Polynomial expansion of diffusion and drift coefficients for classical and quantum statistics

We study by means of a kinetic approach recently proposed [Phys. Rev. E 49, 5103 (1994)] both classical and quantum statistics, stationary solutions of a non-linear Fokker-Planck equation. The non-linear term in this equation has origin in a generalized exclusion-inclusion principle which takes into account the quantum effects due to the bosonic, fermionic or intermediate behaviour of the particles. If the drift and diffusion coefficients are polynomials, in the velocity variable, of degree 2M + 1 and 2N, respectively, only one well-defined statistics can be derived, for any particular pair of values (M, N). In the case of classical particles, in addition to the macroscopic Fokker-Planck equation, we derive the associated microscopic Langevin equation and demonstrate that the pairs (0,0), (1,0) and (0,1) correspond, respectively, to Maxwell-Boltzmann, Druyvenstein and Tsallis statistical distributions. We study also the distribution related to the pair (1, 1) which is the simplest generalization of the Tsallis [1] one. Finally, the above statistics are extended to the quantum case of particles obeying the exclusion-inclusion principle.