## Non-linear kinetics underlying generalized statistics.(English)Zbl 0969.82537

Summary: The purpose of the present effort is threefold. Firstly, it is shown that there exists a principle, that we call kinetical interaction principle (KIP), underlying the nonlinear kinetics in particle systems, independently on the picture (Kramers, Boltzmann) used to describe their time evolution. Secondly, the KIP imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution, both as stationary solution of the nonlinear evolution equation and as the state which maximizes the generalized entropy. Thirdly, the KIP allows, on one hand, to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and, on the other hand, suggests how we can introduce naturally new distributions. Finally, as a working example of the approach to the nonlinear kinetics here presented, a new non-extensive statistics is constructed and studied starting from a one-parameter deformation of the exponential function holding the relation $$f(-x)f(x)=1$$.

### MSC:

 82B40 Kinetic theory of gases in equilibrium statistical mechanics
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