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

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.

Dynamics of dense hard sphere colloidal systems: A numerical analysis

“…Even though a power-law with a small exponent and a logarithm may look similar over restricted time scales, the physical pictures behind them are different [14]. Dense HSC dynamics is usually interpreted in CTRW terms [6,10,11], while logarithmic laws, or equivalently, the fact that macroscopic rates fall off as the inverse of the system age, clearly support the competing RD description [20,24,28,30].…”

“…Cage breakings have been modelled [11,12] using a Continuous Time Random Walk (CTRW) [13], a popular coarse-graining device recently criticized in [14,15]. Other approaches to coarse-graining are e.g.…”

Aging is a log-Poisson process, not a renewal process

“…As noted in Ref. [32], applying this type of description to aging systems violates the system size scaling and self-averaging properties of macro-scopic variables, which are universally observed in nature. The problem is avoided in continuous-time random walks models (CTRW), where each particle in a colloid, say, is now endowed with a power-law distribution of times between displacements.…”

Anomalous scaling of stochastic processes and the Moses effect

“…If, however, a NIP begins anew at certain times, such as after a triggering event, then corresponding time translations may be used to construct a statistical ensemble from a single time-series [23,42,43]. Although, this perhaps can only be done if the time between renewals has a finite average [44], since, more generally, weak ergodicity breaking [45,46] may prevent time averages being equated with ensemble averages in diffusive processes that scale anomalously. The generalization of time series analyses to include NIPs necessitates generalizing the definitions of indices used to characterize scaling in SIPs.…”