Two-time scale subordination in physical processes with long-term memory

TL;DR

This paper introduces a two-time scale subordination approach to model dynamical processes with long-term memory, explaining anomalous diffusion and fractional state equations in continuous media.

Contribution

It develops a novel two-time scale subordination framework to describe long-term memory effects in physical processes, with applications to diffusion and state equations.

Findings

The empirical trapping-reaction law is explained by two-time scale subordination.

A fractional state equation of Bagley-Torvik type is derived from subordinated pressure and density.

The approach unifies anomalous diffusion and fractional calculus in physical media.

Abstract

We use the two-time scale subordination in order to describe dynamical processes in continuous media with a long-term memory. Our consideration touches two physical examples in detail. First we study a temporal evolution of the species concentration for the trapping reaction in which a diffusing reactant is surrounded by a sea of randomly moving traps. The analysis is based on the random-variable formalism of anomalous diffusive processes. We find that the empirical trapping-reaction law, according to which the reactant concentration decreases in time as a product of an exponential and a stretched exponential function, can be explained by the two-time scale subordination of random processes. Another example is connected with a state equation for continuous media with memory. If the pressure and the density of a medium are subordinated in two different random processes, then the ordinary state equation becomes fractional with two time scales. This allows one to arrive at the state equation of Bagley-Torvik type.