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We show that the general two-variable Langevin equations with inhomogeneousnoise and friction can generate many different forms of power-lawdistributions. By solving the corresponding stationary Fokker-Planck equation,we can obtain a condition under which these power-law distributions areaccurately created in a system away from equilibrium. This condition is anenergy-dependent relation between the diffusion coefficient and the frictioncoefficient and thus it provides a fluctuation-dissipation relation fornonequilibrium systems with power-law distributions. Further, we study thespecific forms of the Fokker-Planck equation that correctly leads to suchpower-law distributions, and then present a possible generalization ofKlein-Kramers equation and Smoluchowski equation to a complex system, whosestationary-state solutions are exactly a Tsallis distribution.

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