We present a theory for carrying out homogenization limits for quadratic functions (called "energy densities") of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial) pseudo-differential operators (PDOs). The approach is based on the introduction of phase space Wigner (matrix) measures that are calculated by solving kinetic equations involving the spectral properties of the PDO. The weak limits of the energy densities are then obtained by taking moments of the Wigner measure.The very general theory is illustrated by typical examples like (semi)classical limits of Schrödinger equations (with or without a periodic potential), the homogenization limit of the acoustic equation in a periodic medium, and the classical limit of the Dirac equation. IntroductionWe consider the following type of initial value problems:where ε is a small parameter, u ε (t, x) is a vector-valued L 2 -function on R m x , and P ε is an anti-self-adjoint, matrix-valued (pseudo)-differential operator with a Weyl symbol given by P 0 (x, x/ε, εξ) + O(ε). Here P 0 = P 0 (x, y, ξ) is a smooth function that is periodic with respect to y. By ξ we denote the conjugate variable to the position x; that is, ξ = −i∇ x .The main assumptions are that the data u ε I are bounded in L 2 as ε goes to 0 and that u ε I oscillates at most at frequency 1/ε; for instance,A more general formulation of the assumptions on u ε I is given in definitions (1.26) and (1.27) below.

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In this paper the stability of the Vlasov–Poisson–Fokker–Planck with respect to the variation of its constant parameters, the scaled thermal velocity and the scaled thermal mean free path, is analyzed. For the case in which the scaled thermal velocity is the inverse of the scaled thermal mean free path and the latter tends to zero, a parabolic limit equation is obtained for the mass density. Depending on the space dimension and on the hypothesis for the initial data, the convergence result in L1 is weak and global in time or strong and local in time.

A rigorous derivation of the semiclassical Liouville equation for electrons which move in a crystal lattice (without the influence of an external field) is presented herein. The approach is based on carrying out the semiclassical limit in the band-structure Wigner equation. The semiclassical macroscopic densities are also obtained as limits of the corresponding quantum quantities.