Theory of Adiabatic fluctuations : third-order noise

TL;DR

This paper develops a theoretical framework for analyzing adiabatic fluctuations in dynamical systems, revealing non-Gaussian noise characteristics and providing exact solutions for velocity space fluctuations.

Contribution

It introduces a systematic expansion approach for adiabatic fluctuations, deriving third-order differential equations that describe non-Gaussian noise effects.

Findings

Probability distribution functions obey third-order differential equations.

Exact solutions for adiabatic fluctuations in velocity space are obtained.

Steady states are stable with finite variances and higher moments.

Abstract

We consider the response of a dynamical system driven by external adiabatic fluctuations. Based on the `adiabatic following approximation' we have made a systematic separation of time-scales to carry out an expansion in $\alpha |\mu|^{-1}$, where $\alpha$ is the strength of fluctuations and $|\mu|$ is the damping rate. We show that probability distribution functions obey the differential equations of motion which contain third order terms (beyond the usual Fokker-Planck terms) leading to non-Gaussian noise. The problem of adiabatic fluctuations in velocity space which is the counterpart of Brownian motion for fast fluctuations, has been solved exactly. The characteristic function and the associated probability distribution function are shown to be of stable form. The linear dissipation leads to a steady state which is stable and the variances and higher moments are shown to be finite.