Abstract
A classical gas with short-range interaction in the grand canonical ensemble is studied. Ifp(β, z) denotes the thermodynamic pressure at inverse temperatureβ and activityz, then it follows from the Mayer expansion thatp(β, z) is infinitely differentiable providedβ andβz are sufficiently small. Here it is shown that there existsβ 0>0 such thatp(β, z) is infinitely differentiable ifβ<β 0 andz>0. One can interpret this result as saying that (β 0)−1 is an upper bound on the critical temperature for the system.
Similar content being viewed by others
References
D. Brydges, A short course on cluster expansions, inCritical Phenomena, Random Systems, Gauge Theories, K. Osterwalder and R. Stora, eds. (North-Holland, 1986), pp. 129–182.
D. Brydges and P. Federbush, Debye screening,Commun. Math. Phys. 73:197–246 (1980).
D. Brydges and P. Federbush, Debye screening in classical Coulomb systems, inRigorous Atomic and Molecular Physics, G. Velo and A. Wightman, eds. (Plenum Press, New York, 1981), pp. 371–440.
J. Conlon, The ground state energy of a classical gas,Commun. Math. Phys. 94:439–458 (1984).
J. Conlon, E. Lieb, and H.-T. Yau, TheN 7/5 law for charged bosons,Commun. Math. Phys. 116:417–448 (1988).
P. Federbush, A new approach to the stability of matter problem II,J. Math. Phys. 16:706–709 (1975).
P. Federbush and T. Kennedy, Surface effects in Debye screening,Commun. Math. Phys. 102:361–423 (1985).
J. Fröhlich and T. Spencer, Phase diagrams and critical properties of classical Coulomb systems, inRigorous Atomic and Molecular Physics, G. Velo and A. Wightman, eds. (Plenum Press, New York, 1981), pp. 327–370.
K. Gawedszki and A. Kupiainen, Rigorous renormalization group and asymptotic freedom, inScaling and Self-Similarity in Physics, J. Frohlich, ed. (Birkhäuser, 1983), pp. 227–262.
J. Imbrie, Debye screening for jellium and other Coulomb systems,Commun. Math. Phys. 87:515–565 (1983).
D. Ruelle,Statistical Mechanics—Rigorous Results (Benjamin, 1969).
W.-S. Yang, Debye screening for 2 dimensional Coulomb systems at high temperatures,J. Stat. Phys. 49:1–32 (1987).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Conlon, J.G. An upper bound on the critical temperature for a continuous system with short-range interaction. J Stat Phys 58, 265–293 (1990). https://doi.org/10.1007/BF01020294
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01020294