The RARE model: a generalized approach to random relaxation processes in disordered systems

TL;DR

The paper presents the RARE model, a comprehensive statistical framework for analyzing random relaxation processes in disordered systems, providing analytical insights into their duration, range, and behavior under power-law conditions.

Contribution

It introduces the RARE model, a novel and robust approach for modeling and analyzing relaxation dynamics in disordered systems with detailed stochastic analysis and closed-form solutions.

Findings

Analytic expressions for relaxation duration and range.

Power-law inputs lead to stretched exponential relaxation patterns.

Asymptotic Paretian distribution of relaxation ranges.

Abstract

This paper introduces and analyses a general statistical model, termed the RARE model, of random relaxation processes in disordered systems. The model considers excitations, that are randomly scattered around a reaction center in a general embedding space. The model's input quantities are the spatial scattering statistics of the excitations around the reaction center, and the chemical reaction rates between the excitations and the reaction center as a function of their mutual distance. The framework of the RARE model is robust, and a detailed stochastic analysis of the random relaxation processes is established. Analytic results regarding the duration and the range of the random relaxation processes, as well as the model's thermodynamic limit, are obtained in closed form. In particular, the case of power-law inputs, which turn out to yield stretched exponential relaxation patterns and asymptotically Paretian relaxation ranges, is addressed in detail.