Phase Transition of a Bethe Lattice Gas of Hard Molecules

Runnels, L. K.

doi:10.1063/1.1705123

Cited by 64 publications

(65 citation statements)

References 20 publications

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“…The complete r-ary rooted tree [94,98]. Let T (r) n be the complete rooted tree with branching factor r and depth n. We limit attention to the univariate independent-set polynomial.…”

Section: Examplesmentioning

confidence: 99%

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

2005

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We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {p x } if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p x }. Furthermore, we show that the usual proof of the Lovász local lemma -which provides a sufficient condition for this to occur -corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer [98] and explicitly by Dobrushin [37,38]. We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for "soft" dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

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“…The complete r-ary rooted tree [94,98]. Let T (r) n be the complete rooted tree with branching factor r and depth n. We limit attention to the univariate independent-set polynomial.…”

Section: Examplesmentioning

confidence: 99%

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

2005

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“…This system, which can also interpreted either as 45 o tilted hard-squares of linear size λ = √ 2 or as hard disks of radius √ 2/2, has been extensively studied and here we present some results for the sake of both completeness and comparison. Many different approaches have been used to describe its properties on a square lattice: series expansions [3,5,11,12,13], cluster variational and transfer matrix methods [5,14,15,16,17,18,19,20,21,22,23], renormalization group [24,25], Monte Carlo simulations [26,27,28,29,30,31,32,33], Bethe lattice [5,34,35,36,37,38], and more recently density functional theory [39]. Moreover, this model has also been considered because of its interesting mathematical [40,41,42] and dynamical [43,44,45,46,47,48,49,50,51,…”

Section: A Nearest Neighbor Exclusion (1nn)mentioning

confidence: 99%

Monte Carlo simulations of two-dimensional hard core lattice gases

Fernandes

Arenzon

Levin

2007

The Journal of Chemical Physics

134

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Monte Carlo simulations are used to study lattice gases of particles with extended hard cores on a two dimensional square lattice. Exclusions of one and up to five nearest neighbors (NN) are considered. These can be mapped onto hard squares of varying side length, λ (in lattice units), tilted by some angle with respect to the original lattice. In agreement with earlier studies, the 1NN exclusion undergoes a continuous order-disorder transition in the Ising universality class. Surprisingly, we find that the lattice gas with exclusions of up to second nearest neighbors (2NN) also undergoes a continuous phase transition in the Ising universality class, while the Landau-Lifshitz theory predicts that this transition should be in the universality class of the XY model with cubic anisotropy. The lattice gas of 3NN exclusions is found to undergo a discontinuous order-disorder transition, in agreement with the earlier transfer matrix calculations and the Landau-Lifshitz theory. On the other hand, the gas of 4NN exclusions once again exhibits a continuous phase transition in the Ising universality class -contradicting the predictions of the Landau-Lifshitz theory. Finally, the lattice gas of 5NN exclusions is found to undergo a discontinuous phase transition.

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“…The divergence of Ξ 1 signals an instability toward a cyclic solution of the iteration eqs. [16], which corresponds to the crystal as found for l = 0 by Runnels [18]. Here, we focus on the next instability related to the divergence of the "glass susceptibility" Ξ 2 .…”

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confidence: 99%

Lattice Glass Models

2001

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Motivated by the concept of geometrical frustration, we introduce a class of statistical mechanics lattice models for the glass transition. Monte Carlo simulations in three dimensions show that they display a dynamical glass transition which is very similar to that observed in other off-lattice systems and which does not depend on a specific dynamical rule. A mean-field study shows the existence of a discontinuous glass transition, in agreement with the numerical observations.Understanding the glass transition, and the glass phase, is one of the present major challenges in condensed matter physics. The experimental glass transition is related to a dramatic dynamical slowing down in which the structural relaxation time changes of 14 orders of magnitudes in a relatively small window of temperatures. As a matter of fact, the glass transition temperature T g is empirically defined as that where the structural relaxation time becomes of the order of an hour. In the last fifty years there have been many efforts to try to understand whether this phenomenon is just a dramatic crossover, it is related to a pure dynamical transition, or it is the signature of a true thermodynamic glass transition, called the "ideal glass transition", which would take place at temperature T K below T g but which is kinetically avoided [1]. The possibility of this last scenario in fragile glasses is supported by the closeness of the Vogel-Fulcher temperature (where the extrapolated relaxation time has a divergence) and of the Kauzmann temperature (where the extrapolated excess entropy of the supercooled liquid vanishes) [2], and has been widely explored theoretically [1,[3][4][5][6][7]. The analogy with mean field discontinuous spin glasses [6] has given recently a new boost to this line of research [7]. These models can be solved analytically and display striking similarities with the experimental fragile glasses. However, they predict (because of their mean field character) a dynamic freezing transition at a temperature T c > T g , equal to the mode coupling transition temperature, where real systems still have a finite relaxation time. This difference is generally attributed to the existence in finite dimensions of activated processes which would transform the 'mean field' dynamical transition into a crossover in such a way that the relaxation time increase very fast in the temperature region T K < T < T c and eventually diverges at T K .In this paper we introduce new lattice models of glasses. We study them numerically in three dimensions, and show (1) that they display a dynamic glass transition similar to the one seen in the simulation of glass formers like binary Lennard-Jones [8], (2) that their mean field solution, obtained using the Bethe approximation, predicts the same physical scenario found for mean field discontinuous spin glasses (in particular a dynamic freezing transition preceding the "ideal glass transition").Our lattice glass models are defined as follows. On each node of the lattice (e.g. a cubic lattice), there can be...

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Phase Transition of a Bethe Lattice Gas of Hard Molecules

Cited by 64 publications

References 20 publications

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

Monte Carlo simulations of two-dimensional hard core lattice gases

Lattice Glass Models

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