Abstract
The non-linear Fokker-Planck equation arises in describing the evolution of stochastic system, which is a variant of the Boltzmann equation modeling the evolution of the random system with Brownian motion, where the collision term is replaced by a drift-diffusion operator. This model conserves mass, momentum and energy; the dissipation is much weaker than that in a simplified model we considered before which conserved only mass, thus more difficult to analyze. The macro-micro decomposition of the solution around the local Maxwellian introduced by T.-P. Liu, T. Yang and S.-H. Yu for Boltzmann equation is used, to reformulate the model into a fluid-type system incorporate viscosity and heat diffusion terms, coupled with an equation of the microscopic part. The viscosity and heat diffusion terms can give dissipative mechanism for the analysis of the model.
Highlights
The non-linear Fokker-Planck equations arise in describing the evolution of stochastic system such as the erratic motions of small particles immersed in fluids, fluctuations of the intensity of laser light, velocity distributions of fluid particles in turbulent flows, or the stochastic behavior of exchange rates
The non-linear Fokker-Planck equation arises in describing the evolution of stochastic system, which is a variant of the Boltzmann equation modeling the evolution of the random system with Brownian motion, where the collision term is replaced by a drift-diffusion operator
This model conserves mass, momentum and energy; the dissipation is much weaker than that in a simplified model we considered before which conserved only mass, more difficult to analyze
Summary
The non-linear Fokker-Planck equations arise in describing the evolution of stochastic system such as the erratic motions of small particles immersed in fluids, fluctuations of the intensity of laser light, velocity distributions of fluid particles in turbulent flows, or the stochastic behavior of exchange rates. As a variant of the Boltzmann equation modeling the evolution of the random system with Brownian motion, the collision term is replaced by a drift-diffusion operator Note that this famous model conserves mass, momentum and energy [8] [9], and the right hand side of the equation is strongly non-linear, more precisely, the dissipation is much weaker than that in a simplified model considered in [10], more difficult to analyze. By careful computation, we derived the viscosity and heat diffusion terms in the fluid-type system from the microscopic part, which give the dissipative mechanism for the analysis of the model
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