  Using the kinetic approach, we study the impact of the charged particles
dynamics due to the Schwinger effect on the electric field evolution during
inflation. As a simple model of the electric field generation, we consider the
kinetic coupling of the electromagnetic field to the inflaton via the term
$f^{2}(\phi)F_{\mu\nu}F^{\mu\nu}$ with the Ratra coupling function
$f=\exp(\beta\phi/M_{p})$. The production of charged particles is taken into
account in the Boltzmann kinetic equation through the Schwinger source term.
Produced particles are thermalized due to collisions which we model by using
the collision integral in the self-consistent relaxation time approximation. We
found that the current of created particles exhibits a non-Markovian character
and cannot be described by a simple Ohm's law relation $j\propto E$. On the
contrary, the electric current as well as the electric field are oscillatory
functions of time with decreasing amplitudes and a phase difference due to the
ballistic motion of charged carriers. Our qualitative results are checked by
using a hydrodynamic approach. Deriving a closed system of equations for the
number, current, and energy densities of charged particles and determining its
solution, we find a good agreement with the results obtained in the kinetic
approach.
