Abstract
In this paper, we study the long-time behaviour of solutions to the Vlasov-Fokker-Planck equation where the confining potential is non-convex. This is a nonlocal nonlinear partial differential equation describing the time evolution of the probability distribution of a particle moving under the influence of a non-convex potential, an interaction potential, a friction force and a stochastic force. Using the free-energy approach, we show that under suitable assumptions solutions of the Vlasov-Fokker-Planck equation converge to an invariant probability.
Highlights
1.1 The Vlasov-Fokker-Planck equation This paper concerns with the convergence to equilibrium of the following Vlasov-Fokker-Planck (VFP) equation∂tρ = − divq ρp + divp [ρ (∇qV + ∇qF ∗ ρ + p)] + λ∆pρ. (1.1)for the probability density function ρ(t, q, p) in R+ × Rd × Rd
We study the long-time behaviour of solutions to the Vlasov-FokkerPlanck equation where the confining potential is non-convex
Using the free-energy approach, we show that under suitable assumptions solutions of the Vlasov-FokkerPlanck equation converge to an invariant probability
Summary
1.1 The Vlasov-Fokker-Planck equation This paper concerns with the convergence to equilibrium of the following (dimensionless) Vlasov-Fokker-Planck (VFP) equation. For the probability density function ρ(t, q, p) in R+ × Rd × Rd. In the above equation, subscripts as in divq and ∆p indicate that the differential operators act only on those variables. Equation (1.1) describes the evolution of the probability density ρ(t, q, p) for the following self-stabilizing diffusion process in the phase space dQ(t) = P (t) dt, √. DP (t) = −∇V (Q(t)) dt − ∇F ∗ ρt(Q(t)) dt − P (t) dt + 2λ dW (t) This is a nonlinear diffusion since the own law of the process intervenes in the drift. It models the movement of a particle under a fixed potential V , an interaction potential. Equation (1.1) and system (1.2) have been widely used in chemistry and statistical mechanics such as a model for chemical reactions or as a model for particles interacting through Coulomb or gravitational forces [19, 7]
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