Abstract
We generalize the notion of “ground states” in the Pirogov-Sinai theory of first order phase transitions at low temperatures, applicable to lattice systems with a finite number of periodic ground states to that of “restricted ensembles” with equal free energies. A restricted ensemble is a Gibbs ensemble, i.e. equilibrium probability measure, on a restricted set of configurations in the phase space of the system. When a restricted ensemble contains only one configuration it coincides with a ground state. In the more general case the entropy is also important.
An example of a system we can treat by our methods is theq-state Potts model where we prove that forq sufficiently large there exists a temperature at which the system coexists inq+1 phases;q-ordered phases are small modifications of theq perfectly ordered ground states and one disordered phase which is a modification of the restricted ensemble consisting of all “perfectly disordered” (neighboring sites must have different spins) configurations. The free energy thus consists entirely of energy in the firstq-restricted ensembles and of entropy in the last one.
Our main motivation for this work is to develop a rigorous theory for phase transitions in continuum fluids in which there is no symmetry between the phases, e.g. the liquid-vapour phase transition. The present work goes a certain way in that direction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Peierls, R.: On Ising's model of ferromagnetism. Proc. Camb. Philos. Soc.32, 477 (1936)
Dobrushin, R.L.: Existence of a phase transition in two- and three-dimensional lattice models. Theory Probab. Appl.10, 193 (1965)
Griffiths, R.B.: Peierls' proof of spontaneous magnetization of a two-dimensional Ising ferromagnet. Phys. Rev. A136, 437 (1964)
Ruelle, D.: Existence of a phase transition in a continuous classical system. Phys. Rev. Lett.27, 1040 (1971)
Sinai, Ya. G.: Theory of phase transitions: Rigorous results. New York: Pergamon Press, 1982.
Original papers:Pirogov, S.A., Sinai, Ya. G.: Phase diagram of classical lattice systems I and II: Theor. Math. Phys.25, 1185 (1976) and26, 39 (1976)
Pirogov, S.A., Sinai, Ya. G.: Ground states in two-dimensional Boson quantum field theory. Ann. Phys.109, 393 (1977)
Imbrie, J.Z.: Phase diagrams and cluster expansions for low temperatureP(φ)2 models I and II. Commun. Math. Phys.82, 261 (1981) and82, 305 (1981). For earlier results on phase transitions in quantum field theories, see: Glimm, J., Jaffe, A.: Quantum Physics. A functional integral point of view. New York: Springer 1981 (and references therein)
Slawny, J.: Low temperature expansion for lattice systems with many ground states. J. Stat. Phys.26, 711 (1973). Low temperature properties of classical lattice systems: Phase transitions and phase diagrams. To appear in: “Phase transitions and critical phenomena,” Vol. 10. Domb, C., Lebowitz, J.L. (eds.), New York: Academic Press 1985
Zahradnik, M.: An alternate version of Pirogov-Sinai theory. Commun. Math. Phys.93, 559 (1984)
Kotecky, R., Preiss, D.: An inductive approach to Pirogov-Sinai theory. Proc. Winter School on Abstract Analysis 1983. Suppl. Ai. Rend Circ. Mat. Palermo (1983)
Malyshev, V.A., Minlos, R.A., Petrova, E.N., Teletski, A.: Generalized contour models (in Russian). Surveys of Science and Technology, pp. 3–54. Theory of probability, mathematical statistics and theoretical cybernetics, Vol. 19. Viniti 1982
Bricmont, J., Kuroda, K., Lebowitz, J.L.: The structure of Gibbs states and phase coexistence for non-symmetric continuum Widom-Rowlinson models. Z. Wahrscheinlichkeitstheor. Verw. Geb.67, 121–138 (1984)
Dobrushin, R.L., Zahradnik, M.: Phase diagrams for the continuous spin models. Extension of Pirogov-Sinai theory. Preprint
Dinaburg, E.I., Sinai, Ya.G.: Preprint (in Russian); see also: “An analysis of ANNNI model by Peierls' contour method”. Commun. Math. Phys.98, 119 (1985)
Kotecky, R., Preiss, D.: In preparation
Kotecky, R., Shlosman, S.B.: First-order transitions in large entropy lattice models. Commun. Math. Phys.83, 493 (1982)
Iagolnitzer, D., Souillard, B.: On the analyticity in the potential in classical statistical mechanics. Commun. Math. Phys.60, 131 (1978), and references therein
Bricmont, J., Kuroda, K., Lebowitz, J.L.: Surface tension and phase coexistence for general lattice systems. J. Stat. Phys.33, 59 (1983)
Laanait, L., Messager, A., Ruiz, J.: Phase coexistence and surface tensions for the Potts model. Preprint
Bricmont, J., Lebowitz, J.L., Pfister, C.-E.: Low temperature expansion for continuous spin Ising models. Commun. Math. Phys.78, 117 (1980)
Widom, B., Rowlinson, J.S.: New model for the study of liquid-vapour phase transitions. J. Chem. Phys.52, 1270 (1970)
Lebowitz, J.L., Penrose, O.: Rigorous treatment of the van der Waals-Maxwell theory of the liquid-vapour transition. J. Math. Phys.7, 98 (1966)
Seiler, E.: Gauge theories as a problem of constructive quantum field theory and statistical mechanics. Lecture Notes in Physics, Vol. 159. Berlin, Heidelberg, New York: Springer 1982
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Supported in part by NSF Grant Nr DMR81-14726-02
Rights and permissions
About this article
Cite this article
Bricmont, J., Kuroda, K. & Lebowitz, J.L. First order phase transitions in lattice and continuous systems: Extension of Pirogov-Sinai theory. Commun.Math. Phys. 101, 501–538 (1985). https://doi.org/10.1007/BF01210743
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01210743