Macroscopic Finite Size Effects in Relaxational Processes

TL;DR

This paper investigates how the size of macroscopic systems influences their relaxation dynamics, revealing a crossover from stretched exponential to simple exponential decay governed by system size and a characteristic crossover time.

Contribution

It demonstrates that system size critically affects relaxation behavior, introducing a logarithmic dependence and a crossover time in dynamical processes with competing exponential mechanisms.

Findings

Crossover time depends logarithmically on system size.

Relaxation shifts from stretched exponential to exponential after crossover.

Decay rate also depends logarithmically on system size.

Abstract

We present results on dynamical processes that exhibit a stretched exponential relaxation. When the relaxation is a result of two competing exponential processes, the size of the system, although macroscopic, play a dominant role. There exist a crossover time tx that depends logarithmically on the size of the system, above which, the relaxation changes from a stretched exponential to a simple exponential decay. The decay rate also depends logarithmically on the size of the system. The results are relevant to large-scale Monte-Carlo simulations and should be amenable to experiments in low-dimensional macroscopic systems and mesoscopic systems.