  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.
