Showing papers in "Journal of Statistical Physics in 1978"

Quantitative universality for a class of nonlinear transformations

Abstract: A large class of recursion relations xn+l = Af(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The functions considered all have a unique differentiable maximum 2. With f(2) - f(x) ~ Ix - 21" (for Ix - 21 sufficiently small), z > 1, the universal details depend only upon z. In particular, the local structure of high-order stability sets is shown to approach universality, rescaling in successive bifurcations, asymptotically by the ratio c~ (a = 2.5029078750957... for z = 2). This structure is determined by a universal function g*(x), where the 2"th iterate off, f("~, converges locally to ~-"g*(~nx) for large n. For ithe class of f's considered, there exists a A~ such that a 2"-point stable limit cycle including :7 exists; A~ - ~ ~ ~-" (~ = 4.669201609103... for z = 2). The numbers = and have been computationally determined for a range of z through their definitions, for a variety off's for each z. We present a recursive mechanism that explains these results by determining g* as the fixed-point (function) of a transformation on the class off's. At present our treatment is heuristic. In a sequel, an exact theory is formulated and specific problems of rigor isolated.

Phase Transitions in Quantum Spin Systems with Isotropic and Nonisotropic Interactions

Abstract: We prove the existence of spontaneous magnetization at sufficiently low temperature, and hence of a phase transition, in a variety of quantum spin systems in three or more dimensions. The isotropic spin 1/2 x−y model and the Heisenberg antiferromagnet with spin 1, 3/2,...and with nearest neighbor interactions on a simple cubic lattice are included.

Thermodynamical Proof of the Gibbs Formula for Elementary Quantum Systems

Abstract: An elementary derivation is given of the formula for the thermal equilibrium states of quantum systems that can be described in finite-dimensional Hilbert spaces. The three assumptions made, Passivity, Structural Stability, and Consistency, have phenomenological interpretations. Except at zero temperature, Structural Stability follows already from Passivity and a weak form of Consistency.

Statistical mechanics of a nonlinear stochastic model

Abstract: A multivariable Fokker-Planck equation (FPE) is used to investigate the equilibrium and dynamical properties of a nonlinear stochastic model. The model displays a phase transition. The equilibrium distributions are found to be non-Gaussian; the deviation from Gaussian is especially significant near the transition point. To study the nonequilibrium behavior of the model, a self-consistent dynamic mean field (SCDMF) theory is derived and used to transform the FPE to a systematic hierarchy of equations for the cumulant moments of the time-dependent distribution function. These equations are numerically solved for a variety of initial conditions. During the time evolution of the system from an initial unstable equilibrium state to the final equilibrium state, three distinct time stages are found.

Solution of the Ornstein-Zernike equation with Yukawa closure for a mixture

Abstract: The Ornstein-Zernike equation with Yukawa closure\([c_{ij} (r) = K_{ij} e^{ - z(r - \sigma _{ij} )} /r\) forr > σij] for a mixture is solved. We utilize the Fourier transform or factorization technique introduced by Baxter. The general solution is obtained in the form of algebraic equations.

Open quantum systems with time-dependent Hamiltonians and their linear response

Abstract: We give a rigorous (Hamiltonian) treatment of a quantum system weakly coupled to an infinite free reservoir and subject to an external time-dependent driving potential varying on the scale of dissipation. The linear response of the system initially in thermal equilibrium is determined and compared with the usual expressions of linear response theory.

Variational approximations for square lattice models in statistical mechanics

Abstract: This paper concerns a square lattice, Ising-type model with interactions between the four spins at the corners of each face. These may include nearest and next-nearest-neighbor interactions, and interactions with a magnetic field. Provided the Hamiltonian is symmetric with respect to both row reversal and column reversal, a rapidly convergent sequence of variational approximations is obtained, giving the free energy and other thermodynamic properties. For the usual Ising model, the lowest such approximations are those of Bethe and of Kramers and Wannier. The method provides a new definition of corner transfer matrices.

