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,...,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).