The Quantum Vlasov Equation and its Markov Limit
  The adiabatic particle number in mean field theory obeys a quantum Vlasov equation which is nonlocal in time. For weak, slowly varying electric fields this particle number can be identified with the single particle distribution function in phase space, and its time rate of change is the appropriate effective source term for the Boltzmann-Vlasov equation. By analyzing the evolution of the particle number we exhibit the time structure of the particle creation process in a constant electric field, and derive the local form of the source term due to pair creation. In order to capture the secular Schwinger creation rate, the source term requires an asymptotic expansion which is uniform in time, and whose longitudinal momentum dependence can be approximated by a delta function only on long time scales. The local Vlasov source term amounts to a kind of Markov limit of field theory, where information about quantum phase correlations in the created pairs is ignored and a reversible Hamiltonian evolution is replaced by an irreversible kinetic one. This replacement has a precise counterpart in the density matrix description, where it corresponds to disregarding the rapidly varying off-diagonal terms in the adiabatic number basis and treating the more slowly varying diagonal elements as the probabilities of creating pairs in a stochastic process. A numerical comparison between the quantum and local kinetic approaches to the dynamical backreaction problem shows remarkably good agreement, even in quite strong electric fields, over a large range of times. 
Yuval Kluger, Emil Mottola (Los Alamos), and Judah M. Eisenberg (Tel   Aviv)