# On the "Mean Field" Interpretation of Burgers' Equation

## Details

Serval ID

serval:BIB_643B6957ECBC

Type

**Article**: article from journal or magazin.

Collection

Publications

Institution

Title

On the "Mean Field" Interpretation of Burgers' Equation

Journal

Journal of Statistical Physics

ISSN

0022-4715

Publication state

Published

Issued date

2004

Peer-reviewed

Oui

Volume

116

Number

1-4

Pages

843-853

Language

english

Abstract

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.

Keywords

Mean Field Approximation, Hamilton-Jacobi equation, Morette-Van Hove relation, Hopf-Lax formula

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Create date

07/07/2014 9:32

Last modification date

20/08/2019 15:20