
or heuristically for particular billiard tables.
This section is intended to help fill this gap.
\section{Entropy and Lyapunov exponents}
\label{secELE}
\setcounter{equation}{0}
Billiards belong to a rather general class of smooth dynamical
systems with singularities. An exact definition of this class
and its extensive study may be found in the book by Katok and
Strelcyn \cite{KS}. The main results of that book are the
existence of local stable and unstable
manifolds and exact formulas for the entropy. It was also
shown in that book that the billiard ball map in a planar billiard
table under certain very mild conditions belongs in the class
of smooth maps with singularities. For multidimensional
billiards, the same was shown in \cite{Ch91} under the
assumption that the sectional curvature of the smooth components
of the boundary $\partial Q$ is $C^2$ smooth up to the
singular set $\Gamma^{\ast}$. In particular, the sectional
curvature must be uniformly bounded. This assumption is pretty
mild and possibly can be relaxed.
Thus, all the results of the book \cite{KS} carry over
to generic billiards. These produce the following three theorems.
\begin{theorem}[Lyapunov exponents]
For $\nu$-almost every point $x\in\Omega$ there is a
$DT$-invariant decomposition of the tangent space
$$
{\cal T}_x\Omega = \oplus_{i=1}^{s(x)}H_i(x)
$$
such that, uniformly in the vectors $w\in H_i(x)$,
$||w||=1$, we have the limit
$$
\lim_{n\to\pm\infty} n^{-1}\ln ||DT^n_x(w)||=\chi_i(x)
$$
for $i=1,\ldots, s(x)$. Here $\chi_1(x)<\chi_2(x)<
\cdots <\chi_{s(x)}(x)$ are Lyapunov exponents of the
billiard ball map $T$ at the point $x$.
\end{theorem}
{\em Remark}. This theorem immediately follows from the Oseledec
multiplicative theorem, see \cite{KS}, and the following fact:
$$
\int_{\Omega}\log^+||DT^{\pm}_x||\, d\nu(x)<\infty
$$
where $\log^+a=\max\{\log a,0\}$.\medskip
{\em Remark}. The functions $s(x)$, $\chi_i(x)$ and
dim$\, H_i(x)$ (the multiplicity of
the exponent $\chi_i(x)$) are invariant under both $T$
and $T^{-1}$. In particular, if the billiard ball map
$T$ is ergodic, then these functions are a.e. constant
on $\Omega$. \medskip
As a corollary to the above theorem, there is an a.e.
$DT$-invariant decomposition
$$
{\cal T}_x\Omega = E^u_x\oplus E^s_x\oplus E^0_x
$$
where the subspaces $E^u_x$, $E^s_x$ and $E^0_x$
correspond to positive, negative and zero exponents,
respectively. Any vector $w\in E^u_x$ under the iterations
of $DT$ exponentially grows in the future and exponentially
contracts in the past. A time-symmetric statement holds for
vectors $w\in E^s_x$. The space $E^u_x$ is said to be
unstable, $E^s_x$ stable and $E^0_x$ neutral. We
denote $d^u(x)={\rm dim}\, E^u_x$ and
$d^s(x)={\rm dim}\, E^s_x$.
If positive and negative Lyapunov exponents exist on a
subset of positive measure, but there are zero Lyapunov
exponents as well, the map $T$ is said to be
{\em partially hyperbolic}. An example is a 3-D billiard
table made by placing a vertical cylinder in a 3-D cube, i.e.,
$Q=\{-1\leq x,y,z\leq 1,$ and $x^2+y^2\geq r^2\}$ for
some $r<1$. Here dim$\,\Omega=4$, there are one positive and
one negative Lyapunov exponents a.e., as well as two zero Lyapunov
exponents. There are also billiard trajectories in this domain
that never hit the cylinder $x^2+y^2=r^2$, their all Lyapunov
exponents are zero, but they make a set of measure zero
in the phase space.
If all the Lyapunov exponents are nonzero a.e.,
the map $T$ is said to be (fully) hyperbolic, in which case
$T$ is often said to be {\em chaotic}, quite informally.
At present, the following classes of billiards are known
to be fully hyperbolic: dispersing, some semidispersing,
and planar billiards bounded by the so-called absolutely focusing arcs.
A billiard $Q$ is said to be dispersing (semidispersing)
if its boundary $\partial Q$ is strictly (nonstrictly)
concave outward at all its regular points. All dispersing
billiards are hyperbolic, ergodic, K-mixing and Bernoulli,
see \cite{Si70,SC87,CH}. Semidispersing billiards are
generally hyperbolic, but not necessarily, as the above
example shows.
Planar billiards whose boundary contains concave inward
(focusing) components are sometimes hyperbolic, too.
This happens, e.g., if all the focusing components of $\partial Q$
are arcs of some circles, and all those circles lie
entirely in $Q$, as it was shown by Bunimovich \cite{Bu74,Bu79}.
Much more general classes of planar hyperbolic billiards with
focusing components of the boundary have been later found by
Wojtkowski \cite{W86}, Markarian \cite{M88}, Bunimovich
\cite{Bu91} and Donnay \cite{D}. We will call such components
{\em absolutely focusing arcs}, following \cite{Bu91}.
Normally, hyperbolic billiards are ergodic, but not necessarily.
An example of a fully hyperbolic nonergodic billiard table was
given by Wojtkowski in \cite{W86}.
\begin{theorem}[Stable and unstable manifolds]
For $\nu$-almost every point $x\in\Omega$ there are
smooth submanifolds $W^u_x$ and $W^s_x$ in $\Omega$,
which contain the point $x$ and satisfy the following:
$$
\max_{n\geq 0}\{{\rm diam}\, T^nW^s_x, {\rm diam}\, T^{-n}W^u_x\}\leq C_x\gamma_x^{n}
$$
for some $C_x>0$ and $\gamma_x\in (0,1)$. In other words,
the images of the manifolds $W^s_x$ and $W^u_x$ under the
iterations of $T$ and $T^{-1}$, respectively, exponentially
contract. The tangent spaces to $W^u_x$ and $W^s_x$
at the point $x$ are $E^u_x$ and $E^s_x$, respectively.
In particular, ${\rm dim}\, W^{u,s}_x=d^{u,s}(x)$.
If $d^u(x)=0$ or $d^s(x)=0$, then the corresponding manifold
is degenerate -- it is just one point $x$.
\end{theorem}
{\em Remark}.
The above two theorems hold for the billiard flow $\Phi^t$
with obvious modifications, which we do not elaborate. We will
denote by $\tilde{\chi}_i(x)$ the Lyapunov exponents for the
flow $\Phi^t$, by $\tilde{H}_i(x)$ the corresponding subspaces
of ${\cal T}_xM$, by ${\cal E}^{u,s,0}_x$ the unstable,
stable and neutral subspaces, respectively, and by ${\cal W}^{u,s}_x$
the unstable and stable manifolds. \medskip
The Lyapunov exponents for $T$ are proportional to those
of $\Phi^t$, and their subspaces are related by $(D\tilde{\pi}_-)
\tilde{H}_i(x)=H_i(\tilde{\pi}_-x)$. Since dim$\, M=
$dim$\,\Omega +1$, the flow $\Phi^t$ has an extra Lyapunov exponent.
