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Book review: Gibbs measures and phase transitions

Aernout van Enter, Frank den Hollander
1991 Journal of statistical physics  
Gaussian 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.  ... 
doi:10.1007/bf01057888 fatcat:jus24bflwrg7jheddjjcjscisu

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]

Kay Kirkpatrick, Tayyab Nawaz
2016 arXiv   pre-print
We present a number of results coming from the theory of large deviations 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  ... 
arXiv:1611.02356v1 fatcat:bu6eukfowvaelbpqedv7pag5ie

On the Adam-Gibbs-Kirkpatrick-Thirumalai-Wolynes scenario for the viscosity increase in glasses

Jean-Philippe Bouchaud, Giulio Biroli
2004 Journal of Chemical Physics  
We reformulate the interpretation of the 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.  ... 
doi:10.1063/1.1796231 pmid:15473805 fatcat:x3nfpdtdmjghxdrtcqlhvqg3sq

Power-law correlated phase in random-field XY models and randomly pinned charge-density waves

Ronald Fisch
1997 Physical Review B (Condensed Matter)  
The 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.  ... 
doi:10.1103/physrevb.55.8211 fatcat:yhhqx64jx5a5vd7hkcsuh7cumy

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

X. Xia, P. G. Wolynes
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 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.  ... 
doi:10.1073/pnas.97.7.2990 pmid:10737779 pmcid:PMC16179 fatcat:r55gbs3hbnh3jkn6w25pd7xzea

Perspectives in Statistical Mechanics [article]

Michael Aizenman
2006 arXiv   pre-print
In particular, we focus on rigorous non-perturbative results which provide insight on the spread of 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  ... 
arXiv:math-ph/0607025v1 fatcat:2vuadzqd5vdyfkgje5sco2ofja

Premelting fluctuations [article]

Pavel Golovinski
2020 arXiv   pre-print
A model of the premelting fluctuations is proposed, based on the Landau 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.  ... 
arXiv:2003.09665v1 fatcat:pq7bivy54vhsrdnbrqltmorkzi

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

Aernout C. D. van Enter, Christof Külske
2008 The Annals of Applied Probability  
the increments of the 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.  ... 
doi:10.1214/07-aap446 fatcat:5z5pr4twvvc4zpwpso532irzcq

Critical discontinuous phase transition in the threshold contact process

Evandro F da Silva, Mário J de Oliveira
2011 Journal of Physics A: Mathematical and Theoretical  
We show by 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.  ... 
doi:10.1088/1751-8113/44/13/135002 fatcat:p6yeh75z75fldgalcwvsqyixvi

Critical properties of a three-dimensional p-spin model

S. Franz, G. Parisi
1999 European Physical Journal B : Condensed Matter Physics  
We find evidence that in dimension three, contrary to its 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.  ... 
doi:10.1007/s100510050707 fatcat:dj2rgiirpbe3ncon63ssntafdi

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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