Evolution of collision numbers for a chaotic gas dynamics

Abstract

We put forward a conjecture of recurrence for a gas of hard spheres that collide elastically in a finite volume. The dynamics consists of a sequence of instantaneous binary collisions. We study how the numbers of collisions of different pairs of particles grow as functions of time. We observe that these numbers can be represented as a time-integral of a function on the phase space. Assuming the results of the ergodic theory apply, we describe the evolution of the numbers by an effective Langevin dynamics. We use the facts that hold for these dynamics with probability one, in order to establish properties of a single trajectory of the system. We find that for any triplet of particles there will be an infinite sequence of moments of time, when the numbers of collisions of all three different pairs of the triplet will be equal. Moreover, any value of difference of collision numbers of pairs in the triplet will repeat indefinitely. On the other hand, for larger number of pairs there is but a finite number of repetitions. Thus the ergodic theory produces a limitation on the dynamics.

Emergence of distinguishability of patterns of collisions of particles in a non-equilibrium chaotic system

“…Refs. [6,2]. This assumption is justified in the considered three-dimensional case, since one expects the correlations of ξ ij to decay at large times as t −D/2 , where D is the space dimension [13,14].…”

“…During these periods the particle persistently collides more with certain particles and less with others. This effect generalizes and can be tested on small gaseous systems with arbitrary short-range interactions.Recently it was observed that the numbers of pair collisions N ij (t) in a collection of hard balls can be represented as time integrals of a singular observable on the phase space [2]. The functions N ij (t) count the number of collisions of particles i and j that occurred within time t. This representation is highly relevant because hard balls is one of the few systems for which * Corresponding author.…”

“…To use the integral representation of N ij (t) the fluid state of the collection of hard balls was considered [2]. In this case in the limit of infinite times all pairs of particles collide infinite number of times.…”

“…In this work we produce some evidence of the validity of the Langevin equations by uncovering theoretically and proving numerically the consequence of those equations. We observe that since different N ij (t) have the same average rate of growth then the differences N ij (t) − N kl (t) constitute random walks at large t. Using facts that hold for the random walk with probability one definite predictions were made for individual trajectory in the phase space [2]. Thus the ergodic theory was used to predict properties of evolution of the system whose trajectory in the phase space by itself is incalculable.…”

“…[15], this is not a limitation for the analysis below, that concerns the most probable events. The Langevin equations (3) produce non-trivial physical conclusions [2]. The effective correlations and anti-correlations between the particles are described by the differencesÑ ij,mn (t) ≡ N ij (t) − N mn (t) that obey the equations of the usual Brownian motioṅ…”