Rowlinson Model Research Articles

Translation invariance of two-dimensional Gibbsian systems of particles with internal degrees of freedom

One of the main objectives of equilibrium state statistical physics is to analyze which symmetries of an interacting particle system in equilibrium are broken or conserved. Here we present a general result on the conservation of translational symmetry for two-dimensional Gibbsian particle systems. The result applies to particles with internal degrees of freedom and fairly arbitrary interaction, including the interesting cases of discontinuous, singular, and hard core interaction. In particular we thus show the conservation of translational symmetry for the continuum Widom–Rowlinson model and a class of continuum Potts type models.

ONE-DIMENSIONAL NON-SYMMETRIC WIDOM–ROWLINSON MODEL WITH LONG-RANGE INTERACTION

The absence of phase transitions in one-dimensional Widom–Rowlinson model with long-range interaction is established in the non-symmetric case when different particles have different activity parameters.

Structure and phase equilibria of the Widom–Rowlinson model

The Widom–Rowlinson model plays an important role in the statistical mechanics ofsecond-order phase transitions, and yet there currently exists no theoretical approachcapable of accurately predicting both the microscopic structure and phase equilibria. Weaddress this issue using computer simulation, density functional theory and integralequation theory. A detailed study of the pair correlation functions obtained from computersimulation motivates a closure of the Ornstein–Zernike equations which gives a gooddescription of the pair structure, and locates the critical point to an accuracy of2%.

The statistical properties of the interfaces for the lattice Widom–Rowlinson model

We investigate the statistical properties of the fluctuations of the phase interfaces that separate two phases of the two-dimensional lattice Widom–Rowlinson model. When the chemical potential μ of the W–R model is large enough, we discuss the probability distributions which describe the fluctuations of the phase interfaces, and show the corresponding central limit theory for the two-dimensional lattice W–R model.

Coarse-graining diblock copolymer solutions: a macromolecular version of the Widom–Rowlinson model

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.

SYSTEMS OF CLASSICAL PARTICLES IN THE GRAND CANONICAL ENSEMBLE, SCALING LIMITS AND QUANTUM FIELD THEORY

Euclidean quantum fields obtained as solutions of stochastic partial pseudo differential equations driven by a Poisson white noise have paths given by locally integrable functions. This makes it possible to define a class of ultra-violet finite local interactions for these models (in any space-time dimension). The corresponding interacting Euclidean quantum fields can be identified with systems of classical "charged" particles in the grand canonical ensemble with an interaction given by a nonlinear energy density of the "static field" generated by the particles' charges via a "generalized Poisson equation". A new definition of some well-known systems of statistical mechanics is given by formulating the related field theoretic local interactions. The infinite volume limit of such systems is discussed for models with trigonometric interactions using a representation of such models as Widom–Rowlinson models associated with (formal) Potts models at imaginary temperature. The infinite volume correlation functional of such Potts models can be constructed by a cluster expansion. This leads to the construction of extremal Gibbs measures with trigonometric interactions in the low-density high-temperature (LD-HT) regime. For Poissonian models with certain trigonometric interactions an extension of the well-known relation between the (massive) sine-Gordon model and the Yukawa particle gas connecting characteristic and correlation functionals is given and used to derive infinite volume measures for interacting Poisson quantum field models through an alternative route. The continuum limit of the particle systems under consideration is also investigated and the formal analogy with the scaling limit of renormalization group theory is pointed out. In some simple cases the question of (non-) triviality of the continuum limits is clarified.

The Complexity of Choosing an H-Coloring (Nearly) Uniformly at Random

Cooper, Dyer, and Frieze [J. Algorithms, 39 (2001), pp. 117--134] studied the problem of sampling H-colorings (nearly) uniformly at random. Special cases of this problem include sampling colorings and independent sets and sampling from statistical physics models such as the Widom--Rowlinson model, the Beach model, the Potts model and the hard-core lattice gas model. Cooper et al. considered the family of cautious ergodic Markov chains with uniform stationary distribution and showed that, for every fixed connected nontrivial graph H, every such chain mixes slowly. In this paper, we give a complexity result for the problem. Namely, we show that for any fixed graph H with no trivial components, there is unlikely to be any polynomial almost uniform sampler (PAUS) for H-colorings. We show that if there were a PAUS for the H-coloring problem, there would also be a PAUS for sampling independent sets in bipartite graphs, and, by the self-reducibility of the latter problem, there would be a fully polynomial randomized approximation scheme (FPRAS) for #BIS---the problem of counting independent sets in bipartite graphs. Dyer, Goldberg, Greenhill, and Jerrum have shown that #BIS is complete in a certain logically defined complexity class. Thus, a PAUS for sampling H-colorings would give an FPRAS for the entire complexity class. In order to achieve our result we introduce the new notion of sampling-preserving reduction which seems to be more useful in certain settings than approximation-preserving reduction.

