| Summary: | 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>q</mi> </msub> </semantics> </math> </inline-formula> power-law entropic functionals. Most applications of the connection between the NLFPE and the <inline-formula> <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>q</mi> </msub> </semantics> </math> </inline-formula> 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 <i>q</i>-Gaussian form. These <i>q</i>-Gaussian solutions, which constitute a signature of <inline-formula> <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>q</mi> </msub> </semantics> </math> </inline-formula>-thermostatistics, evolve in a similar but not identical way to the solutions of an appropriate nonlinear, power-law Fokker−Planck equation.
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