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Long time behavior of Fokker-Planck equations for bosons and fermions

Anton Arnold, Marlies Pirner, Gayrat Toshpulatov

TL;DR

The paper analyzes space-inhomogeneous Fokker-Planck dynamics with boson/fermion statistics through a nonlinear $f(1\pm f)$ factor, establishing exponential relaxation to a global quantum equilibrium without requiring near-equilibrium initial data. By blending $L^2$-hypocoercivity with a nonlinear projection onto local equilibria, and constructing a Lyapunov functional that couples relative entropy with a macroscopic flux term, the authors obtain quantitative decay in the weighted $L^2$ space $L^2(\mathbb{T}^d\times\mathbb{R}^d, M^{-1})$. Key ingredients include a nonlinear projection $\Pi f$, a relative entropy functional $H[f|f_\infty]$, a generalized log-Sobolev inequality, and a Picard-type coupling via a Poisson potential $\phi$ solving $-\Delta_x \phi = \rho-\rho_\infty$. The results extend hypocoercivity methods to semi-classical, spatially inhomogeneous quantum kinetic equations, yielding explicit decay rates and a robust framework for long-time behavior of bosonic and fermionic Fokker-Planck dynamics.

Abstract

This paper is concerned with space inhomogeneous quantum Fokker-Planck equations posed on a classical kinetic phase space. The nonlinear factor $f(1\pm f)$ appears both in the transport term and in the collison part of the Fokker-Planck operator, accounting for the inclusion principle of bosons and the exclusion principle of fermions. Assuming that global solutions exist, we prove exponential decay of the solutions to the global equilibrium in a weighted $L^2$-space without a close-to-equilibrium assumption. Our analysis is in the spirit of an $L^2$-hypocoercivity method. Our main Lyapunov functional is constructed from a logarithmic relative entropy and the (nonlinear) projection of the solution to the manifold of local-in-$x$ equilibria.

Long time behavior of Fokker-Planck equations for bosons and fermions

TL;DR

The paper analyzes space-inhomogeneous Fokker-Planck dynamics with boson/fermion statistics through a nonlinear factor, establishing exponential relaxation to a global quantum equilibrium without requiring near-equilibrium initial data. By blending -hypocoercivity with a nonlinear projection onto local equilibria, and constructing a Lyapunov functional that couples relative entropy with a macroscopic flux term, the authors obtain quantitative decay in the weighted space . Key ingredients include a nonlinear projection , a relative entropy functional , a generalized log-Sobolev inequality, and a Picard-type coupling via a Poisson potential solving . The results extend hypocoercivity methods to semi-classical, spatially inhomogeneous quantum kinetic equations, yielding explicit decay rates and a robust framework for long-time behavior of bosonic and fermionic Fokker-Planck dynamics.

Abstract

This paper is concerned with space inhomogeneous quantum Fokker-Planck equations posed on a classical kinetic phase space. The nonlinear factor appears both in the transport term and in the collison part of the Fokker-Planck operator, accounting for the inclusion principle of bosons and the exclusion principle of fermions. Assuming that global solutions exist, we prove exponential decay of the solutions to the global equilibrium in a weighted -space without a close-to-equilibrium assumption. Our analysis is in the spirit of an -hypocoercivity method. Our main Lyapunov functional is constructed from a logarithmic relative entropy and the (nonlinear) projection of the solution to the manifold of local-in- equilibria.
Paper Structure (10 sections, 11 theorems, 156 equations)

This paper contains 10 sections, 11 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Properties of weak solutions
  4. Approximation of weak solutions by smooth functions
  5. Mass conservation
  6. Contraction of the $L^1$ norm and maximum principle
  7. Lyapunov functional and exponential decay
  8. Projection operator
  9. Relative entropy
  10. Proof of Theorem \ref{['main th']}

Key Result

Theorem 2.1

Assume there exist constants $\beta_->0$ and $\, \beta_+>0$ such that $f_0$ satisfies for all $x \in \mathbb{T}^d$ and $p\in \mathbb{R}^d.$ We also require $\beta_{+}<(2\pi )^{d/2}$ when $\kappa=1.$ Let $f\in C([0,\infty), L^1\cap L^{\infty}(\mathbb{T}^d\times \mathbb{R}^d))$ be a weak solution to Eq such that Then there exist explicitly computable constants $\lambda>0$ and $c\geq 1$ (independen

Theorems & Definitions (21)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • ...and 11 more