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Critical Behavior of the Widom-Rowlinson Lattice Model  [PDF]
Ronald Dickman,George Stell
Physics , 1995, DOI: 10.1063/1.468970
Abstract: We report extensive Monte Carlo simulations of the Widom-Rowlinson lattice model in two and three dimensions. Our results yield precise values for the critical activities and densities, and clearly place the critical behavior in the Ising universality class.
Phase Transitions in the Multicomponent Widom-Rowlinson Model and in Hard Cubes on the BCC--Lattice  [PDF]
P. Nielaba,J. L. Lebowitz
Physics , 1997, DOI: 10.1016/S0378-4371(97)00245-8
Abstract: We use Monte Carlo techniques and analytical methods to study the phase diagram of the M--component Widom-Rowlinson model on the bcc-lattice: there are M species all with the same fugacity z and a nearest neighbor hard core exclusion between unlike particles. Simulations show that for M greater or equal 3 there is a ``crystal phase'' for z lying between z_c(M) and z_d(M) while for z > z_d(M) there are M demixed phases each consisting mostly of one species. For M=2 there is a direct second order transition from the gas phase to the demixed phase while for M greater or equal 3 the transition at z_d(M) appears to be first order putting it in the Potts model universality class. For M large, Pirogov-Sinai theory gives z_d(M) ~ M-2+2/(3M^2) + ... . In the crystal phase the particles preferentially occupy one of the sublattices, independent of species, i.e. spatial symmetry but not particle symmetry is broken. For M to infinity this transition approaches that of the one component hard cube gas with fugacity y = zM. We find by direct simulations of such a system a transition at y_c ~ 0.71 which is consistent with the simulation z_c(M) for large M. This transition appears to be always of the Ising type.
Dynamical Widom-Rowlinson model and its mesoscopic limit  [PDF]
Dmitri Finkelshtein,Yuri Kondratiev,Oleksandr Kutoviy,Maria Joao Oliveira
Physics , 2014, DOI: 10.1007/s10955-014-1124-6
Abstract: We consider the non-equilibrium dynamics for the Widom-Rowlinson model (without hard-core) in the continuum. The Lebowitz-Penrose-type scaling of the dynamics is studied and the system of the corresponding kinetic equations is derived. In the space-homogeneous case, the equilibrium points of this system are described. Their structure corresponds to the dynamical phase transition in the model. The bifurcation of the system is shown.
Dominance of most tolerant species in multi-type lattice Widom-Rowlinson models  [PDF]
A. Mazel,Yu. Suhov,I. Stuhl,S. Zohren
Statistics , 2014, DOI: 10.1088/1742-5468/2014/8/P08010
Abstract: We analyse equilibrium phases in a multi-type lattice Widom-Rowlinson model with (i) four particle types, (ii) varying exclusion diameters between different particle types and (iii) large values of fugacity. Contrary to an expectation, it is not the most "aggressive" species, with largest diameters, which dominates the equilibrium measure, but the "most tolerant" one, which has smallest exclusion diameters. Results of numerical simulations are presented, showing densities of species in equilibrium phases and confirming the theoretical picture.
Structure and phase equilibria of the Widom-Rowlinson model  [PDF]
J. M. Brader,R. L. C. Vink
Physics , 2008, DOI: 10.1088/0953-8984/19/3/036101
Abstract: The Widom-Rowlinson model plays an important role in the statistical mechanics of second order phase transitions and yet there currently exists no theoretical approach capable of accurately predicting both the microscopic structure and phase equilibria. We address this issue using computer simulation, density functional theory and integral equation theory. A detailed study of the pair correlation functions obtained from computer simulation motivates a closure of the Ornstein-Zernike equations which gives a good description of the pair structure and locates the critical point to an accuracy of 2 percent.
