Abstract
Let z be the activity of point particles described by classical equilibrium statistical mechanics in \(\mathbf{R}^\nu \). 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
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.
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References
Dashian, S., Nahapetian, B.S.: On the relationship of energy and probability in models of classical statistical physics. Markov Process. Relat. Fields 25, 649–681 (2019)
Dobrushin, R.L.: The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Prob. Appl. 13, 197–224 (1968)
Georgii, H.-O.: Large deviations for hard-core particle systems. In: Kotecký, R. (ed.) Phase Transitions, pp. 108–116. World Scientific, Singapore (1993)
Lanford, O.E., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Commun. Math. Phys. 13, 194–215 (1969)
Ruelle, D.: Statistical Mechanics. Rigorous Results. Benjamin, New York (1969)
Ruelle, D.: Thermodynamic Formalism. Addison-Wesley, Reading (1978)
Simon, B.: Convexity: an analytic viewpoint. Cambridge University Press, Cambridge (2011)
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Communicated by David Ruelle.
This note is dedicated to Joel Lebowitz on his 90-th birthday
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Ruelle, D. Identities for Correlation Functions in Classical Statistical Mechanics and the Problem of Crystal States. J Stat Phys 180, 1002–1009 (2020). https://doi.org/10.1007/s10955-020-02575-3
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DOI: https://doi.org/10.1007/s10955-020-02575-3