It is a zero exponent for the velocity
vector $w=v$ (this vector is the kernel of the projection
$D\tilde{\pi}_-$). It then follows that $(D\tilde{\pi}_-)
{\cal E}^{u,s}_x=E^{u,s}_{\tilde{\pi}_-x}$ and $\tilde{\pi}_-{\cal W}^{u,s}_x
=W^{u,s}_{\tilde{\pi}_-x}$.
\begin{theorem}[Formulas for the entropy]
The measure-theoretic (Kolmogorov-Sinai) entropy of the
billiard ball map $T$ with respect to the measure $\nu$ is given by
\be
h(T) = \int_{\Omega}{\sum}^+\chi_i(x)\,{\rm dim}\, H_i(x)\,d\nu(x)
\label{h1}
\ee
where the sum $\sum^+$ runs over all positive Lyapunov exponents.
It is also given by
\be
h(T) = \int_{\Omega}\ln |DT^u_x|\, d\nu(x)
\label{h2}
\ee
where $|DT^u_x|$ is the Jacobian of the derivative $DT_x$
restricted to the unstable space $E^u_x$ (i.e., this is the
expansion factor of the volume of the unstable space $E^u_x$).
\end{theorem}
The formula (\ref{h1}) is known as Pesin's identity.
It was proved in \cite{Pe77} for general
smooth dynamical systems and in \cite{KS}
for systems with singularities.
Note that the Lyapunov exponents, hence the integral in (\ref{h1}),
do not depend on the choice of Riemannian metric in ${\cal T}(\Omega )$.
The function $\ln |DT^u_x|$ definitely depends on the choice of metric,
but the value of the integral in (\ref{h2}) does not, as it follows from
the invariance of the measure $\nu$ under $T$.
\section{Operator techniques}
\label{secOT}
\setcounter{equation}{0}
For any point $x=(q,v)\in M$ we denote by $dx=(dq,dv)$ tangent
vectors in ${\cal T}_xM$, so that $dq\in{\cal T}_qQ$ and
$dv\in{\cal T}_vS^{d-1}$. Suppose that the segment
$\{\Phi^sx,\, 0\leq s\leq t\}$ does not contain points of reflection.
Then, working with $dx=(dq,dv)$ as with an infinitesimal
vector in $M$, we get
$$
\Phi^t(q,v)=(q+tv,v)\ \ {\rm and}\ \ \Phi^t(q+dq,v+dv)=
(q+dq+t(v+dv),v+dv)
$$
so that
\be
D\Phi^t(dq,dv)=(dq+tdv,dv)
\label{dqdv}
\ee
At a point $x=(q,v)$ of reflection at $\partial Q$, we have
an instantaneous transformation of the velocity vector (\ref{rr}).
This results in an instantaneous transformation
of tangent vectors, $(dq_-,dv_-)\mapsto (dq_+,dv_+)$. To describe
it, denote by $U:\, {\cal T}_qQ\to{\cal T}_qQ$ the reflection across
the hyperplane ${\cal T}_q(\partial Q)$, i.e. $U(w)=w-2\,
(n(q)\cdot w)\, n(q)$
for every $w\in{\cal T}_qQ$. In particular, $U(dv_-)=dv_+$.
Then we have
\be
dq_+=U(dq_-)\ \ \ {\rm and}\ \ \ dv_+=U(dv_-)+\Theta(dq_-)
\label{dqdv-+}
\ee
Here $\Theta$ is a special operator in ${\cal T}_qQ$ associated
with the given reflection. It is defined as follows:\medskip
\noindent
(i) it acts like $U$ on vectors parallel to $v_-$, i.e. it
takes $v_-$ to $v_+$;\\
(ii) denote by $J_-$ and $J_+$ hyperplanes in ${\cal T}_qQ$ perpendicular
to $v_-$ and $v_+$, respectively; then for any vector $w\in J_-$
we have
$$
\Theta(w) = 2\, (v_+\cdot n(q))\, V_+K_qV_-(w)\, \in J_+
$$
Here $V_-$ is the projection of $J_-$ onto ${\cal T}_q(\partial Q)$
parallel to the velocity vector $v_-$, and $V_+$ is the projection of
${\cal T}_q(\partial Q)$ onto $J_+$ parallel to the normal vector
$n(q)$. Also, $K_q$ is the curvature operator of the wall $\partial Q$
at $q$ defined by $n(q+dq)=n(q)+K_q(dq)$ for $dq\in{\cal T}_q(\partial Q)$.
Since $K_q$ is a self-adjoint operator, so is $\Theta U^{-1}$. \medskip
\noindent
The formula (\ref{dqdv-+}) is a multi-dimensional version of the
classical mirror equation in geometrical optics, see a remark in
\cite{W86}. It first appeared, apparently, in \cite{Si78}. Its
proof may be also found in \cite{SiK}, cf. \cite{SC87}.
Equations (\ref{dqdv}) and (\ref{dqdv-+}) completely determine
the action of $D\Phi^t$, for all $-\infty 0$
we have
\be
\frac{d}{dt}\ln\det (I+tB) = {\rm tr}\, B(I+tB)^{-1}
\label{ddt}
\ee
provided the determinant is not zero. Here $I$ is the
identity operator on $L$.
\label{lmdif}
\end{lemma}
The lemma is proved in Appendix. \medskip
{\bf Nonfocusing assumption}. For almost every point $x\in\Omega$
the trajectory segment $\{\Phi^sx,\, 0\leq s\leq\tau(x)\}$
has no focusing points.
Equivalently, for a.e. $x\in\Omega$ and every $0\leq t\leq \tau(x)$ we have
det$\, (I+tB_x)\neq 0$. \medskip
Under this assumption, we can combine Eqs. (\ref{hTS1})
and (\ref{ddt}) as follows:
$$
h(\Phi^t) = c_{\mu}\int_{\Omega}\int_0^{\tau(x)}
{\rm tr} B_x(I+tB_x)^{-1}\, dt\, dq\, dv
$$
We now recall (\ref{Bxt}) and (\ref{change})
and arrive at the following theorem.
\begin{theorem}
Assume the nonfocusing condition stated above.
Then the measure-theoretic (K-S) entropy of the
billiard flow $\Phi^t$ with respect to the measure $\mu$ is given by
\be
h(\Phi^t) = \int_M {\rm tr}\, B_x\, d\mu(x)
\label{h4}
\ee
On the contrary, if the nonfocusing assumption fails, the
integral in (\ref{h4}) diverges and this formula makes no sense.
\label{tmhPb}
\end{theorem}
The formula (\ref{h4}) was first established for 2-D
dispersing billiards in \cite{Si70}. It was later proved in
\cite{Si78,Ch91} for semidispersing billiards in any dimension.
