Asymptotic derivation of Langevin-like equation with non-Gaussian noise and its analytical solution

TL;DR

This paper derives an asymptotic non-Gaussian Langevin equation for stochastic systems with multiple environments, providing a full-order steady distribution formula and demonstrating its application to granular motors.

Contribution

It introduces a novel asymptotic expansion method for non-Gaussian Langevin equations and derives a comprehensive steady distribution formula including higher-order corrections.

Findings

Derived a full-order asymptotic formula for steady distributions

Validated the formula with numerical simulations of a granular motor

Connected higher-order corrections to multiple-kicks effects

Abstract

We asymptotically derive a non-linear Langevin-like equation with non-Gaussian white noise for a wide class of stochastic systems associated with multiple stochastic environments, by developing the expansion method in our previous paper [K. Kanazawa et al., arXiv: 1407.5267 (2014)]. We further obtain a full-order asymptotic formula of the steady distribution function in terms of a large friction coefficient for a non-Gaussian Langevin equation with an arbitrary non-linear frictional force. The first-order truncation of our formula leads to the independent-kick model and the higher-order correction terms directly correspond to the multiple-kicks effect during relaxation. We introduce a diagrammatic representation to illustrate the physical meaning of the high-order correction terms. As a demonstration, we apply our formula to a granular motor under Coulombic friction and get good agreement with our numerical simulations.