Extensive Lyapounov functionals for moment-preserving evolution equations
Comptes Rendus. Mathématique, Volume 334 (2002) no. 5, pp. 429-434.

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.

Nous considérons des solutions stationnaires pour des classes d'équations d'évolution préservant certains moments. Etant donnée une solution stationnaire, nous construisons une fonctionnelle convexe (l'entropie) qui est (dans une classe de fonctions de moments fixés) minimale en ce point. Sous des hypothèses générales, nous montrons qu'une telle entropie canoniquement associée à une distribution stationnaire est, à un changement de variable polynomial près, sa transformée de Legendre. On montre ensuite que, si la fonctionnelle ainsi obtenue est extensive, la solution stationnaire de départ est nécessairement une distribution de Gibbs. Une telle distribution étant donnée par l'exponentielle de la densité d'énergie, ceci clarifie la relation de dualité entre énergie et entropie.

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DOI: 10.1016/S1631-073X(02)02266-5
Collet, Jean François 1

1 Laboratoire J.A. Dieudonné, UMR 6621, Université Nice-Sophia Antipolis, Parc Valrose, 06108 Nice cedex 02, France
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Collet, Jean François. Extensive Lyapounov functionals for moment-preserving evolution equations. Comptes Rendus. Mathématique, Volume 334 (2002) no. 5, pp. 429-434. doi : 10.1016/S1631-073X(02)02266-5. http://www.numdam.org/articles/10.1016/S1631-073X(02)02266-5/

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