The formula (\ref{h3}) was proven in \cite{Ch91} for
semidispersing billiards and Bunimovich-type planar billiards
bounded by circular arcs, see Sect.~\ref{secELE}. It was
later proved in \cite{CM92} for all planar hyperbolic billiards
bounded by absolutely focusing arcs.
As we have just shown, Theorems \ref{tmhTb} and \ref{tmhPb}
hold for generic (not necessarily hyperbolic) billiard
tables in any dimension. In such a generality, these
results have never been published before.
\medskip
{\em Remark}. For 2-D billiard tables our nonfocusing
condition is equivalent to the semidispersing property. This
can be verified by constructing a set of trajectories of
positive measure with focusing points on each. To this end,
one can take
trajectories that make long series of nearly grazing reflections
along a smooth convex component of $\partial Q$. We omit details.
As a conclusion, (\ref{h4}) does not hold for billiards of
Bunimovich, Wojtkowski or Markarian types (even though
(\ref{h3}) still holds). \medskip
{\em Open question.} Is the nonfocusing assumption equivalent
to the semidispersing property in any dimension?
\section{Operator-valued continued fraction}
\label{secOVCF}
\setcounter{equation}{0}
For certain classes of billiards, there is an explicit
formula for the linear operator $B_x$ that looks like
an infinite continued fraction.
For any $x\in\Omega$ with an infinite past trajectory
let $0>t_1>t_2>\cdots$ be all the negative moments of
reflection. Denote by
$U_i$ and $\Theta_i$ for $i=0,1,\ldots$ the operators
associated with those reflections, see Sect.~\ref{secOT}.
Let $\tau_i=t_i-t_{i+1}>0$, $i\geq 0$, be the intercollision
times. Then
\be
B_x=\Theta_0U_0^{-1}+U_0\frac{\textstyle I}{\textstyle \tau_0I+
\frac{\textstyle I}{\textstyle \Theta_1U_1^{-1}+U_1
\frac{\textstyle I}{\textstyle \tau_1I+
\frac{\textstyle I}{\cdot\cdot\cdot}}U_1^{-1}}}U_0^{-1}
\label{BBB}
\ee
where $\frac{I}{A}$ means $A^{-1}$. Here the terms
$\Theta_iU_i$ and $\tau_iI$ alternate. They describe the
contribution of reflections and free paths in between to
the formation of unstable manifolds.
It readily follows from (\ref{Bxt1}) and (\ref{B-+}) that
$$
B_x=\Theta_0U_0^{-1}+U_0\frac{\textstyle I}{\textstyle \tau_0I+
\frac{\textstyle I}{\textstyle B_{T^{-1}x}}}U_0^{-1}
$$
where $I$ following $\tau_0$ is the identity operator
in $V^u_{T^{-1}x}$.
A recursive application of this formula gives (\ref{BBB}).
This is, however, a superficial argument. In order to
prove (\ref{BBB}) mathematically, one has to verify
the convergence of the continued fraction (\ref{BBB}).
There is no proof of the convergence working for generic
billiard tables. One of the most general results in
this direction may be found in \cite{Bu91}. For fully
hyperbolic billiards the invariant cone techniques by
Wojtkowski \cite{W85,W86,LW} normally gives not only
the hyperbolicity but also the convergence of (\ref{BBB}).
On the other hand, one can possibly find a table $Q$ and a
phase point $x\in\Omega$ for which the fraction (\ref{BBB})
diverges. We cite known convergence theorems for two specific
classes of billiard tables.
\begin{proposition}
Let $Q$ be a semidispersing billiard table ($d\geq 2$).
Then the operator-valued continued fraction (\ref{BBB}) converges
at every point $x\in\Omega$ with an infinite past trajectory.
Moreover, if $B_{x,n}$ is a finite continued fraction
obtained from (\ref{BBB}) by truncation at the $n$-th
reflection, we get
$$
||B_x-B_{x,n}||\leq 1/|t_n|
$$
\label{prBB}
\end{proposition}
The proof is based on the fact that all the operators in
(\ref{BBB}) are selfadjoint positive semidefinite, i.e.
$\tau_i>0$ and $\Theta_i\geq 0$.
The first proof was published in \cite{SC82}, see also
\cite{LW}. In a weaker
form the statement was given without proof earlier in
\cite{SiK}. For 2-D billiard tables the statement was
proved in \cite{Si70}. \medskip
{\em Remark}. If $Q$ is a polyhedron of any dimension, then
the wall $\partial Q$ is flat at all its regular points,
and so all $\Theta_i$ in (\ref{BBB}) are zero, hence $B_x=0$
for all $x$. Therefore, all Lyapunov exponents are zero
everywhere, and $h(T)=h(\Phi^t)=0$. For 2-D polygons, this
fact was first proved in \cite{Ka87} by showing that the
topological entropy of the map $T$ vanishes.
\begin{proposition}[see \cite{CM92}]
Let $Q$ be a 2-D billiard table of Bunimovich, Wojtkowski
or Markarian type. Then the continued fraction (\ref{BBB}) converges
at every point $x\in\Omega$ with an infinite past trajectory.
\end{proposition}
Unlike the previous proposition, now the values of $\Theta_i$
in (\ref{BBB}) corresponding to reflections at focusing components
of the boundary are negative. This makes the proof of
convergence in (\ref{BBB}) more difficult. In \cite{CM92},
the convergence of (\ref{BBB}) was proved by using a general
theorem on convergent continued fractions by Bunimovich
\cite{Bu91}.
\section{Entropy of the periodic Lorentz gas}
\label{secEPLG}
\setcounter{equation}{0}
The Lorentz gas is a dynamical system where a pointlike
particle moves freely in space and bounces off some
fixed, immovable obstacles (scatterers). It is a classical
model of electrons in metals. If the scatters are periodically
situated in space, the Lorentz gas is said to be periodic.
In this case one can find a fundamental domain in space
and project the trajectory of the particle onto that domain.
In this way one gets a billiard dynamical system on a torus
with a finite number of obstacles. We will consider only
disjoint convex obstacles. A popular simple example
is a torus with just one ball-like obstacle.
The simplicity of the periodic Lorentz gas has stimulated
numerous computer simulations and theoretical studies of this model.
A full hyperbolicity, ergodicity and K-mixing property
of the periodic Lorentz
gas have been proved in \cite{Si70} in the 2-D case and \cite{SC87}
in any dimension. The Bernoulli property has been established in
\cite{GO} in the 2-D case and in \cite{CH} in any dimensions.
Numerical researches on periodic Lorentz gases have been focused
on the entropy, Lyapunov exponents, the rate of the decay of
correlations and the diffusion coefficients.
We briefly recall here the numerical results
on the Lyapunov exponents and the entropy.
For a 2-D periodic Lorentz gas with a single circular scatterer
of radius $r>0$ on a unit torus the entropy was estimated
\cite{FOK} to be
\be
h(T)\approx -2\ln r
\label{hT2}
\ee
as $r\to 0$, which was later proved in \cite{Ch91}.
