Dynamical transitions and aging in the superdiffusive Pomeau-Manneville map

TL;DR

This paper investigates superdiffusive behavior in the Pomeau-Manneville map, identifying dynamical transitions in diffusion coefficients and showing how aging influences these transitions, combining numerical and analytical approaches.

Contribution

It provides a detailed analysis of superdiffusion and dynamical transitions in the Pomeau-Manneville map, including the effects of aging on these phenomena.

Findings

Identification of two singular dynamical transitions in the generalized diffusion coefficient

Analytic expression for GDC using Lévy walk theory that reproduces the transitions

Aging eliminates the suppression transition, altering superdiffusion behavior

Abstract

The Pomeau-Manneville map is a paradigmatic intermittent dynamical system exhibiting weak chaos and anomalous dynamics. In this paper we analyse the parameter dependence of superdiffusion for the map lifted periodically onto the real line. From numerical simulations we compute the generalised diffusion coefficient (GDC) of this model as a function of the map's nonlinearity parameter. We identify two singular dynamical transitions in the GDC, one where it diverges to infinity, and a second one where it is fully suppressed. Using the continuous-time random walk theory of L\'evy walks we calculate an analytic expression for the GDC and show that it qualitatively reproduces these two transitions. Quantitatively it systematically deviates from the deterministic dynamics for small parameter values, which we explain by slow decay of velocity correlations. Interestingly, imposing aging onto the dynamics in simulations eliminates the dynamical transition that led to suppression of the GDC, thus yielding a non-trivial change in the parameter dependence of superdiffusion. This also applies to a respective intermittent model of subdiffusive dynamics displaying a related transition.