Abstract
For a class of unstable pair interactions in classical continuous systems of identical particles the high-temperature thermodynamic behavior is shown to be normal by extending low-density theorems for the correlation functions. In an example we prove a transition between a translation-invariant phase at high temperatures and low densities and solid with long-range oder at low temperatures. The transition is “catastropic” in the sense that it is accompanied by the divergence of thermodynamic quantities. We also exhibit counterexamples of unstable interactions in any dimension which do not give rise to a low-temperature catastrophe.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Benfatto, An iterated Mayer expansion for the Yukawa gas,J. Stat. Phys. 41:671–684 (1985).
H. J. Brascamp and E. H. Lieb, Some inequalities for Gaussian measures and the longrange order of the one-dimensional plasma, inFunctional Integration and Its Applications, A. M. Arthurs, ed. (Clarendon Press, Oxford, 1975).
D. C. Brydges, Functional integrals and their applications. Troisiéme Cycle de la Physique en Suisse Romande (Summer 1992).
M. Duneau, I. Iagolnitzer, and B. Souillard, Strong cluster properties for classical systems with finite range interaction,Commun. Math. Phys. 35:307–320 (1974).
M. Duneau and B. Souillard, Cluster properties of lattice and continuous systems,Commun. Math. Phys. 47:155–166 (1976).
J. Fröhlich and Y. M. Park, Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems,Commun. Math. Phys. 59:235–266 (1978).
J. Fröhlich and C.-E. Pfister, Absence of crystalline ordering in two dimensions,Commun. Math. Phys. 104:697–700 (1986).
G. Gallavotti and S. Miracle-Sole, Correlation functions of a lattice system,Commun. Math. Phys. 7:274–288 (1968).
G. Gallavotti and S. Miracle-sole, On the cluster property above the critical temperature in lattice gases,Commun. Math. Phys. 12:269–274 (1969).
D. Iagolnitzer and B. Souillard, On the analytical in the potential in classical statistical mechanics,Commun. Math. Phys. 60:131–152 (1978).
T. Kennedy and E. H. Lieb, An itinerant electron model with crstalline or magnetic longrange order,Physica 138A:320 (1986).
H. Kunz, The one-dimensional classical electron gas,Ann. Phys. (NY)85:303–335 (1974).
J. L. Lebowitz and O. Penrose, Analytic and clustering properties of thermodynamic functions and distribution functions for classical lattice and continuous systems,Commun. Math. Phys. 11:99–124 (1968).
J. L. Lebowitz and O. Penrose, On the exponential decay of correlation functions,Commun. Math. Phys. 39:165–184 (1974).
N. D. Mermin, Crystalline order in two dimensions,Phys. Rev. 176:250–254 (1968).
J. Miękisz,Quasicrystals (Leuven University Press, 1993).
O. Penrose, Convergence of the fugacity expansions for fluids and gases,J. Math. Phys. 4:1312–1320 (1963).
O. Penrose, The remainder in Mayer's fugacity series,J. Math. Phys. 4:1488–1494 (1963).
C. Radin, Low temperature and the origin of crystalline symmetry,Int. J. Med. Phys. B 1:1157–1191 (1987).
D. Ruelle,Statistical Mechanics (Benjamin, New York, 1969).
L. Tonks, The complete equation of state of one, two and three dimensional gase of hard elastic spheres,Phys. Rev. 50:955 (1936).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sütő, A. Low-density expanison for unstable interactions and a model of crystallization. J Stat Phys 82, 1541–1573 (1996). https://doi.org/10.1007/BF02183395
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02183395