Quantum maximum entropy principle for a system of identical particles

The quantum maximum entropy principle is proposed here as a rigorous procedure that should be employed when it becomes necessary to treat systems in partially specified quantum-mechanical states. By considering the reduced density matrix, within a second-quantized formalism, we have described a many-body model for identical particles. In this respect we have shown that using the general definition of quantum entropy we have incorporated the indistinguishability principle of a system of identical particles. Thus, by introducing a QMEP, in nonequilibrium conditions we have formulated a nonlocal theory that contains implicitly the Fermi and Bose statistics, a result which was left open since the Wigner seminal papers . We have determined a generalized Wigner equation where the effects of interactions are entirely contained in the definition of the effective potential. In this way we have recovered the quantum Hartree approximation, written in the Wigner formalism, plus some corrections due to fermion and/or boson correlations. We have developed to all orders in powers of a quantum closure procedure for the corresponding QHD system. As simple examples we have reported explicitly, in the cases N=0,1, the closure relations evaluated up to the first quantum approximation. In particular, in the framework of Boltzmann statistics, we have recovered the well-known expression for the quantum chemical potential, for the quantum pressure, and for the quantum closure scheme in the case of the standard drift-diffusion model. When h→0 we recover the framework of classical MEP approach for a fermion or boson system. In closing, we remark that this approach can be further generalized to develop a nonlocal theory for the fractional statistics. In this case, the entropy of the system should be expressed in terms of the statistical weight introduced by Wu and the QMEP should be developed in terms of the reduced density matrix for particles obeying fractional exclusion statistics