Escape Dynamics of Many Hard Disks

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 $n\neq 1$, 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 $P_{n}(t)$.