Monitoring L\'evy-Process Crossovers

TL;DR

This paper introduces two models for generating crossovers among different Le9vy processes, using fractional derivatives with varying orders or coefficients, and analyzes their behavior through semi-analytical solutions.

Contribution

The work proposes novel models for Le9vy process crossovers by varying fractional derivatives, providing a framework for studying diffusive regime transitions in complex systems.

Findings

Models show qualitative similarity far from crossover regimes

Transitions between regimes can occur in different forms for the two models

Semi-analytical solutions effectively track time-dependent behavior

Abstract

The crossover among two or more types of diffusive processes represents a vibrant theme in nonequilibrium statistical physics. In this work we propose two models to generate crossovers among different L\'evy processes: in the first model we change gradually the order of the derivative in the Laplacian term of the diffusion equation, whereas in the second one we consider a combination of fractional-derivative diffusive terms characterized by coefficients that change in time. The proposals are illustrated by considering semi-analytical (i.e., analytical together with numerical) procedures to follow the time-dependent solutions. We find changes between two different regimes and it is shown that, far from the crossover regime, both models yield qualitatively similar results, although these changes may occur in different forms for the two models. The models introduced herein are expected to be useful for describing crossovers among distinct diffusive regimes that occur frequently in complex systems.