Velocity Averaging for the Wigner Kinetic Equation in the Semiclassical Regime
François Golse, Jakob Möller
TL;DR
The paper addresses whether velocity averaging lemmas can be transferred to the Wigner equation in the semiclassical regime. It develops an $L^2$-based velocity averaging framework for mixed states with uniform Hilbert-Schmidt bounds, obtaining Sobolev regularity for velocity-averaged observables, and analyzes the one-dimensional density regularity. For pure states, it characterizes Wigner transforms and shows that semiclassical limits yield monokinetic Wigner measures, which obstruct velocity averaging and provide a natural closure for Madelung hydrodynamics. Consequently, the authors derive Madelung equations rapidly from pure-state dynamics, clarifying when velocity averaging is applicable and highlighting the fundamental differences between mixed and pure quantum states in the classical limit.
Abstract
This paper discusses the possibility of applying the velocity averaging theorems in [F. Golse, P.-L. Lions, B. Perthame, R. Sentis: J. Funct. Anal. 76(1):110--125, 1988] to the Wigner equation governing the quantum evolution of the Wigner transform of quantum density operators. Our first main results address the case of the Wigner function of a special class of density operators associated to mixed states, whose Hilbert-Schmidt norm is of order $\hbar^{d/2}$, where $d$ is the space dimension and $\hbar$ the reduced Planck constant. In space dimension $d=1$, we prove that the density function belongs to the Sobolev space $H^s(\mathbb R)$ for some $s>0$. In the case of pure states, we first obtain a characterization of the Wigner transform of rank-one quantum density operators, and apply this characterization (1) to analyze a rather general setting in which velocity averaging cannot apply to the Wigner functions of a family of rank-one density operators whose evolution is governed by the von Neumann equation, and (2) to obtain a quick derivation of Madelung's system of quantum hydrodynamic equations. This derivation provides a physical explanation of one key assumption used in the proof of the negative result (1) described above.