Applications of the Yang-Lee-Ruelle theory to hard-core lattice gases

The grand-partition-function-zero method is applied to lattice systems of rigid molecules, based on the algebraic technique of Ruelle. Consideration of small collections of lattice molecules, through this approach, provides rigorous delineation of regions of the complex activity plane which are free of zeros of the grand partition function, and hence free of thermodynamic singularities. Two conjectures, as yet unproved, are offered, which greatly reduce the computational effort required in using the technique. A simple proof is provided for the absence of physical phase transitions in monomerdimer systems, and bounds are obtained on the locations of the transitions of other lattice gases.Research supported in part by National Science Foundation Grant GP-17026.     

A remark on the condensation in the hard-core lattice Bose gas

We point out that Bose-Einstein condensation occurs at sufficiently low temperature in a hard-core ^d-lattice Bose gas for d3 and particle density 1/2, by exploiting its equivalence to a spin-1/2XY model.     

Percolation in hard-core lattice gases and a model ferrofluid

A lattice gas on ³ consisting of hard spheres with exclusions extending through third neighbors is proved to undergo a percolation transition. If spins with ferromagnetic couplings are attached to the spheres, spontaneous magnetization is proved to occur. This may provide a model for a ferrofluid, a system which exhibits spontaneous magnetization without crystalline order. Similar results are also obtained for an analogous model on ².     

Phase transition of hard hexagons on a triangular lattice

Systems of hard hexagons on a triangular lattice are investigated. The orientation of the hexagons is kept fixed, while the size of the hexagons is varied. The existence of a phase transition is proved for all sizes by means of the Peierls'argument. The proof does not imply a phase transition in the continuous limit.Work supported by the U. S. Air Force under Contract No. F 44620-71-C-0013.     

Circle theorem for hard-core binary lattice gases

We study the distribution of zeros in a symmetric, two-component Widom-Rowlinson lattice system (any number of dimensions). We show that for sufficiently large mean activity the system partition function cannot vanish if the magnitude of the ratio of the two (complex) activities is different from one.Supported in part by National Science Foundation Grants CHE76-11253 (to LKR) and Phy 78-15920 (to JLL), and by the Petroleum Research Foundation.     

Phase transitions in lattice gases of hard-core molecules having two orientations

A number of hard-core lattice gases in which a lattice site can be occupied by a molecule in either of two possible orientations are proved to undergo order-disorder phase transitions. Examples include lattice gases of trigonal planar molecules on a triangular lattice, tetrahedral molecules on a bodycentered cubic lattice, and linear molecules on a square lattice.This research was supported by The Robert A. Welch Foundation Grant P-446 and by the TCU Research Foundation.     

Nonlinear dynamics of a two-dimensional lattice

The dynamics of the nonlinear excitations in a two-dimensional (2D) φ⁴-diatomic lattice, with nonlinear on-site electron-phonon coupling at the polarizable ion site has been presented, without considering the self consistent phonon approximation. One of the major results obtained from our calculations is in the understanding of continuous structural phase transition, where we have obtained the minimum in soft mode frequency at a soft mode temperatureT _s (>T _c), not at critical temperatureT _c. This occurs due to the anisotropy of such 2D systems.     

Phase transition for a one-dimensional lattice gas with hard core

Existence of a phase transition is proved for a one-dimensional lattice gas with long-range interaction and nearest neighbor exclusion.     

Renormalized equilibria of a Schlögl model lattice gas

A lattice gas model for Schlögl's second chemical reaction is described and analyzed. Because the lattice gas does not obey a semi-detailed-balance condition, the equilibria are non-Gibbsian. In spite of this, a self-consistent set of equations for the exact homogeneous equilibria are described, using a generalized cluster-expansion scheme. These equations are solved in the two-particle BBGKY approximation, and the results are compared to numerical experiment. It is found that this approximation describes the equilibria far more accurately than the Boltzmann approximation. It is also found, however, that it can give rise to spurious solutions to the equilibrium equations.     

Influence of a uniform driving force on tracer diffusion in a one-dimensional hard-core lattice gas

The influence of a uniform driving force on tracer diffusion is investigated for a one-dimensional lattice gas where particles jump stochastically to unoccupied neighboring sites. A new, simple calculation is presented for the diffusion coefficient of a tracer particle with respect to its average drift, obtained recently by rigorous methods by De Masi and Ferrari. A theoretical expression describing the tracer particle mean square displacement approximately for all times is derived and found to be in excellent agreement with the results of Monte Carlo simulations.     

Phase diagram of the one-state potts model on the Cayley tree

The phase diagram of the one-state Potts model on the closed asymmetric Cayley tree with branching ratior=2 is obtained from the Bethe-Peierls map. The route to chaos, via the period doubling cascade, is obtained by considering the antiferromagnetic coupling limit. The connection of the Potts model with the percolation problem is shown by calculating the order parameter, its susceptibility, the internal energy, and the specific heat as well as their asymptotic behavior at the paramagnetic-ferromagnetic critical point. Due to the type of the lattice and to the polynomial character of the map, this is the simplest known example of a McKay-Berker-Kirkpatrick spin-glass.     

