Equilibrium wigner function for fermions and bosons in the case of a general energy dispersion relation

The approach based on the Wigner function is considered as a viable model of quantum transport which allows, in analogy with the semiclassical Boltzmann equation, to restore a description in the phase-space. A crucial point is the determination of the Wigner function at the equilibrium which stems from the equilibrium density function. The latter is obtained by a constrained maximization of the entropy whose formulation in a quantum context is a controversial issue. The standard expression due to Von Neumann, although it looks a natural generalization of the classical Boltzmann one, presents two important drawbacks: it is conserved under unitary evolution time operators, and therefore cannot take into account irreversibility; it does not include neither the Bose nor the Fermi statistics. Recently a diagonal form of the quantum entropy, which incorporates also the correct statistics, has been proposed in Snoke et al. (2012) and Polkovnikov (2011). Here, by adopting such a form of entropy, with an approach based on the Bloch equation, the general condition that must be satisfied by the equilibrium Wigner function is obtained for general energy dispersion relations, both for fermions and bosons. Exact solutions are found in particular cases. They represent a modulation of the solution in the non degenerate situation.