Power-law behaviors from the two-variable Langevin equation: Ito's and Stratonovich's Fokker-Planck equations

TL;DR

This paper investigates power-law distributions arising from two-variable Langevin equations, comparing solutions of Ito's, Stratonovich's, and Zwanzig's Fokker-Planck equations under a unified fluctuation-dissipation framework.

Contribution

It introduces a unified form of power-law solutions for different Fokker-Planck equations derived from two-variable Langevin dynamics, incorporating new parameters.

Findings

All three Fokker-Planck forms yield exact stationary power-law solutions.

The solutions depend on parameters kappa and sigma, characterizing deviation from equilibrium.

Numerical results confirm the theoretical stationary solutions.

Abstract

We study power-law behaviors produced from the stochastically dynamical system governed by the well-known two-variable Langevin equations. The stationary solutions of the corresponding Ito's, Stratonovich's and the Zwanzig's (the backward Ito's) Fokker-Planck equations are solved under a new fluctuation-dissipation relation, which are presented in a unified form of the power-law distributions with a power index containing two parameter kappa and sigma, where kappa measures a distance away from the thermal equilibrium and sigma distinguishes the above three forms of the Fokker-Planck equations. The numerical calculations show that the Ito's, the Stratonovich's and the Zwanzig's form of the power-law distributions are all exactly the stationary solutions based on the two-variable Langevin equations.