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.     

Single-file diffusion in a box

We study diffusion of (fluorescently) tagged hard-core interacting particles of finite size in a finite one-dimensional system. We find an exact analytical expression for the tagged particle probability density function using a Bethe ansatz, from which the mean square displacement is calculated. The analysis shows the existence of three regimes of drastically different behavior for short, intermediate, and large times. The results are in excellent agreement with stochastic simulations (Gillespie algorithm).     

Phase transitions and reflection positivity. II. Lattice systems with short-range and Coulomb interactions

We discuss applications of the abstract scheme of part I of this work, in particular of infrared bounds and chessboard estimates, to proving the existence of phase transitions in lattice systems. Included are antiferromagnets in an external field, hard-core exclusion models, classical and quantum Coulomb lattice gases, and six-vertex models.Research partially supported by Canadian National Research Council Grant A4015 and U.S. National Science Foundation Grants PHY-77-18762, MCS-75-21684-A02, and MCS-78-01885.     

Exotic gapless mott insulators of bosons on multileg ladders

We present evidence for an exotic gapless insulating phase of hard-core bosons on multileg ladders with a density commensurate with the number of legs. In particular, we study in detail a model of bosons moving with direct hopping and frustrating ring exchange on a 3-leg ladder at ν=1/3 filling. For sufficiently large ring exchange, the system is insulating along the ladder but has two gapless modes and power law transverse density correlations at incommensurate wave vectors. We propose a determinantal wave function for this phase and find excellent comparison between variational Monte Carlo and density matrix renormalization group calculations on the model Hamiltonian, thus providing strong evidence for the existence of this exotic phase. Finally, we discuss extensions of our results to other N-leg systems and to N-layer two-dimensional structures.     

On a generalization of the hidden symmetry transformations for the principal chiral model

Nonlocal hidden symmetry transformations with a generalized structure and boundary conditions at spatial infinity for the principal chiral model are proposed. Additional restrictions on these transformations following from the requirement for the existence of an infinite set of conserved nonlocal charges are analyzed. The corresponding Lie algebra is more general than the Kac-Moody one.     

Biased Diffusion with Correlated Noise

The diffusion of hard-core particles subject to a global bias is described by a nonlinear, anisotropic generalization of the diffusion equation with conserved, local noise. Using renormalization group techniques, we analyze the effect of an additional noise term, with spatially long-ranged correlations, on the long-time, long-wavelength behavior of this model. Above an upper critical dimension d _LR, the long-ranged noise is always relevant. In contrast, for d<d _LR, we find a weak noise regime dominated by short-range noise. As the range of the noise correlations increases, an intricate sequence of stability exchanges between different fixed points of the renormalization group occurs. Both smooth and discontinuous crossovers between the associated universality classes are observed, reflected in the scaling exponents. We discuss the necessary techniques in some detail since they are applicable to a much wider range of problems.     

A Cahn-Hilliard model in a domain with non-permeable walls

We consider in this article a Cahn-Hilliard model in a bounded domain with non-permeable walls, characterized by dynamic-type boundary conditions. Dynamic boundary conditions for the Cahn-Hilliard system have recently been proposed by physicists in order to account for the interactions with the walls in confined systems and are obtained by writing that the total bulk mass is conserved and that there is a relaxation dynamics on the boundary. However, in the case of non-permeable walls, one should also expect some mass on the boundary. It thus seems more realistic to assume that the total mass, in the bulk and on the boundary, is conserved, which leads to boundary conditions of a different type. For the resulting mathematical model, we prove the existence and uniqueness of weak solutions and study their asymptotic behavior as time goes to infinity.     

Entropy of classical systems with long-range interactions

We discuss the form of the entropy for classical Hamiltonian systems with long-range interaction using the Vlasov equation which describes the dynamics of a N particle in the limit N-->infinity. The stationary states of the Hamiltonian system are subject to infinite conserved quantities due to the Vlasov dynamics. We show that the stationary states correspond to an extremum of the Boltzmann-Gibbs entropy, and their stability is obtained from the condition that this extremum is a maximum. As a consequence, the entropy is a function of an infinite set of Lagrange multipliers that depend on the initial condition. We also discuss in this context the meaning of ensemble inequivalence and the temperature.     