It was also conjectured in \cite{FOK} that for $d$-dimensional
periodic Lorentz gas with a spherical scatterer of radius $r>0$ it
should be $h(T)\approx -d\ln r$, which turned out to be wrong,
see below. It was also estimated there that in 2-D the difference
\be
\ln\int_{\Omega}\tau(x)\, d\nu(x)\, -\, \int_{\Omega}\ln\tau(x)\, d\nu(x)
\label{lntau}
\ee
remains bounded and has a positive limit ($\approx 0.44\pm 0.01$) as
$r\to 0$. The first part of this conjecture (boundedness)
was later rigorously proved in \cite{Ch91}, see below. The convergence is
still an open problem.
Lyapunov exponents for the billiard ball map $T$ for
multi-dimensional periodic Lorentz gases
with a single spherical scatterer of radius $r$ have been studied
in \cite{BlD}. It was estimated that every positive Lyapunov
exponent $\chi_i>0$, as a function of $r$, increases like
const$\cdot|\ln r|$, as $r\to 0$. Moreover, every positive
exponent but the maximal one was conjectured to be $\approx -1/4\ln (r/2)$.
The maximal Lyapunov exponent was conjectured to be $\approx
-(3d+2)/4\ln r$. The last two conjectures turned out to be wrong,
see (\ref{chimax}) and (\ref{chiall}) below. The first one
is proved below by our (\ref{chiall}).
Baldwin \cite{Ba91} gave a theoretical argument supporting
the following sharpening of the formula (\ref{hT2}):
\be
h(T) = -2\ln r + {\rm const}+ O(r)
\label{hT22}
\ee
His argument was not a mathematical proof, and so his prediction
still remains an open problem.
The following theorem was rigorously proved in \cite{Ch91}.
\begin{theorem}[\cite{Ch91}]
The entropy of the $d$-dimensional periodic Lorentz gas ($d\geq 2$)
with a single spherical scatterer of radius $r>0$ in a unit torus
is given by
\be
h(T) = -d(d-1)\, \ln r + O(1)
\label{hTLg}
\ee
and
$$
h(\Phi^t) = -d(d-1)\, |B^{d-1}|\, r^{d-1}\, \ln r + O(r^{d-1})
$$
as $r\to 0$. The mean free path is
\be
\bar{\tau} = \frac{1-|B^d|\, r^d}{|B^{d-1}|\, r^{d-1}}
= \frac{1}{|B^{d-1}\, |r^{d-1}}+O(r)
\label{tauLg}
\ee
The difference (\ref{lntau}) is always positive
and uniformly bounded in $r$ for every $d$.
\label{tmLg}
\end{theorem}
The proof in \cite{Ch91} is based on the approximation of the
operator $B_x$ in (\ref{h3}) by $\Theta_0U_0^{-1}$, see
(\ref{BBB}). The norm of the error is bounded
$$
||B_x-\Theta_0U_0^{-1}||\leq 1/\tau_0\leq {\rm const}
$$
cf. Proposition~\ref{prBB}. Therefore, the substitution of
$\Theta_0U_0^{-1}$ for $B_x$ in (\ref{h3}) can only change
the integral in (\ref{h3}) by a uniformly bounded amount.
Next, for small $r$ the operator $\Theta_0$
has eigenvalues of order $r^{-1}$, which can be computed
explicitly for spherical scatterers. The details may be
found in \cite{Ch91}. As a result, the integration in
(\ref{h3}) gives
\be
h(T)=(d-1)\left (-\ln r\, +\, \int_{\Omega}\ln\tau(x)\, d\nu(x)
\right ) + H(d) + o(1)
\label{hTe}
\ee
The value of $H(d)$ here comes from the substitution of
$\Theta_0U_0^{-1}$ for $B_x$ in (\ref{h3}). Moreover, it
was computed explicitly in \cite{Ch91}: $H(2)=2$,
$H(3)=\ln 4$, and for $d\geq 4$ we have
$$
H(d)=(d-1)\ln 2 - (d-3)\, |S^{d-2}|\, \int_0^1t^{d-2}\ln\sqrt{1-t^2}\, dt
$$
Lastly, the boundedness of (\ref{lntau}) that was proved in \cite{Ch91}
gives (\ref{hTLg}). The other results of Theorem~\ref{tmLg} then
follow from (\ref{bartau}) and Abramov's formula (\ref{Abramov}).\medskip
It also follows from (\ref{hTe}) that
the existence of the limit of the difference
(\ref{lntau}) is equivalent to the following asymptotical
formula:
$$
h(T) = -d(d-1)\ln r + {\rm const} + o(1)
$$
Both remain, however, open questions, as well as the more
refined prediction (\ref{hT22}).
All the open questions involving the entropy $h(T)$ can be
equivalently restated for the entropy $h(\Phi^t)$, in view
of (\ref{Abramov}) and (\ref{tauLg}).
As for the Lyapunov exponents for $T$, it follows directly
from (\ref{hTLg}) that the maximal one is bounded by
\be
-d\ln r+O(1)\leq \chi_{\max}\leq -d(d-1)\ln r+O(1)
\label{chimax}
\ee
By using again the approximation of $B_x$ by $\Theta_0U_0^{-1}$,
and the asymptotic eigenvalues of the latter, see \cite{Ch91}
for details,
it is easy to estimate {\em every} positive Lyapunov exponent
from below: $\chi_i\geq -d\ln r + O(1)$ for all $\chi_i>0$.
Together with (\ref{hTLg}) this give an asymptotic formula
\be
\chi_i = -d\ln r + O(1)
\label{chiall}
\ee
for every positive Lyapunov exponent $\chi_i>0$.
Therefore, all positive Lyapunov exponents have the same
asymptotics as $r\to 0$.
It was also conjectured in \cite{Ch91} that, due to the
geometrical symmetry of spherical scatterers, all the
positive Lyapunov exponents should be actually equal.
This conjecture is still open. However, it was shown
recently \cite{two}, both analytically and numerically,
that in a 3-D {\em random} Lorentz gas (with a random
configuration of scatterers) the two positive Lyapunov
exponents are distinct!
Two more general results were proved in \cite{Ch91}.
Consider a periodic Lorentz gas with $m$ disjoint spherical
scatterers with radii $r_1,\ldots, r_m$ in a unit torus. Put
$$
Z_0=r_1^{d-1}+\cdots +r_m^{d-1}
$$
and
$$
Z_1=r_1^{d-1}\ln r_1+\cdots +r_m^{d-1}\ln r_m
$$
The entropy of such a Lorentz gas was proved \cite{Ch91} to be
\be
h(T) = -(d-1)[\ln Z_0+Z_1/Z_0] + O(1)
\label{hTLge}
\ee
and
$$
h(\Phi^t) = -(d-1)\, |B^{d-1}|\, [Z_0\ln Z_0+Z_1] + O(Z_0)
$$
as $r_1,\ldots,r_m\to 0$, while the distances between the scatterers
remain bounded away from 0. The mean free path is
$$
\bar{\tau} = \frac{1}{|B^{d-1}|\, Z_0}+O(\max r_i)
$$
Lastly, consider a periodic Lorentz gas with $m$ disjoint convex
scatterers in a unit torus, which are homotetically shrinking with a
common scaling factor $\varepsilon>0$. Let $S_1$ be the total
surface area and $V_1$ the total volume of the scatterers when
$\varepsilon=1$. Then we have, see \cite{Ch91},
$$
h(T) = -d(d-1)\, \ln\varepsilon + O(1)
$$
and
$$
h(\Phi^t) = - d(d-1)\, |B^{d-1}|\, |S^{d-1}|^{-1} S_1
\varepsilon^{d-1}\ln\varepsilon+O(\varepsilon^{d-1})
$$
as $\varepsilon\to 0$.
