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

Technical Summary: Long Time Behavior of Fokker-Planck Equations for Bosons and Fermions

Problem Statement
This paper investigates the long-time behavior of global weak solutions to a nonlinear, spatially inhomogeneous quantum Fokker-Planck equation posed on a classical kinetic phase space $(t, x, p) \in [0, \infty) \times \mathbb{T}^d \times \mathbb{R}^d$. The equation models the statistical evolution of indistinguishable quantum particles (fermions or bosons) and is given by:
$$\partial_t f + p \cdot \nabla_x(f + \kappa f^2) = \text{div}_p(\nabla_p f + p(f + \kappa f^2)),$$
where $\kappa = -1$ corresponds to fermions (Pauli exclusion principle) and $\kappa = 1$ to bosons (inclusion principle). The term $1 \pm f$ appears in both the transport and collision parts, distinguishing this model from classical kinetic equations and other quantum models (such as Wigner-Fokker-Planck or Lindblad equations).

The primary objective is to prove that solutions $f(t)$ converge exponentially to a unique global equilibrium $f_\infty$ (the Fermi-Dirac or Bose-Einstein distribution) without assuming that the initial data is close to equilibrium. This addresses a significant gap in the literature, as previous results for spatially inhomogeneous cases relied on $H^1$ hypocoercivity methods requiring small perturbations from equilibrium [39, 34].

Methodology
The analysis employs an $L^2$-hypocoercivity method, adapted from linear kinetic equations [27, 20, 26, 8, 12] to handle the nonlinearity and the specific structure of the quantum Fokker-Planck operator. The core challenge is that the entropy dissipation functional lacks gradients in the spatial variable $x$, preventing the direct application of standard Sobolev inequalities to derive exponential decay.

The methodology proceeds through the following steps:

1. Properties of Weak Solutions: The authors first establish that global weak solutions conserve mass and satisfy an $L^1$ contraction property (Lemma 3.4). This contraction allows them to prove that if the initial data $f_0$ is bounded between two specific equilibrium-like profiles (determined by constants $\beta_-$ and $\beta_+$), the solution $f(t)$ remains within these bounds for all time (Corollary 3.5). These bounds are crucial for controlling the nonlinearity.

2. Projection Operator: A nonlinear projection operator $\Pi$ is introduced, mapping a solution $f$ to the manifold of local equilibria $\Pi f(t, x, p) = \frac{\beta(t, x)M(p)}{1 - \kappa \beta(t, x)M(p)}$, where $\beta(t, x)$ is chosen to preserve the local mass density $\rho(t, x)$. This projection is nonlinear due to the dependence of the equilibrium form on the local density.

3. Lyapunov Functional Construction: The main technical contribution is the construction of a modified Lyapunov functional $E[f|f_\infty]$. This functional is a perturbation of the relative entropy $H[f|f_\infty]$:
   $$E[f|f_\infty] := H[f|f_\infty] + \delta \int_{\mathbb{T}^d} \nabla_x \phi \cdot j \, dx,$$
   where $j$ is the macroscopic flux, $\phi$ is the solution to a Poisson equation $-\Delta_x \phi = \rho - \rho_\infty$, and $\delta > 0$ is a small parameter.

   • The relative entropy $H[f|f_\infty]$ provides control over the distance between $f$ and its local projection $\Pi f$ (via a generalized log-Sobolev inequality).
   • The cross-term involving $\nabla_x \phi$ and $j$ is designed to generate a negative term controlling the distance between the local projection $\Pi f$ and the global equilibrium $f_\infty$.

4. Hypocoercivity Estimate: By analyzing the time derivative of $E[f|f_\infty]$, the authors show that the "missing" spatial dissipation is recovered through the coupling of the transport and collision operators. Specifically, the time derivative of the cross-term yields a negative contribution proportional to $\|\rho - \rho_\infty\|^2$, which, via elliptic regularity and the projection properties, controls $\|\Pi f - f_\infty\|^2$. Combining this with the entropy dissipation estimate for $\|f - \Pi f\|^2$ allows the application of a Grönwall inequality.

Key Results
The paper proves the following main theorem (Theorem 2.1):
Assuming the existence of global weak solutions and that the initial data $f_0$ satisfies specific upper and lower bounds (related to the existence of a global equilibrium, particularly critical for bosons in dimensions $d \ge 3$), there exist explicitly computable constants $\lambda > 0$ and $c \ge 1$ such that:
$$\|f(t) - f_\infty\| \le c e^{-\lambda t} \|f_0 - f_\infty\|, \quad \forall t \ge 0,$$
where the norm is the weighted $L^2$ norm with weight $M^{-1}$ (the Maxwellian).

Crucially, this exponential convergence holds without the assumption that the initial data is close to the equilibrium in a Sobolev space. The result applies to the full nonlinear problem, provided the solution exists globally and satisfies the necessary regularity and boundedness conditions.

Significance and Claims
The authors claim that this work extends previous hypocoercivity results [39, 34] which were limited to the "close-to-equilibrium" regime. By constructing a Lyapunov functional based on the relative entropy and a nonlinear projection, the paper demonstrates that exponential decay to equilibrium can be established for the full nonlinear quantum Fokker-Planck equation in a global setting.

The analysis highlights the specific difficulties introduced by the nonlinearity of the projection operator $\Pi$ and the need to carefully handle the bounds on the distribution function to ensure the equivalence between the relative entropy and the $L^2$ distance. The paper does not claim to prove the global existence of solutions (which remains an open problem for general initial data) but rather assumes their existence to characterize their long-time asymptotic behavior. The results are strictly theoretical, relying on functional analysis and entropy methods rather than numerical simulations or experimental data.