Non-Equilibrium Noise in V-Shape Linear Well Profiles

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 Lévy flight changes the distribution of the noise from Gaussian to the fat-tailed Lévy distribution. Generalization between Gaussian and Lévy distributions is the $\alpha$-stability distribution, where $1<\alpha \le 2$. 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.

6 pages, 4 figures; Associated Euler scheme on a Mathematica .nb file with adjustable input parameters can be supplied upon request