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 power-law entropic functionals. Most applications of the connection between the NLFPE and the 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 -thermostatistics, evolve in a similar but not identical way to the solutions of an appropriate nonlinear, power-law Fokker–Planck equation.
Highlights
The thermostatistics derived from the Sq non-additive entropies [1,2,3] exhibit interesting links with the statistical properties of (i) systems with long-range interactions, including cases of self-gravitating systems; and (ii) systems described by nonlinear, power-law diffusion or Fokker–Planck equations
We briefly review the application of the power-law nonlinear Fokker–Planck equation to the thermostatistics of systems of confined particles interacting through short-range forces and moving in the overdamped regime [20,21,22,23,24,25]
We have re-visited the derivation of nonlinear Fokker–Planck equations for systems of particles interacting via short-range forces, in order to discuss in detail the criterion for “short-range” behind it, which is central to our present work
Summary
The thermostatistics derived from the Sq non-additive entropies [1,2,3] exhibit interesting links with the statistical properties of (i) systems with long-range interactions, including cases of self-gravitating systems; and (ii) systems described by nonlinear, power-law diffusion or Fokker–Planck equations. We are going to study a particular instance of a system with long-range interactions that, as Newtonian gravitation, follows the inverse square law The dynamics of this system has some similarities with systems governed by power-law nonlinear Fokker–Planck equations. The power-law nonlinear Fokker–Planck equation admits exact time-dependent solutions with a q-Gaussian shape. The connection between this evolution equation and the Sq entropy stimulated the development of a fruitful research field. It has been extended in various directions, and applied to diverse problems in physics and elsewhere [13,14,15,16,17,18,19,20]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
