Classes of N-Dimensional Nonlinear Fokker-Planck Equations Associated to Tsallis Entropy

Mauricio S Ribeiro,Fernando D Nobre,Evaldo M F Curado

doi:10.3390/e13111928

Abstract

Several previous results valid for one-dimensional nonlinear Fokker-Planck equations are generalized to N-dimensions. A general nonlinear N-dimensional Fokker-Planck equation is derived directly from a master equation, by considering nonlinearitiesin the transition rates. Using nonlinear Fokker-Planck equations, the H-theorem is proved;for that, an important relation involving these equations and general entropic forms is introduced. It is shown that due to this relation, classes of nonlinear N-dimensional Fokker-Planck equations are connected to a single entropic form. A particular emphasis is given to the class of equations associated to Tsallis entropy, in both cases of the standard, and generalized definitions for the internal energy.

Highlights

The Boltzmann-Gibbs (BG) theory of statistical mechanics represents one of the most successful theoretical frameworks of physics [1,2]
One may modify the ansatz above to a very general form, as done in [29]; we are mostly interested in N -dimensional nonlinear Fokker-Planck equations (NLFPEs) associated with Tsallis entropy [like the one of Equation (19)], and so, we shall introduce a slight modification in the force term of Equation (23), wk,l (∆, t) = −
In the previous section we have proved the H-theorem, i.e., ≤ 0, for both NLFPEs of Equations (30) and (44), which are associated to the energy definitions of Equations (33) and (42), respectively

Summary

Introduction

The Boltzmann-Gibbs (BG) theory of statistical mechanics represents one of the most successful theoretical frameworks of physics [1,2]. One of the most important equations of non-equilibrium statistical mechanics is the linear Fokker-Planck equation (FPE); its time-dependent solution, for an external harmonic potential, is given by a Gaussian distribution [2]. NLFPEs, deriving relations involving terms of the corresponding NLFPE and the entropic form, using both standard [cf Equation (4)], and generalized definitions for the internal energy.

Results

Conclusion