Power-law distributions and fluctuation-dissipation relation in the stochastic dynamics of two-variable Langevin equations
Abstract
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 lead to such power-law distributions, and then present a possible generalization of the Klein-Kramers equation and the Smoluchowski equation to a complex system, whose stationary-state solutions are exactly a Tsallis distribution.
- Publication:
-
Journal of Statistical Mechanics: Theory and Experiment
- Pub Date:
- February 2012
- DOI:
- 10.1088/1742-5468/2012/02/P02006
- arXiv:
- arXiv:1202.0707
- Bibcode:
- 2012JSMTE..02..006D
- Keywords:
-
- Condensed Matter - Statistical Mechanics;
- Physics - Chemical Physics;
- Physics - Classical Physics;
- Physics - Plasma Physics;
- Physics - Space Physics
- E-Print:
- 18 pages,63 references