Classical model of intermediate statistics

Abstract

In this work we present a classical kinetic model of intermediate statistics. In the case of Brownian particles we show that the Fermi-Dirac (FD) and Bose-Einstein (BE) distributions can be obtained, just as the Maxwell-Boltzmann (MB) distribution, as steady states of a classical kinetic equation that intrinsically takes into account an exclusion-inclusion principle. In our model the intermediate statistics are obtained as steady states of a system of coupled nonlinear kinetic equations, where the coupling constants are the transmutational potentials ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\kappa}}\mathrm{\ensuremath{\kappa}}\ensuremath{'}}$. We show that, besides the FD-BE intermediate statistics extensively studied from the quantum point of view, we can also study the MB-FD and MB-BE ones. Moreover, our model allows us to treat the three-state mixing FD-MB-BE intermediate statistics. For boson and fermion mixing in a D-dimensional space, we obtain a family of FD-BE intermediate statistics by varying the transmutational potential ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. This family contains, as a particular case, when ${\mathrm{\ensuremath{\eta}}}_{\mathrm{B}\mathrm{F}=0}$, the quantum statistics recently proposed by L. Wu, Z. Wu, and J. Sun [Phys. Lett. A 170, 280 (1992)]. When we consider the two-dimensional FD-BE statistics, we derive an analytic expression of the fraction of fermions. When the temperature T\ensuremath{\rightarrow}\ensuremath{\infty}, the system is composed by an equal number of bosons and fermions, regardless of the value of ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. On the contrary, when T\ensuremath{\rightarrow}0, ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$ becomes important and, according to its value, the system can be completely bosonic or fermionic, or composed both by bosons and fermions.