Journal of Statistical Physics

, Volume 48, Issue 1, pp 19–49

Mayer expansions and the Hamilton-Jacobi equation

  • D. C. Brydges
  • T. Kennedy

DOI: 10.1007/BF01010398

Cite this article as:
Brydges, D.C. & Kennedy, T. J Stat Phys (1987) 48: 19. doi:10.1007/BF01010398


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 expansionsrenormalization grouptree graph identities

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • D. C. Brydges
    • 1
  • T. Kennedy
    • 2
  1. 1.Department of MathematicsUniversity of VirginiaCharlottesville
  2. 2.Institute des Hautes Etudes ScientifiquesBures-Sur-YvetteFrance
  3. 3.Department of PhysicsPrinceton UniversityPrinceton