%The mean free path is
%$$
% \bar{\tau} = \frac{|S^{d-1}|\, (1-V_1\varepsilon^d)}{|B^{d-1}|\, S_1
% \varepsilon^{d-1}}
% \label{tauLgm}
%$$
\section{Entropy of stadia and alike}
\label{secESA}
\setcounter{equation}{0}
A stadium is a planar convex billiard table bounded by two parallel
line segments of length $2a$ and two semicircles of radius $r$
that are tangent to the segments at points of contact. Its boundary
is $C^1$ but not $C^2$.
Bunimovich showed \cite{Bu79} that billiards in
stadia are hyperbolic, ergodic and K-mixing. This produced
one of the very first examples of convex tables with
completely chaotic billiard dynamics. This example was
simple and pictorial enough to encourage its further
study and numerical experiments on it.
In particular, the asymptotics of the entropy have been studied
under the conditions that a stadium is deformed approaching
either a circle, which happens as $a/r\to 0$, or a straight segment,
as $r/a\to 0$. In both cases the entropy of the billiard
ball map $T$ approaches zero, and its asymptotics has been
estimated numerically and heuristically, see, e.g., \cite{Ben,Za}.
The only available rigorous estimate for the entropy of
the stadium approaching a circle belongs to Wojtkowski
\cite{W86}.
\begin{theorem}[see \cite{W86}]
There is a constant $c=c(r)>0$ such that the entropy $h(T)$
of the billiard ball map in the stadium obeys $h(T)\geq
c(r)\cdot\sqrt{a}$ as $a\to 0$ and $r>0$ fixed.
\end{theorem}
This is in agreement with an earlier numerical experiment
\cite{Ben} that showed that $h(T)\approx{\rm const}\cdot\sqrt{a}$.
As the stadium approaches a segment ($r/a\to 0$), the following asymptotic
formulas for the entropy were proved in \cite{Ch91}.
\begin{theorem}[see \cite{Ch91}]
Let $r/a\to 0$.
Then the entropy of the billiard ball map in the stadium obeys
\be
h(T) = (r/a) \ln (a/r) + O(r/a)
\label{hTs}
\ee
and the entropy of the billiard flow obeys
$$
h(\Phi^t) = (\pi a)^{-1} \ln (a/r) + O(1/a)
$$
The mean free path is
\be
\bar{\tau} = \frac{\pi r(4a+\pi r)}{4a+2\pi r}
= \pi r+ O(r^2/a)
\label{taus}
\ee
\label{tms}
\end{theorem}
The deformation of the stadium so that it approaches a segment
may seem physically meaningless. However, it can be transformed
into a pictorial model as follows. Take $N\to\infty$ copies of a stadium
with parameters $a>0$ (fixed) and $r\to 0$, so that $N\cdot r$
approaches a constant $b>0$. Put them together side by side
thus making a nearly rectangular
billiard table, $Q_r$, with linear dimensions
$2Nr\approx 2b$ and $\approx 2a$. Two sides of
this big table are straight, and two others are
``scalloped'', made up of a large number of tiny convex
semicircles. It then follows from Theorem~\ref{tms},
as it was shown in \cite{Ch91}, that the
entropy of the billiard ball map in $Q_r$ is
$$
h(T) = \frac{2b}{2a+\pi b} \ln (a/r) + O(1)
$$
and the entropy of the billiard flow is
$$
h(\Phi^t) = (\pi a)^{-1} \ln (a/r) + O(1)
$$
Note that both the entropy of the billiards ball map and that
of the billiard flow in $Q_r$ grow to infinity as $r\to 0$, while
the table $Q_r$ approaches the rectangular billiard table, $Q_0$,
where the entropy is zero, cf. Sect.~\ref{secOVCF} or \cite{Ka87}.
To explain this `mystery', we point out that
the trajectories in $Q_r$ do not approach those in
$Q_0$ as $r\to 0$. In particular, parallel close trajectories remain
parallel after reflections at the boundary in $Q_0$ but such
trajectories diverge drastically in $Q_r$.
The above construction can be generalized as follows. Let
$Q$ be an arbitrary domain with piecewise smooth boundary.
Every piece $\Gamma\subset\partial Q$
of the boundary of $Q$ is then replaced
with a chain of circular arcs of nearly $180^o$
with the same small radius
$r>0$ facing outward and stretching along the curve
$\Gamma$. We thus get another domain, $Q_r^+$, with
``scalloped'' boundary, see Fig.~1. The details of this
construction are not so important, but in order to be
specific, we can take a chain of circles of radius $r$
whose centers all belong to $\partial Q$ such that every
circle is tangent to the neighboring two. Then we erase the
``inner'' arc of every circle (the one that
faces the domain $Q$), and the remaining arcs will
form the boundary of $Q_r^+$. Less formally, we take
a glue and attach identical semicircles to the
boundary of the given domain $Q$. \bigskip
\centerline{Possible location of Figure 1}
.\bigskip
The billiard in $Q_r^+$ satisfies Bunimovich's
conditions \cite{Bu74,Bu79}, and so it is completely
hyperbolic, ergodic and K-mixing.
The entropy of the billiard ball map in $Q_r^+$
is then \cite{Ch91}
\be
h(T) = -\frac {2}{\pi} \ln \frac{r}{{\rm diam}\, Q} + O(1)
\label{hTQ'}
\ee
and the entropy of the billiard flow is
\be
h(\Phi^t) = -\frac{|\partial Q|}{\pi |Q|} \ln \frac{r}{{\rm diam}\, Q}
+ O(1)
\label{hPQ'}
\ee
where $|Q|$ stands for the area of $Q$, and $|\partial Q|$
for the perimeter of $Q$. The above formulas were derived
in \cite{Ch91} for polygonal domains $Q$, but it is easy
to check that the proofs work for arbitrary piecewise
smooth domains as well, so we leave out details. Note
that $h(T)$ does not depend on the domain $Q$ at all.
This is so because the divergence of trajectories in
$Q_r^+$ is determined by the high curvature of the
small semicircles making $\partial Q_r^+$ rather than
by the shape of the original boundary $\partial Q$.
In a recent computer experiment \cite{LBBS}, the logarithmic
dependence of $h(T)$ on $r$ was observed in the case where
$Q$ was a fixed circle.