Direct correlation functions of the Widom–Rowlinson model

We calculate, through Monte Carlo (MC) 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 MC 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.

On Markov Chains for Randomly H-Coloring a Graph

Let H=(W,F) be a graph without multiple edges, but with the possibility of having loops. Let G=(V,E) be a simple graph. A homomorphism c is a map c:V→W with the property that (v,w)∈E implies that (c(v),c(w))∈F. We will often refer to c(v) as the color of v and c as an H-coloring of G. We consider the problem of choosing a random H-coloring of G by Markov chain Monte Carlo. The probabilistic model we consider includes random proper colorings, random independent sets, and the Widom–Rowlinson and Beach models of statistical physics. We prove negative results for uniform sampling and a positive result for weighted sampling when H is a tree.

Entropy-Driven Phase Transitions in Multitype Lattice Gas Models

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. We also analyze the typical geometry of phases, combining percolation techniques with reflection positivity and chessboard estimates.

The complexity of counting graph homomorphisms

The problem of counting homomorphisms from a general graph G to a fixed graph H is a natural generalization of graph coloring, with important applications in statistical physics. The problem of deciding whether any homomorphism exists was considered by Hell and Nesetřil. They showed that decision is NP-complete unless H has a loop or is bipartite; otherwise it is in P. We consider the problem of exactly counting such homomorphisms and give a similarly complete characterization. We show that counting is #P-complete unless every connected component of H is an isolated vertex without a loop, a complete graph with all loops present, or a complete unlooped bipartite graph; otherwise it is in P. We prove further that this remains true when G has bounded degree. In particular, our theorems provide the first proof of #P-completeness of the partition function of certain models from statistical physics, such as the Widom–Rowlinson model, even in graphs of maximum degree 3. Our results are proved using a mixture of spectral analysis, interpolation, and combinatorial arguments. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 260–289, 2000

Graphical representations and cluster algorithms II

We continue the study, initiated in Part I, of graphical representations and cluster algorithms for various models in (or related to) statistical mechanics. For certain models, e.g. the Blume–Emery–Griffths model and various generalizations, we develop Fortuin Kasteleyn-type representations which lead immediately to Swendsen Wang-type algorithms. For other models, e.g. the random cluster model, that are defined by a graphical representation, we develop cluster algorithms without reference to an underlying spin system. In all cases, phase transitions are related to percolation (or incipient percolation) in the graphical representation which, via the IC algorithm, allows for the rapid simulation of these systems at the transition point. Pertinent examples include the (continuum) Widom–Rowlinson model, the restricted 1-step solid-on-solid model and the XY model.

Random-Cluster Analysis of a Class of Binary Lattice Gases

We introduce a class of binary lattice gases which can be viewed as a lattice analogue of the continuum Widom–Rowlinson model, and which also is related to the beach model of Burton and Steif. This new model is shown to exhibit phase transition for large particle intensities. Stochastic monotonicity results of varying strength are derived in various parts of the parameter space. The main tool is a random-cluster representation of the model, analogous to the Fortuin–Kasteleyn representation of the Potts model.

Ordering in many component Widom–Rowlinson models

We examine the changing immiscibility of Widom–Rowlinson-type model mixtures as the number of components is varied. In these mixtures, molecules interact with molecules of a different component via a hard-core interaction but do not interact with other molecules of the same component at all. The interaction between any two molecules of different components is the same, it is that between two parallel hard (hyper-)cubes. Theoretical calculations show that an equimolar mixture of less than 31 components demixes in the fluid phase but if there are 31 or more components demixing is preempted by solidification. These calculations are exact in the limit of infinite dimensionality. The switchover from demixing to solidification as the number of components increases is due to the high entropy cost incurred by a mixture of many components phase separating into a large number of pure phases. A Widom–Rowlinson mixture is shown to be (nearly) equivalent to the Zwanzig model of a liquid crystal at a dimensionality equal to the number of components in the mixture. Using the results obtained for the mixtures we speculate that there is an upper bound to the dimensionality at which liquid crystallinity can exist.

