Stochastic dynamical theory of power-law distributions induced by multiplicative noise

TL;DR

This paper develops a stochastic dynamical framework showing how multiplicative noise in Langevin equations leads to power-law distributions, including Tsallis distributions, in non-equilibrium systems, and derives a generalized fluctuation-dissipation theorem.

Contribution

It introduces a generalized stochastic theory linking multiplicative noise to power-law distributions and derives new equations with stationary solutions as Tsallis distributions.

Findings

Power-law distributions arise from energy-dependent diffusion and friction.

The theory generalizes the fluctuation-dissipation theorem for non-equilibrium systems.

Stationary solutions include Tsallis distributions in strong friction regimes.

Abstract

The two-variable Langevin equations, modeling the Brownian motion of a particle moving in a potential and leading to the Maxwell-Boltzmann distribution of the corresponding Fokker-Planck equation, are shown to give rise to types of stationary power-law distributions through the multiplicative noise. The power-law distributions induced by this inhomogeneous noise are proved to be a result of that the relation of diffusion to friction depends on the energy. We understand the conditions under which the power-law distributions are produced and how they are produced in systems away from equilibrium, and hence derive a generalized fluctuation-dissipation theorem. This leads to a generalized Klein-Kramers equation, and a generalized Smoluchowski equation for the particle moving in a strong friction medium, whose stationary-state solutions are exactly Tsallis distribution.