Another twist of the above construction occurs if,
instead of erasing the ``inner'' arc of every circle,
one erases its ``outer'' arc, the one facing
outward.
Then the remaining (inner) arcs will bound another
domain, $Q_r^-\subset Q_r^+$, which is piecewise
smooth and {\em concave} at every regular point,
see Fig.~1. Less formally, one takes scissors
(instead of a glue) and cuts semicircular cavities
(makes ``cogs'') along the boundary of the given table $Q$.
Obviously, $Q_r^-$ is a Sinai-type (dispersing) billiard table,
so it is completely hyperbolic, ergodic and K-mixing.
It was also shown in \cite{Ch91} that the entropy
of the billiard in $Q_r^-$, to much surprise, satisfies
absolutely the same asymptotic formulas (\ref{hTQ'}) and (\ref{hPQ'}).
This means that the exponential rate of divergence
of trajectories in Sinai's (dispersing) billiard table
$Q_r^-$ and Bunimovich's (focusing) billiard table
$Q_r^+$ is basically the same despite
the seemingly opposite mechanism of divergence,
cf. \cite{Bu74,Bu79,Bu91}.
This points out a universality in the transition from
regular dynamics to chaos as one perturbs the
boundary of any piecewise smooth domain by convex
circular ``scallops'' or concave circular ``cogs''.
Yet another modification of the stadium was studied
by Zaslavski \cite{Za}. Let $Q_{a,r,b}$ be a convex
billiard table bounded by two parallel segments of length $2a$
and two circular arcs of radius $r$ and height $b$,
see Fig.~2.
Let $b\ll r\ll a$. This billiard table satisfies
Bunimovich's conditions, and so it is hyperbolic,
ergodic and K-mixing. Zaslavski \cite{Za}
provided heuristic calculation of the entropy
in $Q_{a,r,b}$ which was later confirmed by a rigorous
argument in \cite{Ch91}. It was shown there that
$$
h(T)=(p/a)\ln(a/r)+O(p/a)
$$
and
$$
h(\Phi^t)=(\pi a)^{-1}\ln (a/r)+O(1/a)
$$
where $p$ is the chord of the arc bounding $Q_{a,r,b}$. \bigskip
\centerline{Possible location of Figure 2}
.\bigskip
Lastly, in \cite{CM92} we modified the above
``scalloped'' tables $Q_r^+$ as follows. Let $\Gamma$ be an
arbitrary $C^4$ convex absolutely focusing curve, see
\cite{Bu91,CM92}.
In particular, it may be a Wojtkowski type \cite{W86}
or a Markarian type \cite{M88} or a Bunimovich type
\cite{Bu91} focusing arc. We shrink $\Gamma$ homotetically
by a small factor $\varepsilon>0$ and attach identical
copies of it to the sides of a given polygon $Q$ so that they
have common points along $\partial Q$. We proved \cite{CM92}
that the entropy of the billiard ball map in the resulting
table is $h(T)=-D\ln\varepsilon + O(1)$, where $D=D(\Gamma)>0$
is a constant.
\section{Mean free path for hard ball gases}
\label{secMFPHB}
\setcounter{equation}{0}
Here we apply the formulas for the mean free path in
Section~\ref{secMFP} to study the mean
intercollision time in systems of hard balls.
We consider a system of $N$ hard balls of diameter $\sigma$
and unit mass in the $k$-dimensional torus $\IT^k_L$ whose
linear dimension is $L>0$. The $k$-dimensional volume of the
torus $\IT^k_L$ is $L^k$.
The balls move freely and collide with each other elastically.
Let $q_{i,1},\ldots,q_{i,k}$ and $p_{i,1},\ldots,p_{i,k}$ be
the coordinates of the position and velocity vector, respectively,
of the $i$-th ball. The configuration space $Q$ of the
system is a subset of the $kN$-dimensional torus $\IT^{kN}_L$,
which correspond to all feasible (nonoverlapping) positions
of the balls. The total kinetic energy of the system is
preserved in time, and
we fix it: $p_{1,1}^2+\cdots +p_{N,k}^2=2EN$, where the constant
$E>0$ is the mean kinetic energy per particle. The phase space is
then $M=Q\times S_1^{kN-1}$ where $S_1^{kN-1}$ is the
$(kN-1)$-dimensional sphere of radius $(2EN)^{1/2}$.
The dynamics of the hard balls with elastic collisions
correspond to the billiard dynamics in the configuration
space $Q$ with specular reflections at the boundary
$\partial Q$. The billiard particle in $Q$ will move at
the speed $(2EN)^{1/2}$ rather than the conventional unit speed,
and we will take this into account later. The boundary
$\partial Q$ consists of $N(N-1)/2$
cylindrical surfaces corresponding to the pairwise collisions of
the balls. We denote by $C_{i,j}$ the open solid cylinder corresponding
to overlapping positions of the balls $i\neq j$. It is given by the
inequality
$$
\sum_{r=1}^k (q_{i,r}-q_{j,r})^2 < \sigma^2\ \ \ \ ({\rm mod}\ L)
$$
The configuration space is then $Q=\IT^{kN}_L\setminus\cup_{i\neq j}C_{i,j}$,
and its boundary is $\partial Q=Q\cap (\cup_{i\neq j}\partial C_{i,j})$.
In order to estimate the mean free path by using Eq.
(\ref{bartau}) we need to compute the volume of the space $Q$
and the surface area of its boundary $\partial Q$. This is
a difficult problem, very hard to solve exactly, since the
cylinders $C_{i,j}$ have plenty of pairwise and multiple
intersections. We will simplify the matter and find
the asymptotic values of both $|Q|$ and $|\partial Q|$
for the gases of hard balls with very low densities.
>From now on we assume that our gas of hard balls is dilute, i.e.