Bifurcations of a lattice gas flow under external forcing

We study the behavior of a Frisch-Hasslacher-Pomeau lattice gas automaton under the effect of a spatially periodic forcing. It is shown that the lattice gas dynamics reproduces the steady-state features of the bifurcation pattern predicted by a properly truncated model of the Navier-Stokes equations. In addition, we show that the dynamical evolution of the instabilities driving the bifurcation can be modeled by supplementing the truncated Navier-Stokes equation with a random force chosen on the basis of the automaton noise.     

Magnetohydrodynamics computations with lattice gas automata

Lattice gas automata have received considerable interest for the last several years and possibly may become a powerful numerical method for solving various partial differential equations and modeling different physical phenomena, because of their discrete and parallel nature and the capability of handling complicated boundaries. In this paper, we present recent studies on the lattice gas model for magnetohydrodynamics. The FHP-type lattice gas model has been extended to include a bidirectional random walk process, which allows well-defined statistical quantities, such as velocity and magnetic field, to be computed from the microscopic particle representation. The model incorporates a new sequential particle collision method to increase the range of useful Reynolds numbers in the model, an improvement that may also be of use in other lattice gas models. In the context of a Chapman-Enskog expansion, the model approximates the incompressible magnetic hydrodynamic equations in the limit of low Mach number and high. Simulation results presented here demonstrate the validity of the model for several basic problems, including sound wave and Alfvén wave propagation, and diffusive Kolmogoroff-type flows.     

Exact analysis of a lattice gas on the 3–12 lattice with two- and three-site interactions

Exact results are obtained for a lattice gas on the 3–12 lattice with two- and three-site interactions. By using a decoration transformation, we map the lattice gas into one on the honeycomb lattice with pure two-site interactions. This procedure permits us to draw exact results for the original 3–12 lattice gas. In particular, we obtain its exact two-phase boundary, and confirm the fact that an experimentally observed anomaly in the critical behavior of the coexistence-curve diameter is present if, and only if, the three-site interactions are present.     

Solvable models of classical lattice gases

We study classical lattice gases at fixed temperature but variable fugacity. It is shown how the thermodynamic functions may be calculated exactly provided the Boltzmann weights are representable as principal minors of a convolution operator. We explicitly construct this operator for the cluster models of Fisher and Felderhof.On leave from the Institute for Theoritical Physics, RWTH Aachen, Germany.     

Entropy of a one-dimensional mixed lattice gas

This paper deals with the grand canonical entropy of a lattice gas mixture. The entropy is a function of the multisite densities corresponding to the interaction pattern of the system in question. It is first evaluated for a nearest-neighborinteraction, one-dimensional simple lattice gas to show how the structure of bulk fluid is locally maintained. Generalization requires one set of interrelations among multisite densities presented in closed form for an arbitrary lattice, and one set between Boltzmann factors and multisite densities which is written down for simply connected lattices. Application is made to two-row lattices, which turn out to have local behavior from this viewpoint, as do all single-row or Bethe lattices with complete range-p interactions. Nonlocal examples are also given, and suggestions made for approximation sequences in general lattices.     

Reynolds stresses in a lattice gas

I use a previously proposed algorithm, based on Lévy walks, to calculate and discuss longitudinal and transverse velocity correlations in turbulent channel flow. The general approach is that of lattice gas hydrodynamics.     

Inhomogeneous Glauber dynamics and the process of crystallization of a lattice gas

The model under consideration is a hard-core lattice gas in an external potential on a Bethe lattice with nonequilibrium time evolution governed by Glauber dynamics. A hierarchical decoupling of nonequilibrium correlations, motivated by and asymptotically providing the exact form of equilibrium multisite correlations in the inhomogeneous potential regime, is proposed. Application is made to the process of lattice gas crystallization, at high activity, from a spatially homogeneous fluid phase to an equilibrium crystal phase with unequal sublattice densities. The first few levels of the hierarchical decoupling give a consistent picture of two kinds of nonequilibrium instabilities—one leading to a sublattice density bifurcation, the other associated with an abrupt increase in densities and correlations in time.     

Analyticity of low-temperature phase diagrams of lattice spin models

The analyticity of all strata of the Pirogov-Sinai phase diagram is proved. As a byproduct of the method, a characterization of typical volumes for which the complex partition function vanishes is given, for a Hamiltonian that is a perturbation of the real-valued one, near the point of a phase transition.     

Lattice gas automata for simple and complex fluids

We review some recent applications of lattice gas automata, including flow through porous media, phase transitions, thermodydrodynamics, and magnetohydrodynamics.