Leaking of trajectories from the phase space of discontinuous dynamics

The escape of particles from the phase space produced by a two-dimensional, nonlinear and area-preserving, discontinuous map is investigated by using both numerical simulations and the explicit solution of the corresponding diffusion equation. The mapping, given in action-angle variables, is parameterized by K, which controls a transition from integrability to non-integrability. We focus on the two dynamical regimes of the map: slow diffusion () and quasilinear diffusion () regimes, separated by the critical parameter value Kc = 1. When a hole is introduced in the action axis, we find the histogram of escape times and the survival probability of particles to be scaling invariant in both the slow and the quasilinear diffusion regimes, with scaling laws proportional to the corresponding diffusion coefficients, namely, proportional to and K2, respectively. Our numerical simulations agree remarkably well with the analytical results obtained from the explicit solution of the diffusion equation, hence giving robustness to the escape formalism.