The Statistics of Curie-Weiss Models

Abstract: LetSn denote the random total magnetization of ann-site Curie-Weiss model, a collection ofn (spin) random variables with an equal interaction of strength 1/n between each pair of spins. The asymptotic behavior for largen of the probability distribution ofSn is analyzed and related to the well-known (mean-field) thermodynamic properties of these models. One particular result is that at a type-k critical point (Sn-nm)/n1−1/2k has a limiting distribution with density proportional to exp[-λs2k/(2k)!], wherem is the mean magnetization per site and A is a positive critical parameter with a universal upper bound. Another result describes the asymptotic behavior relevant to metastability.

Structure of interfaces from uniformity of the chemical potential

Abstract: It is shown that a generalized chemical potential suggested by the potential-distribution theory is uniform even in a nonuniform fluid. Leng, Rowlinson, and Thompson had already observed its uniformity through the liquid-vapor interface in the penetrable-sphere model, in mean-field approximation. Following those authors, we exploit the uniformity of that generalized chemical potential to obtain unified and transparent derivations of the results of Ono and Kondo and of van der Waals on the liquid-vapor interfaces in the lattice-gas model and in the model of attracting hard spheres, respectively, both in mean-field approximation.

Critical Exponents and Large-Order Behavior of Perturbation Theory

Abstract: The principles of the recent calculations of critical exponents from three- and two-dimensional field theory are reviewed. They rely on the Callan-Symanzik equations, diagram calculations, and on the characterization of the asymptotic behavior of perturbation series at large order. We then present new results concerning the normalization of the large-order behavior.

Percolation and the Potts model

Abstract: The Kasteleyn-Fortuin formulation of bond percolation as a lattice statistical model is rederived using an alternate approach. It is shown that the quantities of interest arising in the percolation problem, including the critical exponents, can be obtained from the solution of the Potts model. We also establish the Griffith inequality for critical exponents for the bond percolation problem.

H-theorem for the (modified) nonlinear Enskog equation

Abstract: We construct anH-function suitable for a system of dense hard spheres satisfying the (modified) nonlinear Enskog equation and we show that∂tH ⩽ 0. The equality sign holds only when the system has reached absolute equilibrium, in which caseS=− kBH becomes the exact equilibrium entropy of the hard-sphere fluid.

Percolation and cluster distribution. II. layers, variable-range interactions, and exciton cluster model

Abstract: Monte Carlo simulations for the site percolation problem are presented for lattices up to 64 x 106 sites. We investigate for the square lattice the variablerange percolation problem, where distinct trends with bond-length are found for the critical concentrations and for the critical exponents/~ and 7. We also investigate the layer problem for stacks of square lattices added to approach a simple cubic lattice, yielding critical concentrations as a functional of layer number as well as the correlation length exponent u. We also show that the exciton migration probability for a common type of ternary lattice system can be described by a cluster model and actually provides a cluster generating function.

Essential Singularity in Percolation Problems and Asymptotic Behavior of Cluster Size Distribution

Abstract: It is rigorously proved that the analog of the free energy for the bond and site percolation problem on\(\mathbb{Z}^v \) in arbitrary dimensionΝ (Ν> 1) has a singularity at zero external field as soon as percolation appears, whereas it is analytic for small concentrations. For large concentrations at least, it remains, however, infinitely differentiable and Borel-summable. Results on the asymptotic behavior of the cluster size distribution and its moments, and on the average surface-to-size ratio, are also obtained. Analogous results hold for the cluster generating function of any equilibrium state of a lattice model, including, for example, the Ising model, but infinite-range andn-body interactions are also allowed.

Exact results for the Potts model in two dimensions

Abstract: By considering the zeros of the partition function, we establish the following results for the Potts model on the square, triangular, and honeycomb lattices: (i) We show that there exists only one phase transition; (ii) we give an exact determination of the critical point; (iii) we prove the exponential decay of the correlation functions, in one direction at least, for all temperatures above the critical point. The results are established forq ⩾ 4, whereq is the number of components.

