regime (i.e. under conditions of very strong electric fields and large gradients). The result was a closed effective theory which could reproduce well the results of kinetic Monte Carlo simulations.

In this paper, instead of using the EMP we rely on the EPP, which allows us to find the distribution function that extremizes the entropy production subject to constraints on the conserved currents [@epvm]. We note that the EPP was used in [@christen] to construct a model of nonequilibrium electron transport and in [@christen2] to describe radiative heat transport in a photon gas, obtaining results that agree well with more sophisticated approaches.

The motivation for the choice of the EPP is threefold. This principle is deeply connected to nonequilibrium statistical physics, in particular to the evolution of fluctuations around a stationary state (we shall not discuss this connection here; for details see [@epvm; @min1; @ferz; @prigo]). Moreover, as it is rigorously shown in Ref. [@epvm] for nonrelativistic systems (see also [@ferz]), the distribution function obtained from the EPP actually solves the [*linearized*]{} Boltzmann equation, which means that the closure provided by this variational principle is able, at least in principle, to capture some features of the microscopic dynamics. Finally, the 