Non-Equilibrium Noise in V-Shape Linear Well Profiles

TL;DR

This paper develops a theoretical framework for non-equilibrium noise modeled by alpha-stable distributions in V-shaped potential wells, with computational simulation methods and normalizable probability densities, aimed at systems with self-organized criticality.

Contribution

It introduces a novel approach to modeling non-equilibrium noise in V-shaped wells using alpha-stable distributions within Langevin and fractional Fokker-Planck equations.

Findings

Derived a normalizable probability density function for alpha-stable noise

Presented an Euler scheme for simulating non-equilibrium noise

Focused on theoretical modeling applicable to self-organized critical systems

Abstract

Non-equilibrium noise is characterized as noise realizations where external agitations disrupt the harmonic equilibrium of Brownian motion. Excitations in a particle's random walk into a so-called L\'evy flight changes the distribution of the noise from Gaussian to the fat-tailed L\'evy distribution. Generalization between Gaussian and L\'evy distributions is the $\alpha$-stability distribution, where $1<\alpha\leq2$. In this study, the $\alpha$-stability distributed noise is subjugated into the Langevin and fractional Fokker--Planck equations that correspond to a V-shaped linear potential well $V(x)=F|x|$. From these equations, an Euler scheme for computational simulation via iterations is presented, and a probability density function that is normalizable under any $\alpha\in(1,2]$ is obtained. This study is focused more on the theoretical framework of non-equilibrium noise in V-shaped linear well profiles, which is intended to be applied to systems known to exhibit self-organized criticality.