Nonergodicity and localization of invariant measure for two colliding masses

Abstract

We show evidence, based on extensive and carefully performed numerical experiments, that the system of two elastic hard-point masses in one-dimension is not ergodic for a generic mass ratio and consequently does not follow the principle of energy equipartition. This system is equivalent to a right triangular billiard. Remarkably, following the time-dependent probability distribution in a suitably chosen velocity direction space, we find evidence of exponential localization of invariant measure. For non-generic mass ratios which correspond to billiard angles which are rational, or weak irrational multiples of π, the system is ergodic, in consistence with existing rigorous results. Introduction.-Soon after the Sinai's proof [1] of ergodicity and mixing in the two-dimensional hard disc gas, the question has been raised if ergodicity and mixing may not also occur in an even simpler one-dimensional, unequal mass, hard point gas. In particular, as pointed out by Lebowitz long ago [2], even the simplest unequal mass case having only two moving point particles is not decided. Since those pioneering times, the analysis of simple non-trivial models has allowed a tremendous progress in our understanding of the properties of nonlinear dynamical systems. In particular, the above problem is of special interest since the local dynamical instability is only linear and therefore it is worthwhile to inquire to what extent statistical properties are present in such systems. This problem is also relevant for the understanding of the properties of the diffusion and relaxation process in quantum mechanics. Indeed, unlike the exponentially unstable classical chaotic motion, in the quantum case deviations in the initial conditions propagate only linearly in time and therefore the quantum diffusion and relaxation process takes place in the absence of exponential instability.The dynamics of the one-dimensional unequal mass, hard point gas with reflecting boundary conditions can be reduced to a simple map. Indeed let m 1 , m 2 be the masses of the two particles, x 1 , x 2 and v 1 , v 2 their positions and velocities respectively. After the collision, the new velocities v ′ 1 , v ′ 2 are given by v ′

Impact of symmetry on ergodic properties of triangular billiards

“…Numerical results indicate that irrational billiards are er-godic with respect to Lebesgue measure, while the correlation decay indicates weak and even strong mixing, see [20,21]. However, as pointed out recently [22] the numerical results are not fully conclusive. Here we revisit this strand of research and point out another facet of this problem, namely the role of symmetry.…”

“…A substantial amount of numerical results have been produced for right-angled triangular billiards. A careful examination of those data, (see, e.g., [18]), and recent numerical results [22] cast some doubt on the ergodicity of Lebesgue measure in these systems. In fact, rightangled billiards are closely related to symmetric billiards if one uses a Zemlyakov-Katok construction to unfold the billiard dynamics [23].…”

Mass-Imbalanced Atoms in a Hard-Wall Trap: An Exactly Solvable Model Associated with D6 Symmetry

“…It is interesting to note that two mass-imbalance hard-core particles moving in a 1D box is equivalent to a triangle billiard system (Zhang et al, 2016;Wang et al, 2014;Gorin, 2001), and thus our exact result is helpful for understanding quantum billiard systems from a different perspective.…”

Integrable Families of Hard-Core Particles with Unequal Masses in a One-Dimensional Harmonic Trap

“…Any implementation of the models considered in our article may constitute an efficient experimental realization of spherical triangular (or higher-dimensional, simplex-shaped) quantum billiards [82]. The ergodicity of classical flat triangular billiards is conjectured to strongly depend on the rationality of the billiard angles [58,61]. Numerically, such questions about ergodicity are difficult, requiring long propagation for averages to converge to their infinite time limits.…”