Nonequilibrium Statistical Mechanics of Weakly Ergodic Systems

Abstract

The properties of the Gibbs ensembles of Hamiltonian systems describing the motion along geodesics on a compact configuration manifold are discussed. We introduce weakly ergodic systems for which the time average of functions on the configuration space is constant almost everywhere. Usual ergodic systems are, of course, weakly ergodic, but the converse is not true. A range of questions concerning the equalization of the density and the temperature of a Gibbs ensemble as time increases indefinitely are considered. In addition, the weak ergodicity of a billiard in a rectangular parallelepiped with a partition wall is established.

APPENDIX. ON THE UNIFORM DISTRIBUTION OF A KNUDSEN GAS IN A RECT ANGULAR BOX WITH A PARTITION

Here we consider the property of uniform distribution of a collisionless gas in a container with a partition (Fig. 1). Let the gas be in one of the halves of the box at an initial instant; we do not assume that the gas is in statistical equilibrium at this instant. Then a hole opens abruptly in the partition between the communicating compartments of the container, and the gas begins to flow into the other compartments. It is claimed that, as $t\to+\infty$ (and also as $t\to-\infty$) the Knudsen gas will tend to statistical equilibrium when its density equalizes over the total volume in which the gas of the box is contained.

According to Section 3, this question reduces to that of the weak ergodicity of a billiard in a box with mirror walls and a partition. In a sense, this billiard belongs to completely integrable Hamiltonian dynamical systems. Indeed, for an $n$-dimensional box, a complete set of independent integrals consists of $y_{1}^{2}$, …, $y_{n}^{2}$, where $y_{j}$ is the momentum of the particle along the $j$th edge of the box. Obviously, these integrals are pairwise in involution with respect to the standard symplectic structure of phase space. However, the common $n$-dimensional level surfaces of these integrals in phase space have a complex structure due to the singularities of configuration space.

The singularities of the box itself (and the discontinuity of the phase trajectories of the particle during elastic collisions with the boundary) can be eliminated by a transition to the cover of a rectangular parallelepiped with an $n$-dimensional torus ramified on the boundary (Fig. 2 shows the construction of this cover for $n=2$). As a result, the problem reduces to that of a billiard on an $n$-dimensional Euclidean torus with added “flat” walls.

If we successively reflect the rectangular box along its faces, then we obtain a tiling of the Euclidean space, resulting in the particle moving in straight lines with constant velocity if the partitions are not taken into account. The presence of the partitions leads to a periodic structure of the added walls, from which the particle is elastically reflected (Fig. 3). This is a special case of the so-called Lorenz gas; the role of scatterers is played by the periodically arranged flat partitions. Usually the scatterers are convex regions with a smooth regular boundary. Such a Lorenz gas with periodic boundary conditions (on a Euclidean torus) is an ergodic system and hence, according to Theorem 2, weak ergodicity will take place. In our case, the Lorenz gas is degenerate and, of course, nonergodic.

Thus, we consider a billiard on the Euclidean torus $\mathbb{T}^{n}=\{x_{1}$, $x_{2}$, …, $x_{n}\bmod{2\pi}\}$ with a wall that is a measurable domain $D$ lying in the “flat” section $\{x_{1}=0\}$. The particle undergoes inertial motion (on a rectilinear winding $\mathbb{T}^{n}$) and, if it reaches a point of domain $D$ with velocity $v_{-}=(v_{1},v_{2},\ldots,v_{n})$, then at this instant of time its velocity changes to $v_{+}=(-v_{1},v_{2},\ldots,v_{n})$.

Theorem 7

A billiard on a torus with a wall is a weakly ergodic system.

We recall that the vector $v=(v_{1},\ldots,v_{n})$ is called nonresonant if

$$(m,v)=\sum m_{i}v_{i}\neq 0$$

(A.1)

for all nonzero vectors $m\in\mathbb{Z}^{n}$ with integer components. It is clear that condition (A.1) does not change under the substitution $v_{-}\mapsto v_{+}$.

