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

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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