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Book review: Gibbs measures and phase transitions

1991
*
Journal of statistical physics
*

Gaussian

doi:10.1007/bf01057888
fatcat:jus24bflwrg7jheddjjcjscisu
*random**fields*are shown to be*Gibbs*measures (under a weak kind of Markov property)*for*a pair potential that can be explicitly computed in terms of the*mean**and*covariance*function*. ... The latter are denoted by Ex G(7)*and*represent the pure*phases*of the physical system. Part H deals with Markov*and*with Gauss*random**fields*as*Gibbs*measures. ...##
###
Page 3513 of Mathematical Reviews Vol. , Issue 89F
[page]

1989
*
Mathematical Reviews
*

Specifically, it is demonstrated from the

*phase*diagram on the temperature-*field*plane that both before*and*after the sublattice flopping, the*transition*temperature increases with the increase in*field*... Summary: “A number of*phase*rules of coexistence point sets (*Gibbs**phase*rule)*and*of critical point sets (Griffiths*phase*rule) in one-order parameter systems are available in the literature. ...##
###
Page 1839 of Mathematical Reviews Vol. , Issue 90C
[page]

1990
*
Mathematical Reviews
*

At high tem- peratures

*and**for*arbitrary (possibly*random*)*fields*the pressure exists, susceptibilities (sums of connected*correlation**functions*) are finite*and*translation averages of local observables ... D. van Enter (NL-GRON-P) 90c:82007 82A05 82A25 Gonchar, N.S. (2-AOSUK-P)*Correlation**functions*of some continuous model systems*and*description of*phase**transitions*. Phys. ...##
###
Critical behavior of mean-field XY and related models
[article]

2016
*
arXiv
*
pre-print

We present a number of results coming from the theory of large deviations

arXiv:1611.02356v1
fatcat:bu6eukfowvaelbpqedv7pag5ie
*and*Stein's method, in particular, Cram\'er*and*Sanov-type results, limit theorems with rates of convergence,*and**phase**transition*... We discuss spin models on complete graphs in the*mean*-*field*(infinite-vertex) limit, especially the classical XY model, the Toy model of the Higgs sector,*and*related generalizations. ...*For*the*mean*-*field*Ising model, M 1 = tanh(x),*for*the*mean*-*field*XY model |M 2 | = I 1 (r) I 0 (r) ,*for*the*mean*-*field*Heisenberg model |M 3 | = coth(r) − 1 r ,*and**for*the*mean*-*field*Toy model of the ...##
###
On the Adam-Gibbs-Kirkpatrick-Thirumalai-Wolynes scenario for the viscosity increase in glasses

2004
*
Journal of Chemical Physics
*

We reformulate the interpretation of the

doi:10.1063/1.1796231
pmid:15473805
fatcat:x3nfpdtdmjghxdrtcqlhvqg3sq
*mean*-*field*glass*transition*scenario*for*finite dimensional systems, proposed by Wolynes*and*collaborators. ... This allows us to establish clearly a temperature dependent length xi* above which the*mean*-*field*glass*transition*picture has to be modified. ... Wyart*for*useful discussions. We also thank L. Berthier, J. Kurchan, M. Mézard*and*D. R. ...##
###
Power-law correlated phase in random-field XY models and randomly pinned charge-density waves

1997
*
Physical Review B (Condensed Matter)
*

The

doi:10.1103/physrevb.55.8211
fatcat:yhhqx64jx5a5vd7hkcsuh7cumy
*random**field*is chosen to have infinite strength on a fraction x of the sites of the lattice,*and*to be zero on the remaining sites.*For*x= 1/16 there are two*phase**transitions*. ... At the*transition*between the power-law*correlated**phase**and*the paramagnetic*phase*the magnetic susceptibility diverges,*and*the two-spin*correlations*decay approximately as |k|^(-2.87). ... Acknowledgment The author thanks David Huse*and*Michel Gingras*for*discussions of their work on*random*XY models, which encouraged the author to undertake the calculations described here. ...##
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Page 7417 of Mathematical Reviews Vol. , Issue 98K
[page]

1998
*
Mathematical Reviews
*

., a region where a

*transition*between different*mean*-*field*equilibrium values oc- curs. ... such as the Bethe lattice. g-state Potts models display a first-order*transition*in the*mean**field**for*g > 2, so the thin-graph Potts models provide a useful test case*for*exploring discontinuous*transitions*...##
###
Fragilities of liquids predicted from the random first order transition theory of glasses

2000
*
Proceedings of the National Academy of Sciences of the United States of America
*

