Collisional statistics of the hard-sphere gas

TL;DR

This paper studies the distribution of free flight times and collision counts in a hard-sphere gas, revealing non-Poissonian behavior and establishing a connection with probabilistic ballistic annihilation, supported by analytical, numerical, and simulation results.

Contribution

It introduces a novel analysis of collisional statistics in hard-sphere gases, linking it to ballistic annihilation models and providing efficient computational methods.

Findings

Collision count distribution deviates from Poisson statistics.

Analytical and numerical methods agree with Molecular Dynamics results.

Established a mapping to ballistic annihilation model.

Abstract

We investigate the probability distribution function of the free flight time and of the number of collisions in a hard sphere gas at equilibrium. At variance with naive expectation, the latter quantity does not follow Poissonian statistics, even in the dilute limit which is the focus of the present analysis. The corresponding deviations are addressed both numerically and analytically. In writing an equation for the generating function of the cumulants of the number of collisions, we came across a perfect mapping between our problem and a previously introduced model: the probabilistic ballistic annihilation process [Coppex et al., Phys. Rev. E 69 11303 (2004)]. We exploit this analogy to construct a Monte-Carlo like algorithm able to investigate the asymptotically large time behavior of the collisional statistics within a reasonable computational time. In addition, our predictions are confronted against the results of Molecular Dynamics simulations and Direct Simulation Monte Carlo technique. An excellent agreement is reported.