The non-equilibrium distribution function of particles and anti-particles created in strong fields

We investigate the quantum Vlasov equation with a source term describing the spontaneous particle creation in strong fields. The back-reaction problem is treated by solving this kinetic equation together with the Maxwell equation which determines the induced time-dependent electric field in the system. The evolution of distribution functions for bosons and fermions is studied numerically. We found that the system shows a regular dynamic behavior if the back-reaction is neglected. But if the back-reaction is included, it is not the case and some stochastic features are clearly revealed in the non-equilibrium distribution function..