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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Fröhlich, ed.,Scaling and Self Similarity in Physics—Renormalization in Statistical Mechanics and Dynamics (BirkhÄuser, Boston, 1983).
K. G. Wilson,Phys. Rev. B 4:3174 (1971); J. G. Kogut and K. G. Wilson,Phys. Rep. 12:263 (1974).
J. Polchinski, Renormalization and effective Lagrangians,Nucl. Phys. B 231:269 (1984).
G. Gallavotti, and F. Nicolo, Renormalization theory in four-dimensional scalar fields (I),Commun. Math. Phys. 100:545 (1985).
C. Newman, Unpublished work;J. Stat. Phys. 27:836 (1982).
J. Fröhlich and E. Seiler,Helv. Phys. Acta 49:889 (1976).
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).
G. Battle, and P. Federbush, A note on cluster expansions, tree graph identities, extra 1/N! factors!Lett. Math. Phys. 8:55 (1984).
D. C. Brydges, Convergence of Mayer expansions,J. Stat. Phys. 42:425 (1984).
J. Fröhlich,Commun. Math. Phys. 47:233 (1976).
G. Benfatto, An iterated Mayer expansion for the Yukawa gas,J. Stat. Phys. 41:671 (1985).
M. Göpfert and G. Mack,Commun. Math. Phys. 81:97 (1981);82:545 (1982).
G. Gallavotti, Renormalization theory and ultra-violet stability for scalar fields via renormalisation group techniques,Rev. Mod. Phys. 57:471 (1985).
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).
F. Nicolo, J. Benn, and A. Steinman, On the massive sine Gordon equation in all regions of collapse,Commun. Math. Phys. 105:291 (1986).
G. Benfatto, G. Gallavotti, and F. Nicolo, In preparation.
G. Gallavotti and F. Nicolo, The screening phase transition in the two dimensional Coulomb gas,J. Stat. Phys. 39:133 (1985).
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).
V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978), p. 256.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brydges, D.C., Kennedy, T. Mayer expansions and the Hamilton-Jacobi equation. J Stat Phys 48, 19–49 (1987). https://doi.org/10.1007/BF01010398
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01010398