On the "Mean Field" Interpretation of Burgers' Equation

Détails

ID Serval
serval:BIB_643B6957ECBC
Type
Article: article d'un périodique ou d'un magazine.
Collection
Publications
Titre
On the "Mean Field" Interpretation of Burgers' Equation
Périodique
Journal of Statistical Physics
Auteur⸱e⸱s
Choquard P., Wagner J.
ISSN
0022-4715
Statut éditorial
Publié
Date de publication
2004
Peer-reviewed
Oui
Volume
116
Numéro
1-4
Pages
843-853
Langue
anglais
Résumé
Fruitful analogies, partially first established by C. M. Newman,((1)) between the variables, functions, and equations which describe the equilibrium properties of classical ferro- and antiferromagnets in the Mean Field Approximation (MFA) and those which describe the space-time evolution of compressible Burgers' liquids are developed here for one-dimensional systems. It is shown that the natural analogies are: magnetic field and position coordinate; ferro-/antiferromagnetic coupling constants and negative/positive times; free energy per spin and velocity potential; magnetization and velocity field; magnetic susceptibility and mass density. An unexpected consequence of these analogies is a derivation of the Morette-Van Hove relation. Another novelty is that they necessitate the investigation of weak solutions of Burgers' equation for negative times, corresponding to the Curie-Weiss transition in ferromagnets. This is achieved by solving the "final-value" problem of the homogenous Hamilton-Jacobi equation. Unification of the final- and initial-value problems results in an extended Hopf-Lax variational principle. It is shown that its applicability implies that the velocity potentials at time zero be Lipschitz continuous for the velocity field to be bounded. This is a rather mild condition for the class of physically interesting and functionally compatible velocity potentials, compatible in the sense of satisfying the Morette-Van Hove relation.
Mots-clé
Mean Field Approximation, Hamilton-Jacobi equation, Morette-Van Hove relation, Hopf-Lax formula
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Création de la notice
07/07/2014 9:32
Dernière modification de la notice
20/08/2019 15:20
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