Existence and a priori estimates for Euclidean Gibbs states

Albeverio, Sergio; Kondratiev, Yuri; Pasurek, Tanja; Röckner, Michael

doi:10.1090/s0077-1554-07-00158-6

Cited by 7 publications

(11 citation statements)

References 75 publications

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“…This again implies uniqueness in this case. Finally, following the arguments of [6,11] we prove the uniqueness of EGM on the set of tempered configurations (Section 3).…”

Section: Discussionmentioning

confidence: 96%

Gibbs state uniqueness for an anharmonic quantum crystal with a non-polynomial double-well potential

Rebenko¹,

Zagrebnov²

2006

J. Stat. Mech.

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We construct the Gibbs state for ν-dimensional quantum crystal with site displacements from IR d , d ≥ 1, and with a one-site non-polynomial double-well potential, which has harmonic asymptotic growth at infinity. We prove the uniqueness of the corresponding Euclidean Gibbs measure (EGM) in the lightmass regime for the crystal particles. The corresponding state is constructed via a cluster expansion technique for an arbitrary temperature T ≥ 0. We show that for all T ≥ 0 the Gibbs state (correlation functions) is analytic with respect to external field conjugated to displacements provided that the mass of particles m is less than a certain value m * > 0. The high temperature regime is also discussed.Keywords : quantum crystal model, Gibbs state, Euclidean Gibbs measures, quantum fluctuations, light-mass regime, cluster expansions. Mathematics Subject Classification : 60H30, 82B31 IntroductionIt is generally excepted that for investigation of different physical phenomena in quantum crystals one can consider an infinite system of interacting anharmonic oscillators, which are situated in the sites of ν-dimensional lattice Z Z ν . The heuristic Hamiltonian of such a system (in the case of 2-body interaction) has the following form:where m is the mass of particles, the operator ∆ j corresponds the kinetic energy of the system and, in fact, is d-dimensional Laplace operator in the one-particle Hilbert space, where dq is the Lebesgue measure onν is displacement of a particle from its position in the site j ∈ Z Z ν . For general case d ≤ ν ≥ 1. But the most of results, which where obtained earlier are for d = 1. The particles are confined near their sites by potential W (q j ). A harmonic one-site potential W harm (q j ) = 1 2 aq 2 j , a > 0 together with a harmonic two-particle interaction Q(·, ·) define a well-known harmonic crystal model (1.1). To produce a model describing a (ferroelectric) structural phase transition one usually takes for W (·) a doublewell anharmonic potential, keeping Q(·, ·) harmonic (see e.g. [1,19,24]). For example,or semibounded from below polynomials of higher degree with more than two equal minima. Here q 2 ≡ q · q is the scalar square of displacement vector q ∈ IR d . The proof of existence of phase transition in such kind of systems for harmonic interaction Q(q i , q j ) were obtained in [12,15,26,34,40,47,53] for the case2) whereQ = (Q ij ) i,j∈Z Z ν is a matrix of non-negative harmonic-force constants. The term h · q j = d α=1 h α q α j break the symmetry in the direction of the 1 external field h. The non-uniqueness of Gibbs states for h = 0 is proved when the mass of the particles m is sufficiently large and the temperature T is sufficiently low, or β = (kT ) −1 is sufficiently large. From the other side, another interesting phenomenon, the suppression of the long-range order by strong quantum fluctuations in such systems was experimentally observed (see, e.g. [55]) and was discussing long time ago from the physical point of view, see [51], or the books [1,19]. A rigorous stu...

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“…This again implies uniqueness in this case. Finally, following the arguments of [6,11] we prove the uniqueness of EGM on the set of tempered configurations (Section 3).…”

Section: Discussionmentioning

confidence: 96%

Gibbs state uniqueness for an anharmonic quantum crystal with a non-polynomial double-well potential

Rebenko¹,

Zagrebnov²

2006

J. Stat. Mech.

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“…There are sufficient conditions for the existence and sometimes for the uniqueness of solutions, but not much is known about their properties and connections between infinitesimal invariance and proper invariance with respect to the associated semigroups (the very existence of such semigroups has also been less studied). There exists an extensive literature on stationary distributions of infinite-dimensional diffusions (see, e.g., the book [67]), especially connected with stochastic partial differential equations (see [11], [52], [57], [58], [62], [63], [70], [74], [79], [82], [90], [91], [128], [150], [163]), infinite gradient systems, Gibbs measures, and stochastic quantization (see [5], [6], [56], [83], [85], [86], [96], [138]); in these works numerous additional references can be found. Standard methods of proving the existence of stationary distributions are based on Prohorov's theorem and Lyapunov functions combined with a priori estimates (see, e.g., [59], [113], [114]) or on convergence of transition probabilities (which in turn employs various assumptions of dissipativity and Lyapunov functions).…”

Section: Equations For Measures On Infinite-dimensional Spacesmentioning

confidence: 99%

Elliptic and parabolic equations for measures

Богачев¹,

Крылов²,

Röckner³

2009

Russ. Math. Surv.

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The article gives a detailed account of recent investigations of weak elliptic and parabolic equations for measures with unbounded and possibly singular coefficients. The existence and differentiability of densities are studied, and lower and upper bounds for them are discussed. Semigroups associated with second order elliptic operators in L p-spaces with respect to infinitesimally invariant measures are investigated. Bibliography 181 items.

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“…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%

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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Existence and a priori estimates for Euclidean Gibbs states

Cited by 7 publications

References 75 publications

Gibbs state uniqueness for an anharmonic quantum crystal with a non-polynomial double-well potential

Gibbs state uniqueness for an anharmonic quantum crystal with a non-polynomial double-well potential

Elliptic and parabolic equations for measures

Euclidean Gibbs Measures of Interacting Quantum Anharmonic Oscillators

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