1212.3980 (Guo Ran et al.)

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

Guo Ran, Du Jiulin

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.

View original: http://arxiv.org/abs/1212.3980