By using the kinetic interaction principle, the quons statistics in the framework of kinetic theory is introduced. This is done by properly generalizing the inclusion/exclusion principle of standard boson and fermion statistics within a nonlinear classical model. The related nonlinear Fokker-Planck equation is introduced and the corresponding steady distribution describing quons statistics of type I and type II is derived.
By using the kinetic interaction principle, the quons statistics in the framework of kinetic theory is introduced. This is done by properly generalizing the inclusion/exclusion principle of standard boson and fermion statistics within a nonlinear classical model. The related nonlinear Fokker-Planck equation is introduced and the corresponding steady distribution describing quons statistics of type I and type II is derived.
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.
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 \ensuremath{\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 \ensuremath{\kappa}. When we limit ourselves to Brownian particles, we derive exactly for \ensuremath{\kappa}=-1 the Fermi-Dirac (FD) distribution, for \ensuremath{\kappa}=0 the Maxwell-Boltzmann distribution, and for \ensuremath{\kappa}=1 the Bose-Einstein (BE) distribution. When \ensuremath{\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 \ensuremath{\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.
The quon algebra describes particles, “quons, ” that are neither fermions nor bosons using a label q that parametrizes a smooth interpolation between bosons (q = +1) and fermions (q = −1). We derive “conservation of statistics” relations for quons in relativistic theories, and show that in relativistic theories quons must be either bosons or fermions. There are three reasons to study theories that allow violations of statistics, i.e., that allow particles that are neither bosons nor fermions. One, which may seem frivolous, is to stretch the framework of quantum physics and to find out what possibilities open up when one does so. The second, quite concrete, is to respond to experimental interest in high-precision tests of the symmetrization postulate (that all identical particles occur in one-dimensional representations of the symmetric
We study the quantization of a classical system of interacting particles obeying a recently proposed kinetic interaction principle (KIP) [G. Kaniadakis, Physica A 296, 405 (2001)]. The KIP fixes the expression of the Fokker-Planck equation describing the kinetic evolution of the system and imposes the form of its entropy. In the framework of canonical quantization, we introduce a class of nonlinear Schrödinger equations (NSEs) with complex nonlinearities, describing, in the mean-field approximation, a system of collectively interacting particles whose underlying kinetics is governed by the KIP. We derive the Ehrenfest relations and discuss the main constants of motion arising in this model. By means of a nonlinear gauge transformation of the third kind, it is shown that in the case of constant diffusion and linear drift, the class of NSEs obeying the KIP is gauge-equivalent to another class of NSEs containing purely real nonlinearities depending only on the field rho=|psi|(2) .
I discuss the statistics of "quons" (pronounced to rhyme with muons), particles whose annihilation and creation operators obey the q-deformed commutation relation (the quon algebra or q-mutator) which interpolates between fermions and bosons. Topics discussed include representations of the quon algebra, proof of the TCP theorem and clustering, violation of the usual locality properties, and experimental constraints on violations of the Pauli exclusion principle (i.e., Fermi statistics) and of Bose statistics.