Phase transitions in lattice gas models for water

Water-like lattice gases on the triangular and body-centered cubic lattices are investigated. Molecules may reside on the lattice sites in either of two possible orientations, a hydrogen bond being formed between molecules on neighboring sites if they have the proper orientation with respect to one another. For a range of chemical potential at sufficiently low temperatures, the models are shown to have an ordered phase consisting of an open, hydrogen-bonded, icelike structure. The models are shown to be transitionfree at sufficiently high temperature, indicating the existence of a critical point.Research supported in part by the U.S. National Science Foundation, Grant CHE-7726177, and by The Robert A. Welch Foundation, Grant P-446.     

First-order phase transitions in large entropy lattice models

We show the existence of a first-order phase transition in thev-dimensional Potts model forv≧2, when the number of states of a single spin is big enough. Low-temperature pure phases are proved to survive up to the critical temperature. Also the existence of a first-order transition in thev-dimensional Potts gauge model,v≧3, is obtained if the underlying gauge group is finite but large.     

Space-time phase transitions in driven kinetically constrained lattice models

Kinetically constrained models (KCMs) have been widely used to study and understand the origin of glassy dynamics. These models show an ergodic-nonergodic first-order phase transition between phases of distinct dynamical “activity”. We introduce driven variants of two popular KCMs, the FA model and the (2)-TLG, as models for driven supercooled liquids. By classifying trajectories through their entropy production we prove that driven KCMs display an analogous first-order space-time transition between dynamical phases of finite and vanishing entropy production. We discuss how trajectories with rare values of entropy production can be realized as typical trajectories of a mapped system with modified forces.     

Continuous percolation phase transitions of two-dimensional lattice networks under a generalized Achlioptas process

The percolation phase transitions of two-dimensional lattice networks under a generalized Achlioptas process (GAP) are investigated. During the GAP, two edges are chosen randomly from the lattice and the edge with minimum product of the two connecting cluster sizes is taken as the next occupied bond with a probability p. At p = 0.5, the GAP becomes the random growth model and leads to the minority product rule at p = 1. Using the finite-size scaling analysis, we find that the percolation phase transitions of these systems with 0.5 ≤ p ≤ 1 are always continuous and their critical exponents depend on p. Therefore, the universality class of the critical phenomena in two-dimensional lattice networks under the GAP is related to the probability parameter p in addition.     

Field-induced percolation in a polarized lattice gas

We introduce a lattice gas model with particles carrying a charge either +1 or –1 and drifting in opposite directions due to the presence of an external field. Our numerical simulations show the formation of polarized clusters elongated along the direction of the field. At low enough temperatures the clusters percolate through the system in a similar way as in the strip phase of the driven lattice gas model. A possible application of the model can be found in microemulsions.     

Phase transitions in the two-dimensional ferro- and antiferromagnetic potts models on a triangular lattice

The phase transitions in the two-dimensional ferro- and antiferromagnetic Potts models with q = 3 states of spin on a triangular lattice are studied using cluster algorithms and the classical Monte Carlo method. Systems with linear sizes L = 20–120 are considered. The method of fourth-order Binder cumulants and histogram analysis are used to discover that a second-order phase transition occurs in the ferromagnetic Potts model and a first-order phase transition takes place in the antiferromagnetic Potts model. The static critical indices of heat capacity (α), magnetic susceptibility (γ), magnetization (β), and correlation radius index (ν) are calculated for the ferromagnetic Potts model using the finite-size scaling theory.     

Infinite clusters in percolation models

The qualitative nature of infinite clusters in percolation models is investigated. The results, which apply to both independent and correlated percolation in any dimension, concern the number and density of infinite clusters, the size of their external surface, the value of their (total) surface-to-volume ratio, and the fluctuations in their density. In particular it is shown thatN ₀, the number of distinct infinite clusters, is either 0, 1, or and the caseN ₀= (which might occur in sufficiently high dimension) is analyzed.Alfred P. Sloan Research Fellow, Research supported in part by National Science Foundation grant No. MCS 77-20683 and by the U.S.-Israel Binational Science Foundation.Research supported in part by the U.S.Israel Binational Science Foundation.     

Bond percolation threshold in the simple cubic lattice

In the simple cubic lattice, for bond-percolation our Monte Carlo simulation gives a value of p_∞ = 0.2492 ± 0.0002, in contradiction to a recent speculation of Sahimi et al. (p_∞ = 0.25273) and to the possibility $\text{p}{}_{\mathrm{\infty }}\text{=}\text{1}\text{4}$.     

Dimer percolation and jamming on simple cubic lattice

We consider site percolation of dimers (“needles”) on simple cubic lattice. The percolation threshold is estimated as p_c ^perc ≈ 0.2555 ± 0.0001. The jamming threshold is estimated as p_c ^jamm = 0.799 ± 0.002.     

Spiral model, jamming percolation and glass-jamming transitions

The Spiral Model (SM) corresponds to a new class of kinetically constrained models introduced in joint works with Fisher [9,10] which provide the first example of finite dimensional models with an ideal glass-jamming transition. This is due to an underlying jamming percolation transition which has unconventional features: it is discontinuous (i.e. the percolating cluster is compact at the transition) and the typical size of the clusters diverges faster than any power law, leading to a Vogel-Fulcher-like divergence of the relaxation time. Here we present a detailed physical analysis of SM, see [6] for rigorous proofs.     

Scaling assumption for lattice animals in percolation theory

A scaling assumption for the numberg _ns of different cluster configurations with perimeters and sizen leads to the desired cluster numbers near the percolation threshold. The perimeter distribution function has a mean square width proportional ton for largen. The relation between the average perimeter and the cluster sizen for percolation has three different forms atp _c, belowp _c, and abovep _c and is closely related to the shape of the cluster size distribution.     

One-dimensional site percolation on a finite lattice

The behaviour of the numbers n _s of clusters withs sites each in the case of a chain ofN sites is studied for free and cyclic boundary conditions. Explicit expressions for the n _s, which differ from (1–p)² p ^s=q ² p ^s for the infinite lattice, are given. Also the total numberG(p) of clusters and the mean cluster sizeS(p) are calculated. In the thermodynamic limit the correction terms are found to be of order 1/N. An investigation of the scaling behaviour shows that the scaling of n _s is described by two independent variables in contrast toG(p) andS(p) which require only one variable.     

Thermal phase transitions at the percolation threshold

We present a simple argument, valid in all dimensions for a wide range of quenched dilute spin models, for the equality of the pair correlation function and the pair connectedness function at the point p = p_c, T = 0 is approached along the T axis.     

Jamming transition and new percolation universality classes in particulate systems with attraction

We numerically study the jamming transition in particulate systems with attraction by investigating their mechanical response at zero temperature (T=0). We find three regimes of mechanical behavior separated by two critical transitions--connectivity and rigidity percolation. The transitions belong to different universality classes than their lattice counterparts, due to force balance constraints. We also find that these transitions are unchanged at low temperatures and resemble gelation transitions in experiments on colloidal and silica gels.