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It is long known that the Fokker-Planck equation with prescribed constantcoefficients of diffusion and linear friction describes the ensemble average ofthe stochastic evolutions in velocity space of a Brownian test particleimmersed in a heat bath of fixed temperature. Apparently, it is not so wellknown that the same partial differential equation, but now with constantcoefficients which are functionals of the solution itself rather than beingprescribed, describes the kinetic evolution (in the infinite particle limit) ofan isolated N-particle system with certain stochastic interactions. Here wediscuss in detail this recently discovered interpretation.Comment: Minor revisions and corrections (including the title

The Linear Fokker-Planck Equation for the Ornstein-Uhlenbeck Process as an (Almost) Nonlinear Kinetic Equation for an Isolated N-Particle System