Effect of symmetry in the many-particle Wigner funcion

An analysis of the Wigner function for identical particles is presented. Four situations have been considered. (i) The first is scattering process between two indistinguishable particles described by a minimum uncertainty wave packets showing the exchange and correlation effects in Wigner phase space. (ii) An equilibrium ensemble of N particles in a one-dimensional box and in a one-dimensional harmonic potential is considered second, showing that the reduced one-particle Wigner function, as a function of the energy defined in the Wigner phase space, tends to the Fermi-Dirac or to the Bose-Einstein distribution function, depending on the considered statistics. (iii) The third situation is reduced one-particle transport equation for the Wigner function, in the case of interacting particles, showing the need for the two-particle reduced Wigner function within the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy scheme. (iv) Finally, the electron-phonon interaction in the two-particle case is considered, showing coparticipation of two electrons in the interaction with the phonon bath.