Escape dynamics of many hard disks

Abstract

Many-particle effects in escapes of hard disks from a square box via a hole are discussed in a viewpoint of dynamical systems. Starting from N disks in the box at the initial time, we calculate the probability P_{n}(t) for at least n disks to remain inside the box at time t for n=1,2,...,N. At early times, the probabilities P_{n}(t),n=2,3,...,N-1, are described by superpositions of exponential decay functions. On the other hand, after a long time the probability P_{n}(t) shows a power-law decay ∼t^{-2n} for n≠1, in contrast to the fact that it decays with a different power law ∼t^{-n} for cases without any disk-disk collision. Chaotic or nonchaotic properties of the escape systems are discussed by the dynamics of a finite-time largest Lyapunov exponent, whose decay properties are related with those of the probability P_{n}(t).

Stochastic boundary approaches to many-particle systems coupled to a particle reservoir

“…Since equilibrium statistical-mechanical quantities not only of ideal gases but also of many-hard-disk systems in low-density cases can be calculated analytically based on the grand canonical distribution, we use these systems to show that the systems with the PISB conditions can reproduce correctly equilibrium properties of the systems coupled to a particle reservoir. Furthermore, we apply the PISB method to particle-escape phenomena [37][38][39][40], as a typical example of nonequilibrium phenomena. We consider the systems in which hard disks escaping from a particle reservoir into one end of a tube leave from another end of the tube, and produce a steady particle current inside the tube after a long time.…”