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

We show that the general two-variable Langevin equations with inhomogeneous noise and friction can generate many different forms of power-law distributions. By solving the corresponding stationary Fokker-Planck equation, we can obtain a condition under which these power-law distributions are accurately created in a system away from equilibrium. This condition is an energy-dependent relation between the diffusion coefficient and the friction coefficient and thus it provides a fluctuation-dissipation relation for nonequilibrium systems with power-law distributions. Further, we study the specific forms of the Fokker-Planck equation that correctly leads to such power-law distributions, and then present a possible generalization of Klein-Kramers equation and Smoluchowski equation to a complex system, whose stationary-state solutions are exactly a Tsallis distribution.