Power-law distributions and fluctuation-dissipation relation in the stochastic dynamics oftwo-variable Langevin equations

Abstract

We show that the general two-variable Langevin equations with inhomogeneous noise andfriction can generate many different forms of power-law distributions. By solving thecorresponding stationary Fokker–Planck equation, we can obtain a condition under whichthese power-law distributions are accurately created in a system away from equilibrium.This condition is an energy-dependent relation between the diffusion coefficient and thefriction coefficient and thus it provides a fluctuation-dissipation relation for nonequilibriumsystems with power-law distributions. Further, we study the specific forms of theFokker–Planck equation that correctly lead to such power-law distributions, and thenpresent a possible generalization of the Klein–Kramers equation and the Smoluchowskiequation to a complex system, whose stationary-state solutions are exactly a Tsallisdistribution.