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arXiv · 1212.3980

Power-law behaviors from the two-variable Langevin equation: Ito's and Stratonovich's Fokker-Planck equations

Ran Guo·Jiulin Du
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Abstract

We study power-law behaviors produced from the stochastically dynamical system governed by the well-known two-variable Langevin equations. The stationary solutions of the corresponding Ito's, Stratonovich's and the Zwanzig's (the backward Ito's) Fokker-Planck equations are solved under a new fluctuation-dissipation relation, which are presented in a unified form of the power-law distributions with a power index containing two parameter kappa and sigma, where kappa measures a distance away from the thermal equilibrium and sigma distinguishes the above three forms of the Fokker-Planck equations. The numerical calculations show that the Ito's, the Stratonovich's and the Zwanzig's form of the power-law distributions are all exactly the stationary solutions based on the two-variable Langevin equations.

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Ran Guo, Jiulin Du. 2012-12-18. Power-law behaviors from the two-variable Langevin equation: Ito's and Stratonovich's Fokker-Planck equations. https://doi.org/10.1088/1742-5468%2F2013%2F02%2Fp02015

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