We consider free lattice fermions subjected to a static bounded potential and
a time- and space-dependent electric field. For any bounded convex region
$\mathcal{R}\subset \mathbb{R}^{d}$ ($d\geq 1$) of space, electric fields
$\mathcal{E}$ within $\mathcal{R}$ drive currents. At leading order, uniformly
with respect to the volume $\left| \mathcal{R}\right|$ of $\mathcal{R}$ and
the particular choice of the static potential, the dependency on $\mathcal{E}$
of the current is linear and described by a conductivity distribution. Because
of the positivity of the heat production, the real part of its Fourier
transform is a positive measure, named here (microscopic) conductivity measure
of $\mathcal{R}$, in accordance with Ohm's law in Fourier space. This finite
measure is the Fourier transform of a time-correlation function of current
fluctuations, i.e., the conductivity distribution satisfies Green-Kubo
relations. We additionally show that this measure can also be seen as the
boundary value of the Laplace-Fourier transform of a so-called quantum current
viscosity. The real and imaginary parts of conductivity distributions satisfy
Kramers-Kronig relations. At leading order, uniformly with respect to
parameters, the heat production is the classical work performed by electric
fields on the system in presence of currents. The conductivity measure is
uniformly bounded with respect to parameters of the system and it is never the
trivial measure $0\,\mathrm{d}\nu$. Therefore, electric fields generally
produce heat in such systems. In fact, the conductivity measure defines a
quadratic form in the space of Schwartz functions, the Legendre-Fenchel
transform of which describes the resistivity of the system. This leads to
Joule's law, i.e., the heat produced by currents is proportional to the
resistivity and the square of currents