Fractional diffusion limit for collisional kinetic equations
نویسندگان
چکیده
This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation. Mathematics Subject Classification (2000): 76P05 Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05], 26A33 Fractional derivatives and integrals.
منابع مشابه
Fractional diffusion limit for collisional kinetic equations: a Hilbert expansion approach
We develop a Hilbert expansion approach for the derivation of fractional diffusion equations from the linear Boltzmann equation with heavy tail equilibria. 1 Setting of the result 1.
متن کاملAnomalous diffusion limit for kinetic equations with degenerate collision frequency
This paper is devoted to hydrodynamic limits for collisional linear kinetic equations. It is a classical result that under certain conditions on the collision operator, the long time/small mean free path asymptotic behavior of the density of particles can be described by diffusion type equations. We are interested in situations in which the degeneracy of the collision frequency for small veloci...
متن کاملFractional diffusion limit for collisional kinetic equations: A moments method
This paper is devoted to hydrodynamic limits of linear kinetic equations. We consider situations in which the thermodynamical equilibrium is described by a heavy-tail distribution function rather than a maxwellian distribution. A similar problem was addressed in [14] using Fourier transform and it was shown that the long time/small mean free path behavior of the solution of the kinetic equation...
متن کاملNumerical Schemes for Kinetic Equations in the Anomalous Diffusion Limit. Part I: The Case of Heavy-Tailed Equilibrium
Abstract. In this work, we propose some numerical schemes for linear kinetic equations in the anomalous diffusion limit. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion type equation. However, when a heavy-tailed distribution is considered, another time scale is...
متن کاملGeneralized phase-space kinetic and diffusion equations for classical and dispersive transport
We formulate and solve a physically-based, phase space kinetic equation for transport in the presence of trapping. Trapping is incorporated through a waiting time distribution function. From the phase-space analysis, we obtain a generalized diffusion equation in configuration space. We analyse the impact of the waiting time distribution, and give examples that lead to dispersive or nondispersiv...
متن کامل