Localization and critical diffusion of quantum dipoles in two dimensions

We discuss quantum propagation of dipole excitations in two dimensions. This problem differs from the conventional Anderson localization due to the existence of long-range hops. We find that the critical wave functions of the dipoles always exist which manifest themselves by a scale independent diffusion constant. If the system is T invariant the states are critical for all values of the parameters. Otherwise, there can be a "metal-insulator" transition between this "ordinary" diffusion and the Levy flights (the diffusion constant logarithmically increasing with the scale). These results follow from the two-loop analysis of the modified nonlinear supermatrix σ model.     

The influence of defects in the crystal structure on helium diffusion in quartz

The solubility and diffusion of helium in quartz crystals are investigated as functions of the distribution and density of structural defects. The types of defects in the crystals are identified and their distribution over growth sectors is determined by x-ray diffraction topography and phase radiography with a synchrotron radiation source. The effective solubility and effective diffusion coefficients for helium in quartz are estimated from the experimental data on the amount of helium extracted from samples with different contents of defects. It is revealed that the effective diffusion coefficient of helium depends on the number of dislocations.     

Many-body functions of nonprimitive electrolytes in one dimension

The statistical mechanics of a mixture of hard-core ions and dipoles in one dimension, namely, the one-dimensional version of the so-called nonprimitive model of an electrolyte, is considered by stressing the effect of the charge-dipole interactions and the hard-core repulsions on the thermodynamics and, especially, on the many-body functions of the systems. The adaptation of Baxter's generating function technique to this model lets us express the thermodynamic and structural functions in terms of a non-Hermitian generalized Hill-type Hamiltonian. The eigenvalues and eigenfunctions of this differential operator yield, in closed form, then-body correlation functions in the bulk and near the container's walls. We also comment on the screening of the electric fields by the system ions and study the Donnan equilibrium when one of the ionic species in the mixture cannot diffuse through a semipermeable membrane.     

On Maximal Hard-Core Thinnings of Stationary Particle Processes

The present paper studies existence and distributional uniqueness of subclasses of stationary hard-core particle systems arising as thinnings of stationary particle processes. These subclasses are defined by natural maximality criteria. We investigate two specific criteria, one related to the intensity of the hard-core particle process, the other one being a local optimality criterion on the level of realizations. In fact, the criteria are equivalent under suitable moment conditions. We show that stationary hard-core thinnings satisfying such criteria exist and are frequently distributionally unique. More precisely, distributional uniqueness holds in subcritical and barely supercritical regimes of continuum percolation. Additionally, based on the analysis of a specific example, we argue that fluctuations in grain sizes can play an important role for establishing distributional uniqueness at high intensities. Finally, we provide a family of algorithmically constructible approximations whose volume fractions are arbitrarily close to the maximum.     

Bulk diffusivity of lattice gases close to criticality

We consider lattice gases where particles jump at random times constrained by hard-core exclusion (simple exclusion process with speed change). The conventional theory of critical slowing down predicts that close to a critical point the bulk diffusivity vanishes as the inverse compressibility. We confirm this claim by proving a strictly positive lower bound for the conductivity.     

Asymmetric exclusion processes with site sharing in a one-channel transport system

This Letter investigates two-species totally asymmetric simple exclusion process (TASEP) with site sharing in a one-channel transport system. In the model, different species of particles may share the same sites, while particles of the same species may not (hard-core exclusion). The site-sharing mechanism is applied to the bulk as well as the boundaries. Such sharing mechanism within the framework of the TASEP has been largely ignored so far. The steady-state phase diagrams, currents and bulk densities are obtained using a mean-field approximation and computer simulations. The presence of three stationary phases (low-density, high-density, and maximal current) are identified. A comparison on the stationary current with the Bridge model [M.R. Evans, et al., Phys. Rev. Lett. 74 (1995) 208] has shown that our model can enhance the current. The theoretical calculations are well supported by Monte Carlo simulations.     

Entropy of isolated quantum systems after a quench

A diagonal entropy, which depends only on the diagonal elements of the system's density matrix in the energy representation, has been recently introduced as the proper definition of thermodynamic entropy in out-of-equilibrium quantum systems. We study this quantity after an interaction quench in lattice hard-core bosons and spinless fermions, and after a local chemical potential quench in a system of hard-core bosons in a superlattice potential. The former systems have a chaotic regime, where the diagonal entropy becomes equivalent to the equilibrium microcanonical entropy, coinciding with the onset of thermalization. The latter system is integrable. We show that its diagonal entropy is additive and different from the entropy of a generalized Gibbs ensemble, which has been introduced to account for the effects of conserved quantities at integrability.     

