Understanding the transition to superdiffusion in a simple one-dimensional deterministic map by stochastic Lévy walk models #

Samuel Brevitt, Rainer Klages

14:10 Tuesday in 3Q16.

Part of the Stochastic processes and random walks session.

Abstract #

The one-dimensional Pomeau-Manneville (PM) map is a paradigmatic deterministic dynamical system displaying intermittent dynamics (a mixture of chaotic and laminar dynamical regimes). Parallel to this, the Lévy walk (LW) defines a key stochastic process displaying superdiffusive dynamics (the mean square displacement (MSD) increases superlinearly with time). A diffusive periodic extension of the PM map across the real line is long known to exhibit a time-dependent MSD that can be reproduced by a LW. Here we study the matching of the associated generalised diffusion coefficients (GCD) of both systems. We show that the GDCs of both models exhibit the same non-trivial diffusive transition scenario and aging properties under variation of the map’s nonlinearity parameter. We find, however, that the GDCs of both systems deviate from each other in the range of values where the dynamics become normal diffusive. This deviation cannot be cured by simple improvements of continuous-time random walk theory defining a LW. While it is known that PM map and LW model do not match for typical parameter values, our findings demonstrate that even in the simplest well-studied classical PM setting a matching of their diffusive properties is non-trivial.