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

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

2011-03-07

arxiv.org/abs/1103.1184v2

J. Stat. Mech.(2012)P02006

Physics

Condensed Matter

Statistical Mechanics

9 pages

Scientific paper

10.1088/1742-5468/2012/02/P02006

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.

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