Shocks and Excitation Dynamics in a Driven Diffusive Two-Channel System

We consider classical hard-core particles hopping stochastically on two parallel chains in the same or opposite directions with an inter- and intra-chain interaction. We discuss general questions concerning elementary excitations in these systems, shocks and rarefaction waves. From microscopical considerations we derive the collective velocities and shock stability conditions. The findings are confirmed by comparison to Monte Carlo data of a multi-parameter class of simple two lane driven diffusion models, which have the stationary state of a product form on a ring. Going to the hydrodynamic limit, we point out the analogy of our results to the ones known in the theory of differential equations of two conservation laws. We discuss the singularity problem and find a dissipative term that selects the physical solution.     

A Three State Hard-Core Model on a Cayley Tree

Abstract

We consider a nearest-neighbor hard-core model, with three states , on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as a simple example of a loss network with nearest-neighbor exclusion. The state σ(x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate λ at each site x, each call has an exponential duration of mean 1. If a call finds the node in state 1 or 2 it is lost. If it finds the node in state 0 then things depend on the state of the neighboring sites. If all neighbors are in state 0, the call is accepted and the state of the node becomes 1 or 2 with equal probability 1/2. If at least one neighbor is in state 1, and there is no neighbor in state 2 then the state of the node becomes 1. If at least one neighbor is in state 2 the call is lost. We focus on ‘splitting’ Gibbs measures for this model, which are reversible equilibrium distributions for the above process. We prove that in this model, ? λ > 0 and k ≥ 1, there exists a unique translationinvariant splitting Gibbs measure ^*. We also study periodic splitting Gibbs measures and show that the above model admits only translation - invariant and periodic with period two (chess-board) Gibbs measures. We discuss some open problems and state several related conjectures.     

Diffusion in disordered lattices and related heisenberg ferromagnets

We study the diffusion of classical hard-core particles in disordered lattices within the formalism of a quantum spin representation. This analogy enables an exact treatment of noninstantaneous correlation functions at finite particle densities in terms of single spin excitations in disordered ferromagnetic backgrounds. Applications to diluted chains and percolation clusters are discussed. It is found that density fluctuations in the former exhibit a stretched exponential decay while an anomalous power law asymptotic decay is conjectured for the latter.     

Aging in 1D discrete spin models and equivalent systems

We derive exact expressions for a number of aging functions that are scaling limits of nonequilibrium correlations, R(t(w),t(w)+t) as t(w)-->infinity, t/t(w)-->theta, in the 1D homogenous q-state Potts model for all q with T = 0 dynamics following a quench from T = infinity. One such quantity is (0)(t(w));sigma-->(n)(t(w)+t)> when n/square root of ([t(w))-->z. Exact, closed-form expressions are also obtained when an interlude of T = infinity dynamics occurs. Our derivations express the scaling limit via coalescing Brownian paths and a "Brownian space-time spanning tree," which also yields other aging functions, such as the persistence probability of no spin flip at 0 between t(w) and t(w)+t.     

Kinetic Monte Carlo simulations of electrodeposition: Crossover from continuous to instantaneous homogeneous nucleation within Avrami’s law

The influence of lateral adsorbate diffusion on the dynamics of the first-order phase transition in a two-dimensional Ising lattice gas with attractive nearest-neighbor interactions is investigated by means of kinetic Monte Carlo simulations. For example, electrochemical underpotential deposition proceeds by this mechanism. One major difference from adsorption in vacuum surface science is that under control of the electrode potential and in the absence of mass-transport limitations, local adsorption equilibrium is approximately established. We analyze our results using the theory of Kolmogorov, Johnson and Mehl, and Avrami (KJMA), which we extend to an exponentially decaying nucleation rate. Such a decay may occur due to a suppression of nucleation around existing clusters in the presence of lateral adsorbate diffusion. Correlation functions prove the existence of such exclusion zones. By comparison with microscopic results for the nucleation rate I and the interface velocity of the growing clusters v, we can show that the KJMA theory yields the correct order of magnitude for Iv². This is true even though the spatial correlations mediated by diffusion are neglected. The decaying nucleation rate causes a gradual crossover from continuous to instantaneous nucleation, which is complete when the decay of the nucleation rate is very fast on the time scale of the phase transformation. Hence, instantaneous nucleation can be homogeneous, producing negative minima in the two-point correlation functions. We also present in this paper an n-fold way Monte Carlo algorithm for a square lattice gas with adsorption/desorption and lateral diffusion.