A model of interacting quantum particles performing one-dimensional anharmonic oscillations around their unstable equilibrium positions, which form the lattice , is considered. For this model, two statements describing its equilibrium properties are given. The first theorem states that there exists such that for all values of the particle mass , 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 depended on β in such a way that for β→+∞. The second theorem states that the two-point correlation function has an exponential decay if .
On considère un modèle de particules quantiques en intéraction effectuant des oscillations anharmoniques uni-dimensionelles autour de leur positions d'équilibre sur le réseau . Pour ce modèle, nous énonçons deux résultats décrivant ses propriétés d'équilibre. Le premier théorème affirme l'existence de tel que pour toutes les valeurs de la masse m de la particule inférieures à , 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 dépendait de β de sorte que si β→+∞. Le deuxième théorème dit que la fonction de corrélation a une décroissance exponentielle si .
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Sergio Albeverio 1, 2, 3; Yuri Kondratiev 4, 2, 5; Yuri Kozitsky 6; Michael Röckner 4, 2
@article{CRMATH_2002__335_8_693_0, author = {Sergio Albeverio and Yuri Kondratiev and Yuri Kozitsky and Michael R\"ockner}, title = {Gibbs states of a quantum crystal: uniqueness by small particle mass}, journal = {Comptes Rendus. Math\'ematique}, pages = {693--698}, publisher = {Elsevier}, volume = {335}, number = {8}, year = {2002}, doi = {10.1016/S1631-073X(02)02545-1}, language = {en}, }
TY - JOUR AU - Sergio Albeverio AU - Yuri Kondratiev AU - Yuri Kozitsky AU - Michael Röckner TI - Gibbs states of a quantum crystal: uniqueness by small particle mass JO - Comptes Rendus. Mathématique PY - 2002 SP - 693 EP - 698 VL - 335 IS - 8 PB - Elsevier DO - 10.1016/S1631-073X(02)02545-1 LA - en ID - CRMATH_2002__335_8_693_0 ER -
%0 Journal Article %A Sergio Albeverio %A Yuri Kondratiev %A Yuri Kozitsky %A Michael Röckner %T Gibbs states of a quantum crystal: uniqueness by small particle mass %J Comptes Rendus. Mathématique %D 2002 %P 693-698 %V 335 %N 8 %I Elsevier %R 10.1016/S1631-073X(02)02545-1 %G en %F CRMATH_2002__335_8_693_0
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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