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 $n=1,2,\cdots ,N$. At early times the probabilities $P_n(t)$, $n=2,3,\cdots ,N-1$, 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 $\sim t^{-2n}$ for $n\ne 1$, in contrast to the fact that it decays with a different power law $\sim t^{-n}$ for cases without any disk-disk collision. Chaotic or non-chaotic 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 $P_n(t)$.

arXiv comment14 pages, 11 figures

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