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Extensive Lyapunov functionals for moment-preserving evolution equations. (English. Abridged French version) Zbl 1090.82026

Summary: We consider a certain class of moment-preserving equations from the point of view of their stationary solutions. Starting from a given stationary distribution, we construct a convex entropy functional which is (in a class of functions with prescribed moments) minimal precisely at this point. Under general assumptions, we show that the entropy which is canonically associated to a stationary distribution is, up to a polynomial change of variables, its Legendre-Fenchel transform. We then show that, if this entropy is extensive, necessarily the stationary distribution is a Gibbs state. Such a state being given by the exponential of the energy density, this clarifies the duality relationship between energy and entropy.

MSC:

82C40 Kinetic theory of gases in time-dependent statistical mechanics
35A15 Variational methods applied to PDEs
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References:

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