Thermodynamics of the Widom—Rowlinson Model. Computer Calculation Results Compared with Mean Field and Percus—Yevick Values

Abstract Recent molecular dynamics (MD) calculations showed that the mean field approximation is to be preferred over the Percus-Yevick one for the prediction of the coexistence curve of the Widom-Rowlinson (WR) mixture model. Further MD and Monte-Carlo calculations results are presented here for the pressure and the pair correlation function of the WR mixture in a large region of states. The mean field approximations is found to give reasonable agreement with the computer results in a wide density region, while the Percus-Yevick approximation generates nearly exact results at low densities but overestimates the pressure for denser states. This might explain why the Percus-Yevick theory predicts the demixing density too high compared with mean field and exact computer calculation results.

Static and dynamic properties of the Widom–Rowlinson model mixture. I

The simple and interesting Widom–Rowlinson (WR) mixture model is investigated by molecular-dynamics calculations in comparison with equilibrium and kinetic theories. In the present paper only static properties of the WR model are considered. Particularly, the phase separation of the mixture is studied. The exact molecular-dynamics results are found to be in reasonable agreement with predictions of Percus–Yevick and mean-field approximations.

Selective particle clustering and percolation in binary mixtures of randomly centered spheres

The clustering and percolation of particles in binary mixtures of randomly centered spheres are examined based on a selective particle connectivity criterion in which only particles of different species are allowed to form directly connected bonds. This problem is different from the usually studied ‘‘simple’’ percolation problem in which pairs of particles form directly connected bonds as long as they are separated by a distance σ or less. The percolation threshold and pair-connectedness function of the binary mixture are determined based on the connectivity Ornstein–Zernike integral equation in the Percus–Yevick (PY) approximation. It is shown that, within the PY closure, the present system can be mapped into the Widom–Rowlinson model in the theory of liquid state. The percolation thresholds and the pair–connectedness functions of the particles are numerically computed for a wide range of particle densities and number fractions. It is found that their percolation densities differ considerably from those found in the simple percolation problem for a binary mixture of randomly centered spheres. To our knowledge, this is the first study of selective particle clustering and percolation in multicomponent mixtures of particles.

The structure of Gibbs states and phase coexistence for non-symmetric continuum Widom Rowlinson models

We prove the existence of phase transitions in non-symmetric r-component continuum Widom-Rowlinson models. Our results are based on an extension of the Pirogov-Sinai theory of phase transitions in general lattice spin systems to continuum systems. This generalizes Ruelle's extension of the Peierls argument for lattices to symmetric continuum Widom-Rowlinson models. The Pirogov-Sinai picture of the low temperature phase diagram for spin systems goes over into a phase-diagram of the Widom-Rowlinson model at large fugacities z=(z0,..., z r−1). There is in z-space a point where the system has r-pure phases, lines with r−1 phases, two dimensional surfaces with r−2 phases, etc.

Electric polarisation of water: Monte Carlo studies

Monte Carlo calculations for liquid water in strong static electric fields are used to study dielectric, thermodynamic and structural properties. Emphasis is placed on the importance of long range interactions, particularly effects due to spherical truncation of the pair potential and to the Onsager reaction field. The Rowlinson model for pair interactions is not capable of predicting the high static dielectric constant of water and similar results are to be expected for other point charge models. It is shown that dielectric constants are very sensitive to the Onsager reaction field, whereas thermodynamic properties are somewhat insensitive. Structural effects due to dielectric saturation are demonstrated and the influence of long range interactions on such properties discussed.

Remarks on the global Markov property

The global Markov property is established for the + state and the − state of attractive lattice systems (e.g., the ferromagnetic Ising model and most other systems for which the FKG inequalities are satisfied) and of the (continuum) Widom Rowlinson model.