Gibbs states of a quantum crystal: uniqueness by small particle mass

Sergio Albeverio; Yuri Kondratiev; Yuri Kozitsky; Michael Röckner

doi:10.1016/S1631-073X(02)02545-1

Gibbs states of a quantum crystal: uniqueness by small particle mass
[États de Gibbs de crystaux quantiques: unicité dans le cas d'une petite masse]

Sergio Albeverio ^1,^2,³ ; Yuri Kondratiev ^4,^2,⁵ ; Yuri Kozitsky ⁶ ; Michael Röckner ^4,²

¹ Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany
² Forschungszentrum BiBoS, Universität Bielefeld, 33615 Bielefeld, Germany
³ CERFIM, Locarno and USI, Switzerland
⁴ Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany
⁵ Institute of Mathematics, Kiev, Ukraine
⁶ Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, 20-031 Lublin, Poland

Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 693-698.

Abstract (VO)
Résumé

A model of interacting quantum particles performing one-dimensional anharmonic oscillations around their unstable equilibrium positions, which form the lattice $ℤ^{d}$ , is considered. For this model, two statements describing its equilibrium properties are given. The first theorem states that there exists $m_{*} > 0$ such that for all values of the particle mass $m < m_{*}$ , the set of tempered Euclidean Gibbs measures consists of exactly one element at all values of the temperature β⁻¹. This settles a problem that was open for a long time and is an essential improvement of a similar result proved before by the same authors [1] where the boundary $m_{*}$ depended on β in such a way that $m_{*} (β) \to 0$ for β→+∞. The second theorem states that the two-point correlation function has an exponential decay if $m < m_{*}$ .

On considère un modèle de particules quantiques en intéraction effectuant des oscillations anharmoniques uni-dimensionelles autour de leur positions d'équilibre sur le réseau $ℤ^{d}$ . Pour ce modèle, nous énonçons deux résultats décrivant ses propriétés d'équilibre. Le premier théorème affirme l'existence de $m_{*} > 0$ tel que pour toutes les valeurs de la masse m de la particule inférieures à $m_{*}$ , l'ensemble des mesures euclidiennes tempérées de Gibbs consiste en un seul élément, à toute température β⁻¹. Cela résoud un problème qui est resté ouvert pour longtemps et améliore essentiellement un résultat analogue obtenu par les mêmes auteurs, lorsque $m_{*}$ dépendait de β de sorte que $m_{*} (β) \to 0$ si β→+∞. Le deuxième théorème dit que la fonction de corrélation a une décroissance exponentielle si $m < m_{*}$ .

Reçu le : 2002-08-01
Accepté le : 2002-09-03
Publié le : 2002-10-01

DOI : 10.1016/S1631-073X(02)02545-1

Affiliations des auteurs :

Sergio Albeverio ^{1, 2, 3} ; Yuri Kondratiev ^{4, 2, 5} ; Yuri Kozitsky ⁶ ; Michael Röckner ^{4, 2}

¹ Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany
² Forschungszentrum BiBoS, Universität Bielefeld, 33615 Bielefeld, Germany
³ CERFIM, Locarno and USI, Switzerland
⁴ Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany
⁵ Institute of Mathematics, Kiev, Ukraine
⁶ Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, 20-031 Lublin, Poland

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Sergio Albeverio; Yuri Kondratiev; Yuri Kozitsky; Michael Röckner. Gibbs states of a quantum crystal: uniqueness by small particle mass. Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 693-698. doi : 10.1016/S1631-073X(02)02545-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02545-1/

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Cité par

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Alexei Daletskii; Yuri Kondratiev; Tanja Pasurek Sergio's work in statistical mechanics: from quantum particles to geometric stochastic analysis, Quantum and stochastic mathematical physics. Sergio Albeverio, adventures of a mathematician, Verona, Italy, March 25–29, 2019, Cham: Springer, 2023, pp. 217-246 | DOI:10.1007/978-3-031-14031-0_10 | Zbl:1549.82006
Alina Kargol; Yuri Kondratiev; Yuri Kozitsky Phase transitions and quantum stabilization in quantum anharmonic crystals, Reviews in Mathematical Physics, Volume 20 (2008) no. 5, pp. 529-595 | DOI:10.1142/s0129055x08003353 | Zbl:1151.82014
Alexei L. Rebenko; Valentin A. Zagrebnov Gibbs state uniqueness for an anharmonic quantum crystal with a non-polynomial double-well potential, Journal of Statistical Mechanics: Theory and Experiment, Volume 2006 (2006) no. 9, p. 29 (Id/No p09002) | DOI:10.1088/1742-5468/2006/09/p09002 | Zbl:1456.82071
Sergio Albeverio; Yuri Kondratiev; Yuri Kozitsky; Michael Röckner Small mass implies uniqueness of Gibbs states of a quantum crystal, Communications in Mathematical Physics, Volume 241 (2003) no. 1, pp. 69-90 | DOI:10.1007/s00220-003-0923-4 | Zbl:1149.82307
Sergio Albeverio; Yuri Kondratiev; Yuri Kozitsky; Michael Röckner Quantum Stabilization in Anharmonic Crystals, Physical Review Letters, Volume 90 (2003) no. 17 | DOI:10.1103/physrevlett.90.170603

Cité par 5 documents. Sources : Crossref, zbMATH

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