III. Tsallis Theory, the Maximum Entropy Principle, and Evolution Equations

Abstract

Abstract. The generalized thermostatistics advanced by Tsallis in order to treat nonextensive systems has greatly increased the range of possible applications of statistical mechanics to the description of natural phenomena. Here we consider some aspects of the relationship between Tsallis' theory and Jaynes' maximum entropy (MaxEnt) principle. We review some universal properties of general thermostatistical formalisms based on an entropy extremalization principle. We explain how Tsallis formalism provides a useful tool in order to obtain MaxEnt solutions of nonlinear partial differential equations describing nonextensive physical systems. In particular, we consider the case of the Vlasov-Poisson equations, where Tsallis MaxEnt principle leads in a natural way to the stellar polytrope solutions. We pay special attention to the "P -picture" formulation of Tsallis generalized thermostatistics based on the entropic functionalS q and standard linear constraints.

q-Exponential structure of arbitrary-order reaction kinetics

“…The foregoing rate equations can be further incorporated into a generalised nonlinear advection-dispersion-reaction equation (equivalent to a nonlinear Fokker-Planck equation with reaction) of the following type (c.f. Tsallis & Bukman, 1996;Plastino et al, 2000;Lenzi et al, 2001;Plastino, 2001;Frank, 2002):…”

Applying the Maximum Entropy Technique to the Gaussian Dispersion Plume Model

“…If the time derivatives of the equations of constraint constitute a closed set of differential equations the static solution may be viewed as the solution to the MaxEnt equations at a given time and the time dependent solutions are obtained by replacing λ i by λ i (t) which satisfy the time dependent equations of constraint ( [37]).…”