A Priori Estimates for Symmetrizing Measures and Their Applications to Gibbs States

Albeverio, Sergio; Kondratiev, Yuri; Röckner, Michael; Tsikalenko, T. V.

doi:10.1006/jfan.1999.3482

Cited by 21 publications

(64 citation statements)

References 29 publications

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“…For systems of classical unbounded spins, the problem of deriving such estimates was first posed in [17] (see the discussion following Corollary 4.2). For quantum anharmonic systems, similar estimates were obtained in the so called analytic approach, alternative to the traditional approach based on the DLR equation, see [8,7,13]. In this analytic approach G t is defined as the set of probability measures satisfying an integration-by-parts formula, determined by the model.…”

Section: Commentsmentioning

confidence: 79%

“…. , z n ), analytic in the domain 13) and continuous on its closureD n β ⊂ C n . Foe every n ∈ N, the "imaginary time" subset…”

Section: Quantum Gibbs States In the Euclidean Approachmentioning

confidence: 99%

“…This approach is called Euclidean due to its conceptual analogy with the Euclidean quantum field theory. Its further development was conducted in the papers [2,3,4,5,8,7,11,12,13,14,16,48,49,50,52,54,55,66,67]. Among the achievements one has to mention the settlement in [3,5,6] of a long standing problem of the influence of quantum effects on structural phase transitions in quantum anharmonic crystals, which first was discussed in [77], see also [67,86,87].…”

Section: Introductionmentioning

confidence: 99%

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Euclidean Gibbs Measures of Interacting Quantum Anharmonic Oscillators

Kozitsky

Pasurek

2007

J Stat Phys

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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 L ⊂ R d , possibly irregular; the anharmonic potentials vary from site to site. 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 G t is non-void and compact; (b) every µ ∈ G t obeys an exponential integrability estimate, the same for the whole set G t ; (c) every µ ∈ G t has a Lebowitz-Presutti type support; (d) G t is a singleton at high temperatures. In the case of attractive interaction and ν = 1 we prove that |G t | > 1 at low temperatures. The uniqueness of Gibbs measures due to quantum effects and at a nonzero external field are also proven in this case. Thereby, a qualitative theory of phase transitions and quantum effects, which interprets most important experimental data known for the corresponding physical objects, is developed. The mathematical result of the paper is a complete description of the set G t , which refines and extends the results known for models of this type.

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Section: Commentsmentioning

confidence: 79%

“…. , z n ), analytic in the domain 13) and continuous on its closureD n β ⊂ C n . Foe every n ∈ N, the "imaginary time" subset…”

Section: Quantum Gibbs States In the Euclidean Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Euclidean Gibbs Measures of Interacting Quantum Anharmonic Oscillators

Kozitsky

Pasurek

2007

J Stat Phys

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“…It should be mentioned that this approach was first realized for classical lattice systems in [AKRT99,AKRT00]. However, the abstract scheme, in the form suggested in those papers, is not directly applicable (see also [BR01,BRW01]) to the present case.…”

mentioning

confidence: 86%

“…So far, there are very few rigorous results on Gibbs measures on the path space [C(R)] Λ are well known as the P (φ) 1 -processes and can be looked upon as a special case of Euclidean field theory in space-dimension zero (cf. [Iw85,OS99] AKRT00], however, in the much simpler situation of classical lattice systems such as (2.14) with finite-dimensional spins. But the concrete technique suggested in those papers does not apply to loop spaces, so that a proper (highly nontrivial) modification for the quantum case is necessary.…”

Section: 4mentioning

confidence: 99%

Existence and a priori estimates for Euclidean Gibbs states

Albeverio

Kondratiev

Pasurek

et al. 2007

Trans. Moscow Math. Soc.

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Abstract. We prove a priori estimates and, as a sequel, the existence of Euclidean Gibbs states for quantum lattice systems. For this purpose we develop a new analytical approach, the main tools of which are: first, a characterization of the Gibbs states in terms of their Radon-Nikodým derivatives under shift transformations as well as in terms of their logarithmic derivatives through integration by parts formulae, and second, the choice of appropriate Lyapunov functionals describing stabilization effects in the system. The latter technique becomes applicable since on the basis of the integration by parts formulae the Gibbs states are characterized as solutions of an infinite system of partial differential equations. Our existence results generalize essentially all previous ones. In particular, superquadratic growth of the interaction potentials is allowed and N -particle interactions for N ∈ N ∪ {∞} are included. We also develop abstract frames both for the necessary single spin space analysis and for the lattice analysis apart from their applications to our concrete models. Both types of general results obtained in these two frames should also be of their own interest in infinite-dimensional analysis.

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Correlation asymptotics for non‐translation invariant lattice spin systems

Matte

2008

Mathematische Nachrichten

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A Priori Estimates for Symmetrizing Measures and Their Applications to Gibbs States

Cited by 21 publications

References 29 publications

Euclidean Gibbs Measures of Interacting Quantum Anharmonic Oscillators

Euclidean Gibbs Measures of Interacting Quantum Anharmonic Oscillators

Existence and a priori estimates for Euclidean Gibbs states

Correlation asymptotics for non‐translation invariant lattice spin systems

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