A microscopically motivated theory of glassy dynamics based on an underlying

doi:10.1073/pnas.97.7.2990
pmid:10737779
pmcid:PMC16179
fatcat:r55gbs3hbnh3jkn6w25pd7xzea
*random*first order*transition*is developed to explain the magnitude of free energy barriers*for*glassy relaxation. ... The size of heterogeneous reconfiguring regions in a viscous liquid is inferred*and*the*correlation*of nonexponentiality of relaxation with fragility is qualitatively explained. ... We also thank John Portman*and*David Oxtoby*for*critically reading the manuscript. This work was supported by National Science Foundation Grant CHE-9530680. ...##
###
Perspectives in Statistical Mechanics
[article]

2006
*
arXiv
*
pre-print

In particular, we focus on rigorous non-perturbative results which provide insight on the spread of

arXiv:math-ph/0607025v1
fatcat:2vuadzqd5vdyfkgje5sco2ofja
*correlations*in*Gibbs*equilibrium states*and*yield information on*phase**transitions**and*critical phenomena ... Without attempting to summarize the vast*field*of statistical mechanics, we briefly mention some of the progress that was made in areas which have enjoyed Barry Simon's interests. ... Structures Emerging from Local Interactions •*Random*path /*random*cluster representations •*Random*surface models,*and*their*phase**transitions*The spin*correlations*which exist in the equilibrium states ...##
###
Premelting fluctuations
[article]

2020
*
arXiv
*
pre-print

A model of the premelting fluctuations is proposed, based on the Landau

arXiv:2003.09665v1
fatcat:pq7bivy54vhsrdnbrqltmorkzi
*mean**field*theory applied to a first-order*phase**transition*. ... Using the thermodynamic potential, the nonlinear Langevin equation*for*the order parameter is formulated, which describes the dynamics of the*phase**transition*considering microscopic thermal fluctuations ... Bityutskaya*for*interpretation of experimental data. Disclosure statement No potential conflict of interest was reported by the author. ...##
###
Page 1973 of Mathematical Reviews Vol. 57, Issue 5
[page]

1979
*
Mathematical Reviews
*

terms of cumulant averages of micro- scopic

*random*variables, is presented in the*mean**field*approxima- tion. ... as a*function*of the magnetic*field*h implies the existence of a unique translation-invariant*Gibbs*state,*and*if furthermore h =0 then all*Gibbs*states are invariant by rotation of the spins. (2) If ...##
###
Nonexistence of random gradient Gibbs measures in continuous interface models in d =2

2008
*
The Annals of Applied Probability
*

the increments of the

doi:10.1214/07-aap446
fatcat:5z5pr4twvvc4zpwpso532irzcq
*field*, as was shown by Funaki*and*Spohn. ... In d=3 where*random*gradient*Gibbs*measures are expected to exist, our method provides a lower bound of the order of the inverse of the distance on the decay of*correlations*of*Gibbs*expectations w.r.t ... Theorem 1.2 (Slow decay of*correlations*in d = 3) Suppose that d = 3*and*that µ [η](dϕ ) is a*random*gradient*Gibbs*measure. ...##
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Critical discontinuous phase transition in the threshold contact process

2011
*
Journal of Physics A: Mathematical and Theoretical
*

We show by

doi:10.1088/1751-8113/44/13/135002
fatcat:p6yeh75z75fldgalcwvsqyixvi
*means*of Monte Carlo simulations that the process undergoes a discontinuous*phase**transition*at a definite value of the annihilation parameter, in accordance with the*Gibbs**phase*rule,*and*that ... the discontinuous*transition*exhibits critical behavior. ... However, as usually happens with*mean*-*field*approaches, the discontinuous*transition*occurs along a certain range of the parameters, in disagreement with the*Gibbs**phase*rule. ...##
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Critical properties of a three-dimensional p-spin model

1999
*
European Physical Journal B : Condensed Matter Physics
*

We find evidence that in dimension three, contrary to its

doi:10.1007/s100510050707
fatcat:dj2rgiirpbe3ncon63ssntafdi
*mean**field*limit, the glass*transition*is associated to a diverging susceptibility (*and**correlation*length). ... Differently from*mean**field*there is a*correlation*length growing in the system, that suggests a second order*phase**transition*. ... INTRODUCTION Two different*transition*mechanisms are known in spin glass*mean*-*field*theory [1, 2] , according to the form of the*random*Hamiltonian. ...##
###
Page 777 of Mathematical Reviews Vol. , Issue 2000a
[page]

2000
*
Mathematical Reviews
*

As the volume A, grows to infinity, this

*Gibbs*measure can have very complicated behavior. The paper under review studies this behavior*for*a class of*mean*-*field*models. ... Let wa, (7) denote the finite-volume*Gibbs*measure of a disordered system, as a*function*of the*random*disorder parameter 7. ...
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