Journal of Statistical Physics

, Volume 48, Issue 1–2, pp 19–49 | Cite as

Mayer expansions and the Hamilton-Jacobi equation

  • D. C. Brydges
  • T. Kennedy


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Fröhlich, ed.,Scaling and Self Similarity in Physics—Renormalization in Statistical Mechanics and Dynamics (BirkhÄuser, Boston, 1983).Google Scholar
  2. 2.
    K. G. Wilson,Phys. Rev. B 4:3174 (1971); J. G. Kogut and K. G. Wilson,Phys. Rep. 12:263 (1974).Google Scholar
  3. 3.
    J. Polchinski, Renormalization and effective Lagrangians,Nucl. Phys. B 231:269 (1984).Google Scholar
  4. 4.
    G. Gallavotti, and F. Nicolo, Renormalization theory in four-dimensional scalar fields (I),Commun. Math. Phys. 100:545 (1985).Google Scholar
  5. 5.
    C. Newman, Unpublished work;J. Stat. Phys. 27:836 (1982).Google Scholar
  6. 6.
    J. Fröhlich and E. Seiler,Helv. Phys. Acta 49:889 (1976).Google Scholar
  7. 7.
    D. C. Brydges, A short course on cluster expansions, Appendix A, inCritical Phenomena, Random Systems, Gauge Theories, K. Osterwalder and R. Stora, eds. (Elsevier, 1986).Google Scholar
  8. 8.
    G. Battle, and P. Federbush, A note on cluster expansions, tree graph identities, extra 1/N! factors!Lett. Math. Phys. 8:55 (1984).Google Scholar
  9. 9.
    D. C. Brydges, Convergence of Mayer expansions,J. Stat. Phys. 42:425 (1984).Google Scholar
  10. 10.
    J. Fröhlich,Commun. Math. Phys. 47:233 (1976).Google Scholar
  11. 11.
    G. Benfatto, An iterated Mayer expansion for the Yukawa gas,J. Stat. Phys. 41:671 (1985).Google Scholar
  12. 12.
    M. Göpfert and G. Mack,Commun. Math. Phys. 81:97 (1981);82:545 (1982).Google Scholar
  13. 13.
    G. Gallavotti, Renormalization theory and ultra-violet stability for scalar fields via renormalisation group techniques,Rev. Mod. Phys. 57:471 (1985).Google Scholar
  14. 14.
    G. Benfatto, G. Gallavotti, and F. Nicolo, On the massive sine Gordon equation in the first few regions of collapse,Commun. Math. Phys. 83:387 (1982).Google Scholar
  15. 15.
    F. Nicolo, J. Benn, and A. Steinman, On the massive sine Gordon equation in all regions of collapse,Commun. Math. Phys. 105:291 (1986).Google Scholar
  16. 16.
    G. Benfatto, G. Gallavotti, and F. Nicolo, In preparation.Google Scholar
  17. 17.
    G. Gallavotti and F. Nicolo, The screening phase transition in the two dimensional Coulomb gas,J. Stat. Phys. 39:133 (1985).Google Scholar
  18. 18.
    J. Imbrie, Iterated Mayer expansions and their applications to Coulomb systems, inScaling and Self-Similanty in Physics—Renormalization in Statistical Mechanics and Dynamics, J. Frölich, ed. (BirkhÄuser, Boston, 1983).Google Scholar
  19. 19.
    V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978), p. 256.Google Scholar

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

Personalised recommendations