Incollection,
H-Theorem from Nonlinear Fokker-Planck Equations: Associated Entropies
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
Among the most investigated systems exhibiting nonlinear phenomena, one should
mention those characterized by anomalous diffusion. Nonlinear Fokker-Planck
equations (NFPEs) provide a powerful tool to treat such systems.
Recently, a general NFPE has been proposed by two of the authors through a direct derivation from the master equation1. The introduction of nonlinear transition rates into the
master equation lead to this NFPE which generalizes, for instance, the one
related to Tsallis' thermostatistics and provides a general tool to model
nonlinear diffusion in the presence of external forces. Here, we calculate the
stationary state distributions of such a NFPE in some cases. We derive a
generalized entropy for which the NFPE satisfies the H-theorem.
The results implicate some restrictions to the parameters of the NFPE.
Some particular cases of the system are discussed.
Additionally, we consider more general transition rates in the master equation and derive a
second NFPE. Within this second approach, the
functional forms of the transition rates in the original master equation are related
to generalized entropies, in order to satisfy the H-theorem. These entropies may be
the same for different functional forms of the transition rates. Hence, we can say that, in
this scenario, the dynamics of a system chooses its entropy.
1) Curado and Nobre, Phys. Rev. E 67, 021107, 2003