its density
$$
\rho=\frac{|B^k|\cdot \sigma^k N}{(2L)^k}
$$
is low, $\rho\to 0$. (Here again $|B^k|$ is the volume
of the unit ball in $\IR^k$.) The quantity $\rho$
measures the fraction of volume of $\IT^k_L$ occupied by
all the balls together. Technically, our further calculations will be
valid under either of the following regimes:\medskip
\noindent
{\em Regime A}. The number of balls $N$ is fixed and $\rho\to 0$;\\
{\em Regime B}. $N\to\infty$ and $\rho\to 0$ in such a way that
$\rho N\to 0$;\medskip
For the $kN$-dimensional volume of $Q$, we have then
$$
|Q|=L^{kN}(1-O(\rho N))=L^{kN}(1-o(1))
$$
For the $(kN-1)$-dimensional volume of $\partial Q$ we have
$$
|\partial Q|=\frac{N(N-1)}{2}\cdot |\partial C_{1,2}|\cdot (1-o(1))
$$
To compute the area of the cylindrical surface $\partial C_{1,2}$
we use an orthogonal change of variables:
$q_r'=(q_{1,r}-q_{2,r})/\sqrt{2}$ and $q_r''=(q_{1,r}+q_{2,r})/\sqrt{2}$
for $1\leq r\leq k$, leaving the other coordinates $q_{i,r}$,
with $i\geq 3$, unchanged. Then the equation of $\partial C_{1,2}$ becomes
$$
\sum_{r=1}^k (q_r')^2=\sigma^2/2\ \ \ \ ({\rm mod}\ L/\sqrt{2})
$$
This shows that the base of the cylindrical surface $\partial C_{1,2}$
is a $(k-1)$-dimensional sphere of radius
$\sigma/\sqrt{2}$. The other coordinates vary as follows:
$0\leq q_r''\leq \sqrt{2}L$ and $0\leq q_{i,r}\leq L$ for
$i\geq 3$ and all $1\leq r\leq k$. Therefore,
\begin{eqnarray*}
|\partial C_{1,2}| &=& (\sigma/\sqrt{2})^{k-1}\cdot |S^{k-1}|
\cdot (\sqrt{2} L)^k\, L^{(N-2)k}\, (1+o(1))\\
&=& \sqrt{2}\, \sigma^{k-1}\cdot |S^{k-1}| \cdot
L^{kN-k}\, (1+o(1))
\end{eqnarray*}
This gives the following:
\begin{eqnarray*}
|\partial Q| &=& \frac{N(N-1)}{\sqrt{2}}\cdot |S^{k-1}|\cdot
\sigma^{k-1}\, L^{kN-k}\, (1+o(1))\\
&=& \frac{N-1}{\sqrt{2}}\cdot \frac{2^k\, k\rho}{\sigma}
\cdot L^{kN}\, (1+o(1))
\end{eqnarray*}
The mean free path of the billiard particle in the domain $Q$
is then
\begin{eqnarray}
\bar{\tau}&=&\frac{|Q|\cdot |S^{kN-1}|\cdot (kN-1)}{|\partial Q|\cdot
|S^{kN-2}|}\nonumber\\
&=&\frac{\sqrt{2}\,\sigma (kN-1)\cdot |S^{kN-1}|}
{2^k\, k\rho (N-1) \cdot |S^{kN-2}|}
\cdot (1+o(1))
\label{btb}
\end{eqnarray}
This formula for the mean free path is correct but not good
enough, however, because
the billiard system in $Q$ is not ergodic. Indeed, the total
momentum ${\bf P}=(P_1,\ldots, P_k)$, where $P_r=\sum_i p_{i,r}$,
is invariant under the dynamics.
Those phase trajectories whose total momentum ${\bf P}$ is large
will display slow relative motion of the balls, and thus the mean free
path between reflections in $\partial Q$ along such trajectories
will be larger than $\bar{\tau}$ in (\ref{btb}).
On the contrary, the mean free path along trajectories with
zero or small ${\bf P}$ will be below $\bar{\tau}$.
The value of $\bar{\tau}$ in (\ref{btb}) only gives the phase space
average of the mean free paths taken over individual trajectories.
We focus on the case
of a particular physical interest, that of zero total momentum,
${\bf P}={\bf 0}$, where the gas is ``at equilibrium''.
In this case, the ergodicity of the gas of hard
balls is known as Boltzmann-Sinai ergodic hypothesis.
It has been proven in many particular cases, see
a survey \cite{Sz96}, and believed to be true in general.
If so, the mean free path along all typical trajectories with
${\bf P}={\bf 0}$ will be the same. We denote it by $\bar{\tau}_0$
and will compute it next (without assuming ergodicity).
Pick any point $q'\in Q$ and consider
$$
Q_0(q') = \left\{q\in Q:\, \sum_{i=1}^N(q_{i,r}-q_{i,r}')=0\ \
({\rm mod}\ L)\ \ \ {\rm for}\ {\rm all}\ \ r=1,\ldots,k\right\}
$$
The hard ball dynamics with ${\bf P}={\bf 0}$ and initial
configuration point $q'$ corresponds to billiard dynamics in the
$(kN-k)$-dimensional domain $Q_0(q')$ with specular
reflections at its boundary $\partial Q_0(q')$. Therefore,
$$
\bar{\tau}_0=\frac{|Q_0(q')|\cdot |S^{kN-k-1}|\cdot (kN-k-1)}
{|\partial Q_0(q')|\cdot |S^{kN-k-2}|}
$$
Note that the domains $Q_0(q')\subset Q$ are isomorphic
for all $q'\in Q$, and so $|Q_0(q')|/|\partial Q_0(q')|=
|Q|/|\partial Q|$. Combining this with (\ref{btb}) gives
\begin{eqnarray}
\bar{\tau}_0&=&\frac{\sqrt{2}\,\sigma\, (kN-k-1)\cdot |S^{kN-k-1}|}
{2^k\, k\,\rho\, (N-1) \cdot |S^{kN-k-2}|}
\cdot (1+o(1))\nonumber\\
&=&\frac{\sqrt{2\pi}\,\sigma \cdot
\Gamma\left(\frac{kN-k+1}{2}\right )}
{2^k\, \rho \cdot
\Gamma\left(\frac{kN-k+2}{2}\right )}
\cdot (1+o(1))
\label{btb0}
\end{eqnarray}
One can `translate' this result into physically sensible
terms as follows. The speed of the billiard particle in $Q$
is $(2EN)^{1/2}$, and so the mean intercollision time
(in the whole system) is $\bar{t}_{\rm sys}=\bar{\tau}_0\,
(2EN)^{-1/2}$. The mean intercollision time for every
individual particle is simply $\bar{t}_{\rm par}=
\bar{t}_{\rm sys}\cdot N/2$, since every collision involves
two particles. This gives
\be
\bar{t}_{\rm par}
=\frac{\pi^{1/2} \cdot
\Gamma\left(\frac{kN-k+1}{2}\right )\cdot N\sigma}
{2^{k+1} \cdot
\Gamma\left(\frac{kN-k+2}{2}\right )
\cdot (EN)^{1/2}\, \rho }
\cdot (1+o(1))
\label{btbpar}
\ee
We now take the limit in (\ref{btbpar}) as $N\to\infty$
by using a handy formula $\Gamma(N)/\Gamma(N-1/2)=
\sqrt{N}(1+o(1))$:
$$
\bar{t}_{\rm par}(N\to\infty)
=\frac{\pi^{1/2}\, \sigma}
{2^{k+1}\, (Ek/2)^{1/2}\, \rho}
\cdot (1+o(1))
$$
In particular, for
$k=2$ and $k=3$ we recover the so-called Boltzmann mean free
time for hard disks and hard balls in the dilute mode,
see, e.g., \cite{CC,EW}:
\be
\bar{t}_{\rm Boltz}(k=2)
=\frac{\pi^{1/2}\, \sigma}{8\, E^{1/2}\, \rho}
=\frac{1}{2 \sigma\, n\, \sqrt{\pi k_B T}}
\label{Boltz2}
\ee
and
\be
\bar{t}_{\rm Boltz}(k=3)
=\frac{\pi^{1/2}\, \sigma}{8\, (6E)^{1/2}\, \rho}
=\frac{1}{(2 \sigma)^2\, n\, \sqrt{\pi k_BT}}
\label{Boltz3}
\ee
Here $n=N/L^k$ is the
number density of the gas, $k_B$ is the Boltzmann
constant and $T$ is the temperature of the gas related
to $E$ by classical formulas: $E=k_BT$ for 2-D disks
and $E=\frac 32 k_BT$ for 3-D balls. To our best
knowledge, this is the first mathematically exact
derivation of the Boltzmann free time formulas based
solely on the Liouville equilibrium distribution
for finite systems of hard balls.
