Nonlinear Fokker–Planck Equation Approach to Systems of Interacting Particles: Thermostatistical Features Related to the Range of the Interactions

Abstract

Nonlinear Fokker–Planck equations (NLFPEs) constitute useful effective descriptions of some interacting many-body systems. Important instances of these nonlinear evolution equations are closely related to the thermostatistics based on the S q power-law entropic functionals. Most applications of the connection between the NLFPE and the S q entropies have focused on systems interacting through short-range forces. In the present contribution we re-visit the NLFPE approach to interacting systems in order to clarify the role played by the range of the interactions, and to explore the possibility of developing similar treatments for systems with long-range interactions, such as those corresponding to Newtonian gravitation. In particular, we consider a system of particles interacting via forces following the inverse square law and performing overdamped motion, that is described by a density obeying an integro-differential evolution equation that admits exact time-dependent solutions of the q-Gaussian form. These q-Gaussian solutions, which constitute a signature of S q -thermostatistics, evolve in a similar but not identical way to the solutions of an appropriate nonlinear, power-law Fokker–Planck equation.

On the construction of a family of anomalous-diffusion Fokker–Planck−Kolmogorov’s equations based on the Sharma–Taneja–Mittal entropy functional

“…Such contributions can be associated, in particular, with non-extensive statistical mechanics. In the scientific literature, situations where diffusion contributions are written as a degree of probability density have been studied in most detail (see, for example, [21,24,[46][47][48][49][50][51][52][53][54]).…”

Multispecies effects in the equilibrium and out-of-equilibrium thermostatistics of overdamped motion

“…There are compelling reasons for rewriting the system of coupled nonlinear Fokker-Planck equations (37) in the more abstract form (40). First, some of the most fundamental features of the evolution equation (40), such as it admitting an H theorem, do not depend on the detailed, particular form (38) of the function . Those essential features follow directly from the general structure encapsulated in Eqs.…”

“…Those essential features follow directly from the general structure encapsulated in Eqs. (39) and (40). Consequently, the results obtained exploiting this general structure are not restricted to the evolution equations (37).…”

“…For example, within the general setting described by Eqs. ( 39) and (40), it may be possible to consider multispecies extensions of the systems considered in Ref. [12], consisting of particles interacting through a force obeying the power law…”

Generalized entropies, density of states, and non-extensivity

“…A variety of generalized entropic functionals have been introduced to phenomenologically extend statistical mechanics to specific non-ergodic or strongly interacting systems, both within and outside the realm of physics including spin-like systems [9][10][11], cosmic ray energy spectra [12], multifractals [13], networks [14], quantum information [15][16][17], special relativity [18], anomalous diffusive processes [19][20][21][22][23], superstatistics [24,25], time series analysis [26][27][28] and artificial neural networks [29].…”