On the "Mean Field" Interpretation of Burgers' equation

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 ldquofinal-valuerdquo 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.