CLASSICAL-MODEL OF BOSON AND FERMIONS

In a previous work [Phys. Rev. E 48, 4263 (1993)] we have derived a nonlinear one-dimensional kinetic equation for the distribution function of particles obeying an exclusion principle. In the present work, on the same grounds, we extend this kinetics to D-dimensional continuous or discrete space, in order to study the distribution function of particles obeying a generalized exclusion-inclusion Pauli principle (EIP). This exclusion or inclusion principle is introduced into the classical transition rates by means of an inhibition or an enhancement factor, which contains a parameter kappa, whose values range between -1 and +1 and can balance the effect of the full or partial validity of EIP. After deriving the kinetic equation we obtain a general expression of the stationary distribution function depending on the value we give to the parameter kappa. When we limit ourselves to Brownian particles, we derive exactly for kappa = -1 the Fermi-Dirac (FD) distribution, for kappa = 0 the Maxwell-Boltzmann distribution, and for kappa = 1 the Bose-Einstein (BE) distribution. When kappa assumes an intermediate value, except zero, between the extreme values -1 and +1, we obtain statistical distributions different from the FD and BE ones. We attribute to the parameter kappa the meaning of the degree of indistinguishability of identical particles, the degree of antisymmetrization, or the symmetrization of the wave function of the particle system.