H-theorem and generalized entropies within the framework of nonlinear kinetics

In the present effort we consider the most general nonlinear particle kinetics within the framework of the Fokker-Planck picture. We show that the kinetics imposes the form of the generalized entropy and subsequently we demonstrate the H-theorem. The particle statistical distribution is obtained, both as stationary solution of the nonlinear evolution equation and as the state which maximizes the generalized entropy. The present approach allows to treat the statistical distributions already known in the literature in a unifying scheme. As a working example we consider the kinetics, constructed by using the κ-exponential exp_{κ}(x)=( 1+κ ²x ²+κx) ^1/κ recently proposed, which reduces to the standard exponential as the deformation parameter κ approaches to zero and presents the relevant power law asymptotic behaviour exp_{κ}(x) limit_∼x→±∞|2κx| ^±1/|κ|. The κ-kinetics obeys the H-theorem and in the case of Brownian particles, admits as stationary state the distribution f= Z^-1exp _{{ κ}} [-( βmv²/2- μ)] which can be obtained also by maximizing the entropy S _κ=∫d ⁿv [c(κ)f ^1+κ+c(-κ)f ^1-κ] with c( κ)=- Z^κ/[2 κ(1+ κ)] after properly constrained.