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Journal ArticleDOI

Identities for Correlation Functions in Classical Statistical Mechanics and the Problem of Crystal States

01 Sep 2020-Journal of Statistical Physics (Springer Science and Business Media LLC)-Vol. 180, Iss: 1, pp 1002-1009
TL;DR: In this article, it was shown that the correlation functions of point particles described by classical equilibrium statistical mechanics are real analytic functions of the point particle activity of a point particle in a crystal state, assuming a suitable cluster property (decay of correlations).
Abstract: Let z be the activity of point particles described by classical equilibrium statistical mechanics in $$\mathbf{R}^ u $$ . The correlation functions $$\rho ^z(x_1,\dots ,x_k)$$ denote the probability densities of finding k particles at $$x_1,\dots ,x_k$$ . Letting $$\phi ^z(x_1,\dots ,x_k)$$ be the cluster functions corresponding to the $$\rho ^z(x_1,\dots ,x_k)/z^k$$ we prove identities of the type $$\begin{aligned}&\phi ^{z_0+z'}(x_1,\dots ,x_k)\\&\quad =\sum _{n=0}^\infty {z'^n\over n!}\int dx_{k+1}\dots \int dx_{k+n}\,\phi ^{z_0}(x_1,\dots ,x_{k+n}) \end{aligned}$$ It is then non-rigorously argued that, assuming a suitable cluster property (decay of correlations) for a crystal state, the pressure and the translation invariant correlation functions $$\rho ^z(x_1,\dots ,x_k)$$ are real analytic functions of z.
References
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Book
01 Jan 1969
TL;DR: The problem of phase transition group invariance of physical states has been studied in the literature as discussed by the authors, where the thermodynamic limit for thermodynamic functions has been investigated in the context of statistical mechanics.
Abstract: Thermodynamic behaviour - ensembles the thermodynamic limit for thermodynamic functions - lattice systems the thermodynamic limit for thermodynamic functions - continuous systems low density expansions and correlation functions the problem of phase transitions group invariance of physical states the states of statistical mechanics Appendix: some mathematical tools

2,036 citations

Journal ArticleDOI
TL;DR: In this paper, the authors show that a representation of an algebra of local observables has short-range correlations if any observable which can be measured outside all bounded sets is a multiple of the identity, and that a state has finite range correlations if the corresponding cyclic representation does.
Abstract: We say that a representation of an algebra of local observables has short-range correlations if any observable which can be measured outside all bounded sets is a multiple of the identity, and that a state has finite range correlations if the corresponding cyclic representation does. We characterize states with short-range correlations by a cluster property. For classical lattice systems and continuous systems with hard cores, we give a definition of equilibrium state for a specific interaction, based on a local version of the grand canonical prescription; an equilibrium state need not be translation invariant. We show that every equilibrium state has a unique decomposition into equilibrium states with short-range correlations. We use the properties of equilibrium states to prove some negative results about the existence of metastable states. We show that the correlation functions for an equilibrium state satisfy the Kirkwood-Salsburg equations; thus, at low activity, there is only one equilibrium state for a given interaction, temperature, and chemical potential. Finally, we argue heuristically that equilibrium states are invariant under time-evolution.

600 citations

Book
30 Jun 2011
TL;DR: Convexity is important in theoretical aspects of mathematics and also for economists and physicists as discussed by the authors, and a comprehensive overview of convex sets and functions including the infinite-dimensional case and emphasizing the analytic point of view is provided.
Abstract: Convexity is important in theoretical aspects of mathematics and also for economists and physicists. In this monograph the author provides a comprehensive insight into convex sets and functions including the infinite-dimensional case and emphasizing the analytic point of view. Chapter one introduces the reader to the basic definitions and ideas that play central roles throughout the book. The rest of the book is divided into four parts: convexity and topology on infinite-dimensional spaces; Loewner's theorem; extreme points of convex sets and related issues, including the Krein–Milman theorem and Choquet theory; and a discussion of convexity and inequalities. The connections between disparate topics are clearly explained, giving the reader a thorough understanding of how convexity is useful as an analytic tool. A final chapter overviews the subject's history and explores further some of the themes mentioned earlier. This is an excellent resource for anyone interested in this central topic.

127 citations

Posted Content
TL;DR: A new point of view on the mathematical foundations of statistical physics of infinite volume systems is presented, based on the newly introduced notions of transition energy function, transition energy field and one-point transitionEnergy field, which establishes a straightforward relationship between the probabilistic notion of (Gibbs) random field and the physical notion of(transition) energy.
Abstract: In this paper we present a new point of view on the mathematical foundations of statistical physics of infinite volume systems. This viewpoint is based on the newly introduced notions of transition energy function, transition energy field and one-point transition energy field. The former of them, namely the transition energy function, is a generalization of the notion of relative Hamiltonian introduced by Pirogov and Sinai. However, unlike the (relative) Hamiltonian, our objects are defined axiomatically by their natural and physically well-founded intrinsic properties. The developed approach allowed us to give a proper mathematical definition of the Hamiltonian without involving the notion of potential, to propose a justification of the Gibbs formula for infinite systems and to answer the problem stated by D. Ruelle of how wide the class of specifications, which can be represented in Gibbsian form, is. Furthermore, this approach establishes a straightforward relationship between the probabilistic notion of (Gibbs) random field and the physical notion of (transition) energy, and so opens the possibility to directly apply probabilistic methods to the mathematical problems of statistical physics.

8 citations