Compactness and the maximal Gibbs state for random Gibbs fields on a lattice

Jean Bellissard; Raphael H�egh-Krohn

doi:10.1007/bf01208480

Compactness and the maximal Gibbs state for random Gibbs fields on a lattice

Communications in Mathematical Physics ◽

10.1007/bf01208480 ◽

1982 ◽

Vol 84 (3) ◽

pp. 297-327 ◽

Cited By ~ 37

Author(s):

Jean Bellissard ◽

Raphael H�egh-Krohn

Keyword(s):

Gibbs State ◽

Gibbs Fields

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Simple Systems

10.1093/oso/9780199595068.003.0004 ◽

2017 ◽

Author(s):

Jochen Rau

Keyword(s):

Magnetic Field ◽

Low Temperatures ◽

General Framework ◽

Gibbs State ◽

Macroscopic System ◽

Basic Model ◽

System A ◽

Paramagnetic Salt ◽

High And Low Temperatures ◽

Simple Systems

Even though the general framework of statistical mechanics is ultimately targeted at the description of macroscopic systems, it is illustrative to apply it first to some simple systems: a harmonic oscillator, a rotor, and a spin in a magnetic field. These applications serve to illustrate how a key function associated with the Gibbs state, the so-called partition function, is calculated in practice, how the entropy function is obtained via a Legendre transformation, and how such systems behave in the limits of high and low temperatures. After discussing these simple systems, this chapter considers a first example where multiple constituents are assembled into a macroscopic system: a basic model of a paramagnetic salt. It also investigates the size of energy fluctuations and how—in the case of the paramagnet—these fluctuations scale with the number of constituents.

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Free transport for convex potentials

New Zealand Journal of Mathematics ◽

10.53733/102 ◽

2021 ◽

Vol 52 ◽

pp. 259-359

Author(s):

Yoann Dabrowski ◽

Alice Guionnet ◽

Dima Shlyakhtenko

Keyword(s):

Tensor Product ◽

Stochastic Analysis ◽

Gibbs State ◽

Gibbs States ◽

Convexity Condition ◽

Semicircle Law ◽

Free Transport ◽

Transport Maps

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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