A proper nonlocal formulation of maximum entropy principle for Fermi, Bose and Fractional Eclusion Statistics

By considering the Wigner formalism the quantum maximum entropy principle (QMEP) is here asserted as the fundamental principle of quantum statistical mechanics when it becomes necessary to treat systems in partially specified quantum mechanical states. From one hand, the main difficulty in QMEP is to define an appropriate quantum entropy that explicitly incorporates quantum statistics. From another hand, the availability of rigorous quantum hydrodynamic (QHD) models is a demanding issue for a variety of quantum systems like, interacting fermionic and bosonic gases, confined carrier transport in semiconductor heterostrucures, anyonic systems, etc. We present a rigorous nonlocal formulation of QMEP by defining a quantum entropy that includes Fermi, Bose and, more generally, fractional exclusion statistics. In particular, by considering anyonic systems satisfying fractional exclusion statistic, all the results available in the literature are generalized in terms of both the kind of statistics and a nonlocal description for excluson gases. Finally, gradient quantum corrections are explicitly given at different levels of degeneracy and classical results are recovered when \hbar tends to 0.