Analysis of Two-Layered Random Interfaces for Two Dimensional Widom-Rowlinson's Model  [PDF]
Jun Wang,Jiguang Shao,Bingtuan Wang
Abstract and Applied Analysis , 2011, DOI: 10.1155/2011/858725
Abstract: The statistical behaviors of two-layered random-phase interfaces in two-dimensional Widom-Rowlinson's model are investigated. The phase interfaces separate two coexisting phases of the lattice Widom-Rowlinson model; when the chemical potential of the model is large enough, the convergence of the probability distributions which describe the fluctuations of the phase interfaces is studied. In this paper, the backbones of interfaces are introduced in the model, and the corresponding polymer chains and cluster expansions are developed and analyzed for the polymer weights. And the existence of the free energy for two-layered random-phase interfaces of the two-dimensional Widom-Rowlinson model is given. 1. Introduction We investigate the statistical behaviors of random interfaces between the two coexisting phases of the Widom-Rowlinson model (W-R model) when the chemical potential is large enough; especially we consider the two-layered interfaces behaviors of the model in this paper. The lattice system interfaces in two dimensions are known to fluctuate widely, for example, see [1–4] for the W-R model and [5–13] for the Ising spin system. There are two types of particles (either or ) in the lattice W-R system, and there is a strong repulsive interaction between particles of the different types. Namely, a particle cannot occupy a site within distance from a site where an particle has occupied and vice versa. This means that different types of particles are separated by the empty sites. In [2], under some special conditions for the interfaces (with specified values of the area enclosed below interfaces and the height difference of two endpoints) and the chemical potential large enough, it shows the weak convergence of the probability distributions (which describe the fluctuations of such interfaces) to certain conditional Gaussian distribution. According to the dynamic system of the W-R model and the results of [2], the thickness of the random interface (or the intermediate “belt”) between the two coexisting phases of the model is expected to become thinner as becomes larger, so we believe that the interfaces of the W-R model behave like those of the Ising model to some extent. We are also interested in the fluctuation behaviors of two or more random interfaces that one interface lies above the other one, such model presents the coexistence of three or more phases, which is the multilayer interacting interface model. In the present paper, the two-layered lattice W-R model is considered, and the convergence of the probability distributions which describe the
Coarse-graining diblock copolymer solutions: a macromolecular version of the Widom-Rowlinson model  [PDF]
C. I. Addison,J. P. Hansen,V. Krakoviack,A. A. Louis
Physics , 2005, DOI: 10.1080/00268970500186086
Abstract: We propose a systematic coarse-grained representation of block copolymers, whereby each block is reduced to a single ``soft blob'' and effective intra- as well as intermolecular interactions act between centres of mass of the blocks. The coarse-graining approach is applied to simple athermal lattice models of symmetric AB diblock copolymers, in particular to a Widom-Rowlinson-like model where blocks of the same species behave as ideal polymers (i.e. freely interpenetrate), while blocks of opposite species are mutually avoiding walks. This incompatibility drives microphase separation for copolymer solutions in the semi-dilute regime. An appropriate, consistent inversion procedure is used to extract effective inter- and intramolecular potentials from Monte Carlo results for the pair distribution functions of the block centres of mass in the infinite dilution limit.
Direct correlation functions of the Widom-Rowlinson model  [PDF]
R. Fantoni,G. Pastore
Physics , 2003, DOI: 10.1016/j.physa.2003.10.012
Abstract: We calculate, through Monte Carlo numerical simulations, the partial total and direct correlation functions of the three dimensional symmetric Widom-Rowlinson mixture. We find that the differences between the partial direct correlation functions from simulation and from the Percus-Yevick approximation (calculated analytically by Ahn and Lebowitz) are well fitted by Gaussians. We provide an analytical expression for the fit parameters as function of the density. We also present Monte Carlo simulation data for the direct correlation functions of a couple of non additive hard sphere systems to discuss the modification induced by finite like diameters.
Entropy-driven phase transitions in multitype lattice gas models  [PDF]
H. -O. Georgii,V. Zagrebnov
Physics , 2000,
Abstract: In multitype lattice gas models with hard-core interaction of Widom--Rowlinson type, there is a competition between the entropy due to the large number of types, and the positional energy and geometry resulting from the exclusion rule and the activity of particles. We investigate this phenomenon in four different models on the square lattice: the multitype Widom-Rowlinson model with diamond-shaped resp. square-shaped exclusion between unlike particles, a Widom-Rowlinson model with additional molecular exclusion, and a continuous-spin Widom-Rowlinson model. In each case we show that this competition leads to a first-order phase transition at some critical value of the activity, but the number and character of phases depend on the geometry of the model. Our technique is based on reflection positivity and the chessboard estimate.
Ising meets Ornstein and Zernike, Debye and Hückel, Widom and Rowlinson, and others
Dickman, Ronald;
Brazilian Journal of Physics , 2000, DOI: 10.1590/S0103-97332000000400014
Abstract: the name ising has come to stand not only for a specific model, but for an entire universality class - arguably the most important such class - in the theory of critical phenomena. i review several examples, both in and out of equilibrium, in which ising universality appears or is pertinent. the "ornstein-zernike" connection concerns a thermodynamically self-consistent closure of the eponymous relation, which lies at the basis of the modern theory of liquids, as applied to the ising lattice gas. debye and hückel founded the statistical mechanics of ionic solutions, which, despite the long-range nature of the interaction, now appear to exhibit ising-like criticality. the model of widom and rowlinson involves only excluded-volume interactions between unlike species, but again belongs to the ising universality class. far-from-equilibrium models of voting behavior, catalysis, and hysteresis provide further examples of this ubiquitous universality class.
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