On survival dynamics of classical systems. Non-chaotic open billiards

Abstract

We report on decay problem of classical systems. Mesoscopic level consideration is given on the basis of transient dynamics of non-interacting classical particles bounded in billiards. Three distinct decay channels are distinguished through the long-tailed memory effects revealed by temporal behavior of survival probability t − α : (i) the universal (independent of geometry, initial conditions and space dimension) channel with α=1 of Brownian relaxation of non-trapped regular parabolic trajectories and (ii) the non-Brownian channel α<1 associated with subdiffusion relaxation motion of irregular nearly trapped parabolic trajectories. These channels are common of non-fully chaotic systems, including the non-chaotic case. In the fully chaotic billiards the (iii) decay channel is given by α>1 due to “highly chaotic bouncing ball” trajectories. We develop a statistical approach to the problem, earlier proposed for chaotic classical systems (Physica A 275 (2000) 70). A systematic coarse-graining procedure is introduced for non-chaotic systems (exemplified by circle and square geometry), which are characterized by a certain finite characteristic collision time. We demonstrate how the transient dynamics is related to the intrinsic dynamics driven by the preserved Liouville measure. The detailed behavior of the late-time survival probability, including a role of the initial conditions and a system geometry, is studied in detail, both theoretically and numerically.