  We study the probability distribution of the number of particle and
antiparticle pairs produced via the Schwinger effect when a uniform but
time-dependent electric field is applied to noninteracting scalars or spinors
initially at a thermodynamic equilibrium. We derive the formula for the
characteristic function by employing techniques in mesoscopic physics,
reflecting a close analogy between the Schwinger effect and mesoscopic
tunneling transports. In particular, we find that the pair production in a
medium is enhanced (suppressed) for scalars (spinors) due to the Bose
stimulation (Pauli blocking). Furthermore, in addition to the production of
accelerated pairs by the electric field, the annihilation of decelerated pairs
is found to take place in a medium. Our formula allows us to extract the
probability distributions in various situations, such as those obeying the
generalized trinomial statistics for spin-momentum resolved counting and the
bidirectional Poisson statistics for spin-momentum unresolved counting.
