Free transport for convex potentials

We construct non-commutative analogs of transport maps among free Gibbs state satisfying a certain convexity condition. Unlike previous constructions, our approach is non-perturbative in nature and thus can be used to construct transport maps between free Gibbs states associated to potentials which are far from quadratic, i.e., states which are far from the semicircle law. An essential technical ingredient in our approach is the extension of free stochastic analysis to non-commutative spaces of functions based on the Haagerup tensor product.

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EUCLIDEAN GIBBS STATES OF QUANTUM LATTICE SYSTEMS

Reviews in Mathematical Physics ◽

10.1142/s0129055x02001545 ◽

2002 ◽

Vol 14 (12) ◽

pp. 1335-1401 ◽

Cited By ~ 15

Author(s):

S. ALBEVERIO ◽

YU. KONDRATIEV ◽

YU. KOZITSKY ◽

M. RÖCKNER

Keyword(s):

Critical Point ◽

Long Range ◽

Gibbs State ◽

Classical Model ◽

Gibbs Measures ◽

Gibbs States ◽

Multiplication Operators ◽

Approximation Technique ◽

Lattice Systems ◽

Euclidean Gibbs States

An approach to the description of the Gibbs states of lattice models of interacting quantum anharmonic oscillators, based on integration in infinite dimensional spaces, is described in a systematic way. Its main feature is the representation of the local Gibbs states by means of certain probability measures (local Euclidean Gibbs measures). This makes it possible to employ the machinery of conditional probability distributions, known in classical statistical physics, and to define the Gibbs state of the whole system as a solution of the equilibrium (Dobrushin–Lanford–Ruelle) equation. With the help of this representation the Gibbs states are extended to a certain class of unbounded multiplication operators, which includes the order parameter and the fluctuation operators describing the long range ordering and the critical point respectively. It is shown that the local Gibbs states converge, when the mass of the particle tends to infinity, to the states of the corresponding classical model. A lattice approximation technique, which allows one to prove for the local Gibbs states analogs of known correlation inequalities, is developed. As a result, certain new inequalities are derived. By means of them, a number of statements describing physical properties of the model are proved. Among them are: the existence of the long-range order for low temperatures and large values of the particle mass; the suppression of the critical point behavior for small values of the mass and for all temperatures; the uniqueness of the Euclidean Gibbs states for all temperatures and for the values of the mass less than a certain threshold value, dependent on the temperature.

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The weak Bernoulli property for matrix Gibbs states

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.129 ◽

2018 ◽

Vol 40 (8) ◽

pp. 2219-2238 ◽

Cited By ~ 1

Author(s):

MARK PIRAINO

Keyword(s):

General Result ◽

Gibbs State ◽

Dimensional Space ◽

Gibbs States ◽

Sufficient Condition ◽

Infinite Dimensional Space ◽

Infinite Dimensional ◽

Ergodic Properties ◽

Finite Dimensional ◽

Novel Approach

We study the ergodic properties of a class of measures on $\unicode[STIX]{x1D6F4}^{\mathbb{Z}}$ for which $\unicode[STIX]{x1D707}_{{\mathcal{A}},t}[x_{0}\cdots x_{n-1}]\approx e^{-nP}\Vert A_{x_{0}}\cdots A_{x_{n-1}}\Vert ^{t}$, where ${\mathcal{A}}=(A_{0},\ldots ,A_{M-1})$ is a collection of matrices. The measure $\unicode[STIX]{x1D707}_{{\mathcal{A}},t}$ is called a matrix Gibbs state. In particular, we give a sufficient condition for a matrix Gibbs state to have the weak Bernoulli property. We employ a number of techniques to understand these measures, including a novel approach based on Perron–Frobenius theory. We find that when $t$ is an even integer the ergodic properties of $\unicode[STIX]{x1D707}_{{\mathcal{A}},t}$ are readily deduced from finite-dimensional Perron–Frobenius theory. We then consider an extension of this method to $t>0$ using operators on an infinite- dimensional space. Finally, we use a general result of Bradley to prove the main theorem.

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Classical approximations to Gibbs states

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100056383 ◽

1979 ◽

Vol 86 (3) ◽

pp. 521-527

Author(s):

E. B. Davies

Keyword(s):

Phase Space ◽

Coherent States ◽

Gibbs State ◽

Gibbs States ◽

Microcanonical Ensemble ◽

Trace Norm ◽

Classical Approximation

AbstractWe obtain two approximate representations of a one-particle Gibbs state, both of which become asymptotically exact in trace norm as m → ∞. The second representation is an integral of pure coherent states over phase space, and can therefore be regarded as a classical approximation to the Gibbs state. We also obtain a version of the second representation applicable to the microcanonical ensemble.

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Examples of Gibbs States of Mechanical Systems with Symmetries

Journal of Geometry and Symmetry in Physics ◽

10.7546/jgsp-58-2020-55-79 ◽

2020 ◽

Vol 58 ◽

pp. 55-79

Author(s):

Charles-Michel Marle ◽

Keyword(s):

Gibbs State ◽

Three Dimensional ◽

Cross Product ◽

Symplectic Manifolds ◽

Gibbs States ◽

Vector Spaces ◽

Two Dimensional ◽

Dimensional Vector ◽

Hamiltonian Action ◽

French Mathematician

In a previous paper, the notion of Gibbs state for the Hamiltonian action of a Lie group on a symplectic manifold was given, together with its applications in Statistical Mechanics, and the works in this field of the French mathematician and physicist Jean-Marie Souriau were presented. Using an adaptation of the cross product for pseudo-Euclidean three-dimensional vector spaces, we present several examples of such Gibbs states, together with the associated thermodynamic functions, for various two-dimensional symplectic manifolds, including the pseudo-spheres, the Poincar{\'e} disk and the Poincar{\'e} half-plane.

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