Theorem 8

Let domain $D$ be Jordan measurable and let $\varphi\colon\mathbb{T}^{n}\to\mathbb{R}$ be any function that is Riemann integrable. Then for any motion of the particle, $t\mapsto x(t)$ with the nonresonant velocity vector

$$\lim_{\tau\to\infty}\frac{1}{\tau}\int\limits_{0}^{\tau}\varphi\big{(}x(t)\big{)}dt=\frac{1}{(2\pi)^{n}}\int\limits_{\mathbb{T}^{n}}\varphi(x)d^{n}x.$$

(A.2)

In the absence of domain $D$, Theorem 8 becomes the classical Weyl theorem on uniform distribution. In particular, let $\varphi$ be a characteristic function of domain $\Phi\subset\mathbb{T}^{n}$. Then, according to (A.2), the average time the particle stays in domain $\Phi$ is equal to the ratio $\mathop{\rm vol}\nolimits\Phi/\mathop{\rm vol}\nolimits\mathbb{T}^{n}$ — the fraction which domain $\Phi$ occupies in the volume of the entire torus $\mathbb{T}^{n}$. Theorems 7 and 8 are closely related to each other. For the case of an empty domain $D$ this question is discussed, for example, in [3].

The substantial part of the proof of Theorem 8 is to consider the case where

$$\varphi(x)=e^{i(m,x)},\quad m\in\mathbb{Z}^{n}\setminus\{0\}.$$

(A.3)

We need to prove Eq. (A.2) with a zero constant on the right-hand side. Let $x=vt+x_{0}$ be the motion of the particle between collisions. Then

$$\int\limits_{\tau_{1}}^{\tau_{2}}\varphi(vt+x_{0})dt=\left.\frac{e^{i(m,x)}}{i(m,v)}\right|_{x^{1}}^{x^{2}},$$

(A.4)

where $x^{1}=x(\tau_{1})$ and $x^{2}=x(\tau_{2})$.

Let $\widehat{x}\in D$ be the location where the particle collides with a flat wall $D$, $v_{-}$ and let $v_{+}$ be its velocity before and after the impact. Then, according to (A.4), the integral of $\varphi$ as a function of time obtains the increment

$$-ie^{i(m,\widehat{x})}\left[\frac{1}{(m,v_{-})}-\frac{1}{(m,v_{+})}\right]\!.$$

(A.5)

The expression in square brackets is in absolute value the same for all impacts, but its sign depends on from which side the particle has reached the point of the domain $D$.

Let $t_{1}$, $t_{2}$, …, $t_{p}$ be the instants of consecutive collisions of the particle with domain $D$. It is clear that the differences $(t_{j+1}-t_{j})$ are multiples of $2\pi/v_{1}$. We note that ${v_{1}\neq 0}$, otherwise the velocity vector $(\pm v_{1},v_{2},\ldots,v_{n})$ would be resonant. Let $\Delta_{j}$ denote the sign of the expression in the square brackets in (A.5) at the instant of the $j$th collision: $\Delta_{j}=+1(-1)$ if this expression is positive (negative).

Thus, our problem reduces to calculating the limit of the ratio

$$\frac{1}{t_{p}-t_{1}}\sum\Delta_{j}e^{i(m,\widehat{x}_{j})}$$

(A.6)

as $p\to\infty$, where $\widehat{x}_{j}\in D$ is the point of the $j$th collision with the wall $D$.

It is clear that

$$\widehat{x}_{j+1}-\widehat{x}_{j}=(t_{j+1}-t_{j})\widehat{v},\quad\text{where}\widehat{v}=(v_{2},\ldots,v_{n})={\rm const}.$$

(A.7)

Next, if $\Delta_{j}=+1$ (or $-1$), then $\Delta_{j+1}=-1$ (resp. $+1$). This follows immediately from the following obvious observation: if a particle collides with the wall $D$ on the left (right), then next time it will collide on the right (left). But then the sum (A.6) splits into two sums: with even and odd numbers $j$. Each such sum, divided by $(t_{p}-t_{1})$, tends (with its sign), as $p\to\infty$, to

$$\frac{v_{1}}{2(2\pi)^{n}}\int\limits_{\mathbb{T}^{n-1}}\widehat{\varphi}\psi dx_{2}\ldots dx_{n},$$

where $\psi$ is a characteristic function of the Jordan measurable domain $D\subset\mathbb{T}^{n-1}=\{x\in\mathbb{T}^{n}\colon$ $x_{1}=0\}$. This is a consequence of the classical Weyl theorem on uniform distribution (more precisely, its discrete version) when applied to the product of the functions $\widehat{\varphi}\psi$ and to the map (A.7). The main nonresonance condition has been satisfied here. But then the limit of the ratio (A.6) tends to zero as $p\to\infty$. Further reasoning repeats the proof of the continuous version of the Weyl theorem.

