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Euclidean Gibbs Measures of Interacting Quantum Anharmonic Oscillators

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Abstract

A rigorous description of the equilibrium thermodynamic properties of an infinite system of interacting ν-dimensional quantum anharmonic oscillators is given. The oscillators are indexed by the elements of a countable set \({\mathbb L}\subset {\mathbb R}^d\), possibly irregular; the anharmonic potentials vary from site to site and the interaction has infinite range. The description is based on the representation of the Gibbs states in terms of path measures—the so called Euclidean Gibbs measures. It is proven that: (a) the set of such measures \(\mathcal{G}^{\rm t}\) is non-void and compact; (b) every \(\mu \in \mathcal{G}^{\rm t}\) obeys an exponential integrability estimate, the same for the whole set \(\mathcal{G}^{\rm t}\); (c) every μ \(\mu \in \mathcal{G}^{\rm t}\) has a Lebowitz-Presutti type support; (d) \(\mathcal{G}^{\rm t}\) is a singleton at high temperatures. The case of attractive interaction and \(\nu=1\) is studied in more detail. We prove that: (a) \(|\mathcal{G}^{\rm t}|>1\) at low temperatures; (b) \(|\mathcal{G}^{\rm t}|=1\) due to quantum effects and at a nonzero external field. Thereby, a qualitative theory of phase transitions and quantum effects, which interprets most important experimental data known for the corresponding physical objects, is developed.

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  1. Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, PL 20-031, Lublin, Poland

    Yuri Kozitsky

  2. Fakultät für Mathematik und Forschungszentrum BiBoS, Universität Bielefeld, D-33615, Bielefeld, Germany

    Tatiana Pasurek

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  1. Yuri Kozitsky
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1991 Mathematics Subject Classification: 82B10, 60J60, 60G60, 46G12, 46T12.

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Kozitsky, Y., Pasurek, T. Euclidean Gibbs Measures of Interacting Quantum Anharmonic Oscillators. J Stat Phys 127, 985–1047 (2007). https://doi.org/10.1007/s10955-006-9274-9

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