Classical model of bosons and fermions

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Abstract
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 κ, 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 κ. When we limit ourselves to Brownian particles, we derive exactly for κ=-1 the Fermi-Dirac (FD) distribution, for κ=0 the Maxwell-Boltzmann distribution, and for κ=1 the Bose-Einstein (BE) distribution. When κ 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 κ 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.