As for Theorem 7, it can be proved using the density of linear combinations of the functions (A.3) in spaces of Lebesgue integrable functions.

By similar reasoning one can prove the weak ergodicity of somewhat more general billiards on $\mathbb{T}^{n}$, when, in addition to $D$, there is another “wall” in the form of a measurable domain $D^{\prime}$ from the section $\mathbb{T}^{n-1}=\{x\in\mathbb{T}^{n}\colon$ $x_{1}=\pi\}$.

Finally, by slightly complicating the construction, one can prove the weak ergodicity of the billiard on an $n$-dimensional Euclidean torus with walls in the sections $\mathbb{T}_{j}^{n-1}=\{x\in\mathbb{T}^{n}\colon$ $x_{j}=0$ or $x_{j}=\pi\}$. The thing is that, in a sense, the dynamics of a particle in such billiards breaks down into “independent” motions for each of the angular coordinates $x_{1}$, …, $x_{n}$.

We conclude by mentioning another interesting problem of the Knudsen gas in a container with a hole which is represented by some set of positive measure on the boundary of this container. Particles can leave the container through this hole without difficulty, moving farther in straight lines with constant velocity. The question is whether the Knudsen gas will ultimately escape from the container? In other words, will almost all gas particles leave the container?

The answer to this question depends greatly on the shape of the container. For example, the gas will escape almost entirely from a circle with an indefinitely small hole, whereas it will, generally speaking, not escape from a container in the form of a sphere. The answer is also positive for a container in the form of a rectangular parallelepiped. This is a consequence of the Weyl theorem on uniform distribution. In exactly the same way, the Knudsen gas will escape from a parallelepiped with a hole that has, in addition, a partition (as shown in Fig. 1). This can be easily inferred from Theorem 8.

In general, this problem is associated with the conditions for ergodicity (and weak ergodicity) of a billiard. For example, suppose we are given a Lyapunov stable two-sectional periodic trajectory (this is a segment that is orthogonal at its end points to the boundary of the billiard). Then this billiard is nonergodic and the Knudsen gas will, as a rule, not escape entirely from the container with such a shape.

Suppose there is a billiard in the bounded domain $\Pi$ of an $n$-dimensional Euclidean space with a piecewise smooth and regular boundary. Let the hyperplane $p$ divide $\Pi$ into two (generally speaking, disconnected) parts $\Pi_{1}$ and $\Pi_{2}$ of a positive volume, and let the intersection $p\cap\Pi$ be a “hole” in the new containers $\Pi_{1}$ and $\Pi_{2}$ (Fig. 4).

Theorem 9

If the billiard in $\Pi$ is a weakly ergodic system, then almost the whole Knudsen gas with any integrable initial density of distribution over velocities and coordinates will escape from each of containers $\Pi_{1}$ and $\Pi_{2}$ .

Indeed, by Theorem 1, almost all trajectories are everywhere dense in $\Pi$. Consequently, the particle, which was at the initial time in domain $\Pi_{1}$, will after some time almost certainly find itself in domain $\Pi_{2}$. Thus, almost all particles of the Knudsen gas from container $\Pi_{1}$ will leave this domain after traversing the hole $p\cap\Pi$. This is the required result.

Of course, Theorem 9 gives only sufficient conditions for complete escape of the Knudsen gas. One can consider a similar problem for more “realistic” models of an ideal gas, for example, for the Boltzmann – Gibbs gas, which consists of a large number of identical small balls which elastically collide with each other and with the walls of the container. Is it true that for almost all initial states of such a gas all balls will eventually escape from the container through a hole whose size is admissible? Since this is a nonequilibrium process, it is probably not sufficient to merely refer to the ergodicity of the Boltzmann – Gibbs gas. In addition, the ergodicity of this system has not been completely proved so far (see [17, 18]). Moreover, for a container in the form of a sphere there is no ergodicity at all. Recalling the words of S. Ulam cited in Section 1, one would like to see in a textbook on statistical mechanics a rigorous analysis of the adiabatic expansion of a gas into vacuum under sufficiently general assumptions.