Long-Time Tails in a Chaotic System

The long-time tail in the diffusive behavior in a strongly chaotic system (stadium billiard) is studied from the point of view of "stickiness" of invariant lines. Diffusion near the invariant line in the phase space can be described by a one-dimensional continuous random-walk problem with pausing time which is inversely proportional to the distance from the line. The problem is studied by numerical methods and it is shown that the first-passage-time distribution has an algebraic decay tail in the form $\sim t^{-v}$ with the estimated exponent $v\simeq 2.933\pm 0.005$.