1 October 2001
Physics Letters A 288 (2001) 283–291
H-theorem and generalized entropies within the framework
of nonlinear kinetics
Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Istituto Nazionale di Fisica della Materia, Unitá del Politecnico di Torino, Torino, Italy
Received 13 March 2001; received in revised form 24 July 2001; accepted 1 August 2001
Communicated by C.R. Doering
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-
(x) = (
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
The κ-kinetics obeys the H-theorem and in the case of Brownian particles, admits as stationary state the distribution f =
/2 − µ)] which can be obtained also by maximizing the entropy S
with c(κ) =−Z
/[2κ(1 + κ)] after properly constrained.
2001 Elsevier Science B.V. All rights reserved.
PACS: 05.10.Gg; 05.20.-y
In the last few decades there has been an inten-
sive discussion on nonconventional classical or quan-
tum statistics. For instance, there are several exper-
imental evidences of distributions with tails which
exhibit a power law decay. Up to now several en-
tropies with the ensuing statistics have been consid-
ered. A question which arises spontaneously is if it
is possible to obtain the stationary statistical distribu-
tion of the various physical systems within the frame-
E-mail address: firstname.lastname@example.org (G. Kaniadakis).
work of a time dependent scheme. The problem of
the nonlinear kinetics from a more general point of
view in the Fokker–Planck picture has been consid-
ered only in 1994. In Ref.  an evolution equation
has been proposed, describing a generic nonlinear ki-
netics. Subsequently, some properties of this kinetics
have been studied in Refs. [2–4]. After 1995, in the
Fokker–Planck picture, the anomalous diffusion has
been linked with the time dependent Tsallis statisti-
cal distribution [5–11]. Finally, the kinetics described
by nonlinear Fokker–Planck equationshas been recon-
sidered in Refs. [12,13].
Recently, in paper  it has been proposed the fol-
lowing new, one-parameter deformation of the expo-
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.