**6. Conclusions**

In this paper we have proposed an algebraic approach to study many body particle systems obeying a non-conventional statistics, in the semiclassical picture. A nonlinear Fokker-Planck equation, describing the kinetic of collectively interacting particles, has been obtained according to a kinetic interaction principle. The particle current is fixed by the two functions *a*(*f*) and *b*(*f*) that regulate the transition probability from the departing site to the arrival site in a way that depends only on the population of the initial and final sites, respectively. In this formalism, bosons-like and fermions-like particles follow from a very easy assumption on the function *a*(*f*) and *b*(*f*) by means of a generalized version of the inclusion/exclusion principle given by *b*(*f*) -∓ *a*(*f*) = 1, with *b*(*f*) = *a*(*c* ± *f*) for a generalized composition law that fixes the form of the functions *a*(*f*), and then *b*(*f*), and consequently fixes the steady particle distribution.

Following this approach, we have studied boson-like and fermion like quons of type II [26], whose underling algebra is related with the generalized sum (26), as well as boson-like and fermion like quons of type I [13], whose underlying algebra is defined by the generalized sum (44). It has been shown that the kinetic of type II quons is described by a nonlinear Fokker-Planck equation with a nonlinear drift current and a linear diffusive current like in the case of standard Bose and Fermi particles; otherwise, the kinetic of type I quons is described by a nonlinear Fokker-Planck equation with a nonlinear drift current and a nonlinear diffusive current.

Finally, let us remark that, following the same approach described in this work, several non conventional statistics in the classical picture can be obtained employing different composition laws. For instance, in addition to the *κ*-sum and the *q*-sum studied in this paper, other examples can be found in the framework of the generalized statistical mechanics [48,49].

**Author Contributions:** The authors have equally contributed to the manuscript. They all have read and approved its final version.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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