Coarse-graining complex dynamics: Continuous Time Random Walks vs. Record Dynamics

TL;DR

This paper compares Continuous Time Random Walks and Record Dynamics, arguing that CTRW are inadequate for macroscopic relaxation modeling and demonstrating how RD better captures complex relaxation behaviors and sub-diffusive dynamics.

Contribution

The paper introduces Record Dynamics as a superior alternative to CTRW for modeling complex relaxation processes in systems with hierarchical traps.

Findings

CTRW are inadequate for macroscopic relaxation modeling.

RD captures stretched exponential, power-law, and logarithmic relaxations.

RD explains sub-diffusive behavior in complex environments.

Abstract

Continuous Time Random Walks (CTRW) are widely used to coarse-grain the evolution of systems jumping from a metastable sub-set of their configuration space, or trap, to another via rare intermittent events. The multi-scaled behavior typical of complex dynamics is provided by a fat-tailed distribution of the waiting time between consecutive jumps. We first argue that CTRW are inadequate to describe macroscopic relaxation processes for three reasons: macroscopic variables are not self-averaging, memory effects require an all-knowing observer,and different mechanisms whereby the jumps affect macroscopic variables all produce identical long time relaxation behaviors. Hence, CTRW shed no light on the link between microscopic and macroscopic dynamics. We then highlight how a more recent approach, Record Dynamics (RD) provides a viable alternative, based on a very different set of physical ideas: while CTRW make use of a renewal process involving identical traps of infinite size, RD embodies a dynamical entrenchment into a hierarchy of traps which are finite in size and possess different degrees of meta-stability. We show in particular how RD produces the stretched exponential, power-law and logarithmic relaxation behaviors ubiquitous in complex dynamics, together with the sub-diffusive time dependence of the Mean Square Displacement characteristic of single particles moving in a complex environment.