A REMARK ON DIFFERENTIABILITY OF THE PRESSURE FUNCTIONAL

1995 ◽  
Vol 07 (06) ◽  
pp. 959-977 ◽  
Author(s):  
AERNOUT VAN ENTER ◽  
BOGUSLAW ZEGARLINSKI
Keyword(s):  
High Temperatures ◽  
Stochastic Dynamics ◽  
Gibbs Measures ◽  
Logarithmic Sobolev Inequality ◽  
Short Review ◽  
Spin Systems ◽  
Spin Interactions ◽  
Lattice Spin Systems ◽  
Strong Differentiability ◽  
Natural Way

We give a short review of results on equilibrium description and description by stochastic dynamics for spin systems on a lattice. We remark also that some coercive inequalities for the generators of stochastic dynamics, as e.g. the Logarithmic Sobolev inequality, can be used in a direct and natural way to prove strong differentiability properties of the pressure functional for lattice spin systems with multiparticle interactions at high temperatures. Motivated by this, we exhibit also a class of examples of multiparticle interactions which do not belong to the space [Formula: see text] of spin interactions, but for which the Gibbs measures exist and are unique at high temperatures.

scholarly journals Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations

2021 ◽  
Author(s):  
Ivan Bardet ◽  
Ángela Capel ◽  
Cambyse Rouzé
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebras ◽  
Logarithmic Sobolev Inequality ◽  
Spin Systems ◽  
Von Neumann ◽  
Strong Subadditivity ◽  
Finite Dimensional ◽  
Lattice Spin ◽  
Lattice Spin Systems ◽  
Conditional Expectations

AbstractIn this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. This generalisation, referred to as approximate tensorization of the relative entropy, consists in a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.

QUANTUM STOCHASTIC DYNAMICS II

1996 ◽  
Vol 08 (05) ◽  
pp. 689-713 ◽  
Author(s):  
ADAM W. MAJEWSKI ◽  
BOGUSLAW ZEGARLINSKI
Keyword(s):  
High Temperatures ◽  
Stochastic Dynamics ◽  
Quantum Spin ◽  
Spin Systems ◽  
Finite Range ◽  
Quantum Spin Systems ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Spin Flip ◽  
Nikodym Theorem

We shortly review the progress in the domain of stochastic dynamics for quantum spin systems on a lattice. We also present some new results obtained in the framework of noncommutative [Formula: see text] spaces. In particular, using noncommutative Radon-Nikodym theorem of A. Connes we construct Markov generators of stochastic dynamics of spin flip type for systems at high temperatures or on one-dimensional lattice and with interactions of finite range at arbitrary temperatures.

scholarly journals Geometric criterion for solvability of lattice spin systems

2020 ◽  
Vol 102 (24) ◽  
Author(s):  
Masahiro Ogura ◽  
Yukihisa Imamura ◽  
Naruhiko Kameyama ◽  
Kazuhiko Minami ◽  
Masatoshi Sato
Keyword(s):  
Spin Systems ◽  
Geometric Criterion ◽  
Lattice Spin ◽  
Lattice Spin Systems

Mixing in time and space for lattice spin systems: A combinatorial view

2004 ◽  
Vol 24 (4) ◽  
pp. 461-479 ◽  
Author(s):  
Martin Dyer ◽  
Alistair Sinclair ◽  
Eric Vigoda ◽  
Dror Weitz
Keyword(s):  
Spin Systems ◽  
Time And Space ◽  
Lattice Spin ◽  
Lattice Spin Systems

scholarly journals Interpolation between low and high temperatures of the specific heat for spin systems

2017 ◽  
Vol 95 (4) ◽  
Author(s):  
Heinz-Jürgen Schmidt ◽  
Andreas Hauser ◽  
Andre Lohmann ◽  
Johannes Richter
Keyword(s):  
Specific Heat ◽  
High Temperatures ◽  
Spin Systems

scholarly journals Nonconventional averages along arithmetic progressions and lattice spin systems

2012 ◽  
Vol 23 (3) ◽  
pp. 589-602 ◽  
Author(s):  
Gioia Carinci ◽  
Jean-René Chazottes ◽  
Cristian Giardinà ◽  
Frank Redig
Keyword(s):  
Spin Systems ◽  
Arithmetic Progressions ◽  
Lattice Spin ◽  
Lattice Spin Systems ◽  
Nonconventional Averages

scholarly journals EQUILIBRIUM KAWASAKI DYNAMICS OF CONTINUOUS PARTICLE SYSTEMS

2007 ◽  
Vol 10 (02) ◽  
pp. 185-209 ◽  
Author(s):  
YURI KONDRATIEV ◽  
EUGENE LYTVYNOV ◽  
MICHAEL RÖCKNER
Keyword(s):  
Stochastic Dynamics ◽  
A Priori ◽  
Scaling Limit ◽  
Glauber Dynamics ◽  
Particle Systems ◽  
Death Process ◽  
Kawasaki Dynamics ◽  
Interacting Particles ◽  
Lattice Spin Systems ◽  
Diffusion Dynamics

We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a priori explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birth-and-death process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).

scholarly journals The infinite word problem and limit sets in Fuchsian groups

1981 ◽  
Vol 1 (3) ◽  
pp. 337-360 ◽  
Author(s):  
Caroline Series
Keyword(s):  
Gibbs Measures ◽  
Fuchsian Groups ◽  
Countable Number ◽  
Exponent Of Convergence ◽  
Limit Sets ◽  
Infinite Word ◽  
Dimensional Measure ◽  
Fixed Set ◽  
Infinite Sequences ◽  
Natural Way

AbstractLet Г be a finitely generated non-elementary Fuchsian group acting in the disk. With the exception of a small number of co-compact Г, we give a representation of g ∈ Г as a product of a fixed set of generators Гo in a unique shortest ‘admissible form’. Words in this form satisfy rules which after a suitable coding are of finite type. The space of infinite sequences Σ of generators satisfying the same rules is identified in a natural way with the limit set Λ of Г by a map which is bijective except at a countable number of points where it is two to one. We use the theory of Gibbs measures onΣ to construct the so-called Patterson measure on Λ [8], [9]. This measure is, in fact, Hausdorff 5-dimensional measure on Λ, where S is the exponent of convergence of Г.

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