Articles

Journal of Statistical Physics

, Volume 48, Issue 1, pp 19-49

First online:

Mayer expansions and the Hamilton-Jacobi equation

  • D. C. BrydgesAffiliated withDepartment of Mathematics, University of Virginia
  • , T. KennedyAffiliated withInstitute des Hautes Etudes Scientifiques

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

Abstract

We review the derivation of Wilson's differential equation in (infinitely) many variables, which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal “integration out” is performed. We show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic. The resulting series solutions are generalizations of the Mayer expansion in statistical mechanics. The differential equation approach gives a remarkable identity for “connected parts” and precise estimates which include criteria for convergence of iterated Mayer expansions. Applications include the Yukawa gas in two dimensions past theΒ=4π threshold and another derivation of some earlier results of Göpfert and Mack.

Key words

Multiscale Mayer expansions renormalization group tree graph identities