  Energy-transport equations for the transport of fermions in optical lattices
are formally derived from a Boltzmann transport equation with a periodic
lattice potential in the diffusive limit. The limit model possesses a formal
gradient-flow structure like in the case of the energy-transport equations for
semiconductors. At the zeroth-order high temperature limit, the
energy-transport equations reduce to the whole-space logarithmic diffusion
equation which has some unphysical properties. Therefore, the first-order
expansion is derived and analyzed. The existence of weak solutions to the
time-discretized system for the particle and energy densities with periodic
boundary conditions is proved. The difficulties are the nonstandard degeneracy
and the quadratic gradient term. The main tool of the proof is a result on the
strong convergence of the gradients of the approximate solutions. Numerical
simulations in one space dimension show that the particle density converges to
a constant steady state if the initial energy density is sufficiently large,
otherwise the particle density converges to a nonconstant steady state.