Lastly, a little numerical experiment reported below
in Table~1 shows that the above formulas are fairly
accurate for hard disks at low densities. The first column shows
the value of the product $\bar{t}_{\rm par}\cdot\rho$
computed according to (\ref{btbpar}) for $k=2$ and some
finite $N$. The other columns show experimentally
estimated mean free times per particle for three
particular values of $\rho$. The last row of the table shows
the Boltzmann mean free time (\ref{Boltz2}) and the so-called
Enskog mean free times for hard disk fluids. The latter take
into account the non-zero density $\rho$ of the fluid,
which is incorporated into the Enskog scaling factor $\chi$.
Precisely,
\be
\bar{t}_{\rm Enskog}(k=2)
=\frac{\pi^{1/2}\, \sigma}{8\, E^{1/2}\, \rho\, \chi}
=\frac{1}{2 \sigma\, n\, \chi\, \sqrt{\pi k_B T}}
\label{Enskog}
\ee
see, e.g., \cite{Gass}, with
$$
\chi\approx 1+0.782\cdot 2\rho+0.5327\cdot (2\rho)^2
$$
see, e.g., \cite{CL97}.
Table 1 actually illustrates two phenomena. First, our theoretical
formula (\ref{btbpar}) works well as $\rho\to 0$
for {\em every} particular value of $N$. Second, the
Enskog formula (\ref{Enskog}) approximates $\bar{t}_{\rm par}$
as $N\to\infty$ for every particular value of $\rho$.
Our experiment was performed on the SPARC workstation
at the University of Alabama at Birmingham. In every
run, molecular dynamics have been simulated up to
$10^7$ interparticle collisions, with a random choice
of the initial state. We have chosen $\sigma=1$ (this sets
the unit of length) and $E=1/2$ (this simply sets the unit
of time). To ensure that the total kinetic energy
($=2EN=N$) and total momentum ($={\bf 0}$) do not
deteriorate due to round-off errors, our program
resets these values periodically, after every 100 collisions
between particles. \bigskip
\centerline{Possible location of Table 1.}
.\bigskip
{\bf Acknowledgements}. The author was partially supported
by NSF grant DMS-9622547. It is also a pleasure to thank
R.~Markarian for helping with the references to the mean
free path formula (\ref{bartau}) and R.~Dorfman for a
helpful discussion of problems studied in \cite{two}.
% \appendix
\section*{Appendix}
\setcounter{section}{1}
\setcounter{theorem}{0}
\renewcommand{\thetheorem}{\Alph{section}.\arabic{theorem}}
\renewcommand{\theequation}{\Alph{section}.\arabic{equation}}
\setcounter{equation}{0}
Here we prove Lemma~\ref{lmdif}. We will need another lemma.
\begin{lemma}
Let $L,E$ be linear subspaces of the Euclidean vector
space $\IR^m$, $m\geq 1$.
For any linear operator $A:\, L\to E$ and $s>0$
we have
\be
\frac{d}{ds}\mid_{s=0}\det (I+sA) = {\rm tr}\, A
\label{ddt1}
\ee
where $d/ds\mid_{s=0}$ means the value of the derivative at $s=0$.
Here $I$ is the identity operator on $L$.
\label{lmA}
\end{lemma}
In the case $L=E=\IR^m$, this lemma is an easy consequence of
the fact that tr$\, A$ is the second leading coefficient of the
characteristic polynomial of the matrix $A$.
In the general case, we pick an orthonormal basis $e_1,\ldots ,
e_k$ in $L$, where $k=$dim$L$. Then the determinant in (\ref{ddt1})
equals the volume of the $k$-dimensional parallelepiped $K_s$ spanned
by the vectors $f_i=e_i+sAe_i$, $1\leq i\leq k$. It is clear
that $f_i=[1+s(e_i\cdot Ae_i)]e_i+sg_i$, where $g_i$ is some
vector perpendicular to $e_i$. For small $s$, the component $sg_i$
of $f_i$ contributes to the volume of $K_s$ a quantity of order $s^2$.
Therefore,
$$
{\rm vol}\, K_s = \prod_{i=1}^k [1+s(e_i\cdot Ae_i)]\, +\, O(s^2)
$$
Now Lemma~\ref{lmA} follows, see our convention in Section~\ref{secOT}.
The proof of Lemma~\ref{lmdif} consists in the following calculation:
\begin{eqnarray*}
\frac{\frac{d}{dt}\det (I+tB)}{\det (I+tB)} &=&
\frac{\frac{d}{ds}\mid_{s=0}\det (I+tB+sB)}{\det (I+tB)}\\
&=&\frac{d}{ds}\mid_{s=0}\frac{\det (I+tB+sB)}{\det (I+tB)}\\
&=&\frac{d}{ds}\mid_{s=0}\det \left ( (I+tB+sB)(I+tB)^{-1}\right )\\
&=&\frac{d}{ds}\mid_{s=0}\det \left ( I_1+sB(I+tB)^{-1}\right ) \\
&=&\, {\rm tr}\, B(I+tB)^{-1}
\end{eqnarray*}
Here $I_1$ is the identity operator on the linear space
$(I+tB)L$, i.e. on the image of $L$ under the operator
$I+tB$.
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\newpage
\begin{center}
\begin{tabular} {||l||c||c|c|c||}\hline\hline
& theory & \multicolumn{3}{c||}{experiment}\\ \cline{3-5}
& by (\ref{btbpar})
& $\rho=0.001$ & $\rho=0.01$ & $\rho=0.1$ \\ \hline\hline
$N=3$ & 0.3607 & 0.3600 & 0.3540 & 0.2955 \\
$N=5$ & 0.3396 & 0.3391 & 0.3329 & 0.2821 \\
$N=10$ & 0.3257 & 0.3255 & 0.3197 & 0.2736 \\
$N=50$ & 0.3157 & 0.3151 & 0.3109 & 0.2671 \\
$N=100$ & 0.3145 & 0.3142 & 0.3091 & 0.2660 \\
\hline
$N=\infty$ & 0.3133 & 0.3128 & 0.3084 & 0.2661 \\
& (Boltz.) & (Enskog) & (Enskog) & (Enskog) \\
\hline\hline
\end{tabular}
\end{center}
\begin{center}
Table 1. Theoretical and experimental values of
$\bar{t}_{\rm par}\cdot\rho$. Here $k=2$, $\sigma=1$
and $E=1/2$.
\end{center}
\end{document}
\end