  Electric resistance in conducting media is related to heat (or entropy)
production in presence of electric fields. In this paper, by using Araki's
relative entropy for states, we mathematically define and analyze the heat
production of free fermions within external potentials. More precisely, we
investigate the heat production of the non-autonomous C*-dynamical system
obtained from the fermionic second quantization of a discrete Schr\"odinger
operator with bounded static potential in presence of an electric field that is
time- and space-dependent. It is a first preliminary step towards a
mathematical description of transport properties of fermions from thermal
considerations. This program will be carried out in several papers. The regime
of small and slowly varying in space electric fields is important in this
context, and is studied the present paper. We use tree-decay bounds of the
$n$-point, $n\in 2\mathbb{N}$, correlations of the many-fermion system to
analyze this regime. We verify below the 1st law of thermodynamics for the
system under consideration. The latter implies, for systems doing no work, that
the heat produced by the electromagnetic field is exactly the increase of the
internal energy resulting from the modification of the (infinite volume) state
of the fermion system. The identification of heat production with an energy
increment is, among other things, technically convenient. We initially focus
our study on non-interacting (or free) fermions, but our approach will be later
applied to weakly interacting fermions.
