Abstract
We consider particles in ℝd, d≥2, interacting via attractive pair and repulsive four-body potentials of the Kac type. Perturbing about mean-field theory, valid when the interaction range becomes infinite, we prove rigorously the existence of a liquid–gas phase transition when the interaction range is finite but long compared to the interparticle spacing for a range of temperature.
Similar content being viewed by others
REFERENCES
R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London/New York, 1982).
J. Bricmont, K. Kuroda, and J. L. Lebowitz, First order phase transitions in lattice and continuous systems: Extension of Pirogov–Sinai theory, Comm. Math. Phys. 101:501–538 (1985).
T. Bodineau and E. Presutti, Phase diagram of Ising systems with additional long range forces, Comm. Math. Phys. 189:287–298, (1996).
A. Bovier and M. Zahradnik, The low temperature phase of Kac–Ising models, J. Stat. Phys., to appear.
J. T. Chayes, L. Chayes, and R. Kotecky, The analysis of the Widom–Rowlinson model by stochastic geometric methods, Comm. Math. Phys. 172:551–569 (1995).
M. Cassandro, E. Olivieri, A. Pellegrinotti, and E. Presutti, Existence and unique ness of DLR measures for unbounded spin systems, Z. Wahr. verv. Geb. 41:313–334 (1978).
M. Cassandro and E. Presutti, Phase transitions in Ising systems with long but finite range interactions, Markov Processes and Related Fields 2:241–262 (1996).
R. L. Dobrushin, Existence of phase transition in two-and three-dimensional Ising models, Th. Prob. Appl. 10:193–313 (1965).
R. L. Dobrushin, Prescribing a system of random variables by conditional distributions, Th. Prob. Appl. 15:456–458 (1970).
R. L. Dobrushin, Estimates of semi-invariants for the Ising model at low tem peratures, topics in statistical and theoretical physics, Amer. Math. Soc. Transl. (2) 177:59–81 (1996).
E. I. Dinaburg and Ya. G. Sinai, Contour models with interaction and their applications, Sel. Math. Sov. 7:291–315 (1988).
R. L. Dobrushin and M. Zahradnik, Phase diagrams for continuous spin models. Extension of Pirogov–Sinai theory, in Mathematical Problems of Statistical Mechanics and Dynamics, R. L. Dobrushin, ed. (Kluwer Academic Publishers, Dordrecht, Boston, 1986), pp. 1–123.
M. V. Fedoryuk, Asymptotic: Integrals and Series (Nauka, Moscow, 1987).
B. U. Felderhof and M. E. Fisher, Phase transitions in one-dimensional cluster-interaction fluids. IA. Thermodynamics, IB. Critical behavior, II. Simple logarithmic model, Annals of Phys. 58:176–216, 217–267, 268–280 (1970).
H.-O. Georgii and Haggstrom, Phase transitions in continuum Potts models, Comm. Math. Phys. 181:507–528 (1996).
R. B. Griffiths, Peierls' proof of spontaneous magnetization in a two-dimensional Ising ferromagnet, Phys. Rev. A 136:437–439 (1964).
K. Johansson, On separation of phases in one-dimensional gases, Comm. Math. Phys. 169:521–561 (1995).
N. G. van Kampen, Condensation of a classical gas with long-range attraction, Phys. Rev. A 135:362 (1964).
R. Kotecky and D. Preiss, Cluster expansion for abstract polymer models, Comm. Math. Phys. 103:491–498 (1986).
M. Kac, G. Uhlenbeck, and P. C. Hemmer, On the Van der Waals theory of vapor–liquid equilibrium, J. Mat. Phys. 4:216–228, 229–247 (1963); 5:60–74 (1964).
J. L. Lebowitz and E. H. Lieb, Phase transition in a continuum classical system with finite interactions, Phys. Lett. A 39:98–100 (1972).
J. L. Lebowitz, A. E. Mazel, and E. Presutti, Rigorous proof of a liquid–vapor phase transition in a continuum particle system, Phys. Rev. Lett. 80:4701–4704 (1998).
J. L. Lebowitz, A. E. Mazel, and E. Presutti, in preparation.
J. L. Lebowitz and O. Penrose, Rigorous treatment of the Van der Waals–Maxwell theory of the liquid–vapor transition, J. Mat. Phys. 7:98–113 (1966).
A. E. Mazel and Yu. M Suhov, Ground states of boson quantum lattice model, Amer. Math. Soc. Transl. (2) 171:185–226 (1996).
L. Onsager, Crystal statistics I. A two-dimensional model with an order–disorder transition, Phys. Rev. 65:117–149 (1944).
R. Peierls, On Ising's model of ferromagnetism, Proc. Camb. Phil. Soc. 32:477–481 (1936).
E. Presutti, Notes on phase transition in the continuum, Lectures at IHP, Paris, June–July 1998, preprint.
S. A. Pirogov and Ya. G Sinai, Phase diagrams of classical lattice systems, Theor. Math. Phys. 25:358–369, 1185–1192 (1975).
D. Ruelle, Existence of a phase transition in a continuous classical system, Phys. Rev. Lett. 27:1040–1041 (1971).
D. Ruelle, Probability estimates for continuous spin systems, Comm. Math. Phys. 50:189–194 (1976).
Ya. G. Sinai, Theory of Phase Transitions (Academia Kiado and Pergamon Press, Budapest, London, 1982).
E. Seiler, Gauge theories as a problem of constructive quantum field theory and statistical mechanics, Lect. Notes in Physics, Vol. 159 (Springer-Verlag, Berlin, 1982).
B. Widom and J. S. Rowlinson, New model for the study of liquid–vapor phase transitions, J. Chem. Phys. 52:1670–1684 (1970).
M. Zahradnik, An alternate version of Pirogov–Sinai theory, Comm. Math. Phys. 93:559–581 (1984).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lebowitz, J.L., Mazel, A. & Presutti, E. Liquid–Vapor Phase Transitions for Systems with Finite-Range Interactions. Journal of Statistical Physics 94, 955–1025 (1999). https://doi.org/10.1023/A:1004591218510
Issue Date:
DOI: https://doi.org/10.1023/A:1004591218510