Transport properties of the Lorentz gas: Fourier's law

Abstract: We investigate the stationary nonequilibrium (heat transporting) states of the Lorentz gas. This is a gas of classical point particles moving in a region gL containing also fixed (hard sphere) scatterers of radiusR. The stationary state considered is obtained by imposing stochastic boundary conditions at the top and bottom of Λ, i.e., a particle hitting one of these walls comes off with a velocity distribution corresponding to temperaturesT1 andT2 respectively,T1

Statistical Mechanics of a Dynamical System Based on Conway's Game of Life

Abstract: We study a discrete dynamical system whose evolution is governed by rules similar to those of Conway's game of Life but also include a stochastic element (parametrized by a “temperature”). Statistical properties that are examined are density as a function of temperature and entropy (suitably defined). A phase transition and a certain “thermodynamic” constant of the motion are observed.

Kinetic theory of nonlinear viscous flow in two and three dimensions

Abstract: On the basis of a nonlinear kinetic equation for a moderately dense system of hard spheres and disks it is shown that shear and normal stresses in a steady-state, uniform shear flow contain singular contributions of the form ¦X¦3/2 for hard spheres, or ¦X¦ log ¦X¦ for hard disks. HereX is proportional to the velocity gradient in the shear flow. The origin of these terms is closely related to the hydrodynamic tails t−d/2 in the current-current correlation functions. These results also imply that a nonlinear shear viscosity exists in two-dimensional systems. An extensive discussion is given on the range ofX values where the present theory can be applied, and numerical estimates of the effects are given for typical circumstances in laboratory and computer experiments.

Note on Time Evolution of Non-Markov Processes

Abstract: We discuss the question of the construction of a linear semigroup for the time evolution of the single-event probabilities of general non-Markov processes. It is shown that such a linear semigroup may not exist for all finite times. The consequences are sketched for the description of equilibrium and nonequilibrium systems. Further, the relationship with nonstationary Markov processes is investigated, and some confusion in recent works is cleared up using the example of free Brownian motion.

Solution of the mean spherical approximation for hard ions and dipoles of arbitrary size

Abstract: The general solution of the mean spherical approximation (MSA) for an arbitrary mixture of hard spherical ions and dipoles, in which the ions can be of different size, is found. This solution is given in terms of three parameters that are calculated by solving an algebraic equation. Two of these parameters are scaling parameters required to satisfy the general symmetry of the pair correlation functions, and are similar to the one introduced in the solution of the MSA for an ionic mixture in earlier work. For equal size and low ionic concentration, we get a rather explicit solution of the MSA, which is formally similar to the Waisman-Lebowitz solution of the restricted primitive model, but with a concentration-dependent dielectric constant.

Long-time correlation effects on displacement distributions

Abstract: The distribution of displacements in a fluid of hard disks is found by molecular dynamics to be non-Gaussian in the long-time limit, as surmised from the moments of the distribution that yield divergent diffusion and Burnett coefficients. On the other hand, for the Lorentz gas of hard disks, the distribution of displacements is Gaussian in the long-time limit and the diffusion coefficient exists, though the autocorrelation functions have power law tails, which lead to divergent Burnett coefficients.

Exact analytical formulas for the distribution functions of charged hard spheres in the mean spherical approximation

Abstract: Exact analytical expressions are derived for the electrostatic part of the mean spherical approximation (MSA) radial distribution functions (RDFs) for a system of charged hard spheres. These expressions are valid for all distances. In addition, it is shown that these same expressions arise in the MSA description of the charge profile of charged hard spheres near a charged hard wall. We also derive analytical expressions for the nonelectrostatic part of the profile in this case, valid forz ⩽ 5σ, and discuss a numerical method for obtaining results forz > 5σ. Some simple approximate expressions are also considered.

Critical temperature for two-dimensional ising ferromagnets with quenched bond disorder

Abstract: The configuration-averaged free energy of a quenched, random bond Ising model on a square lattice which contains an equal mixture of two types of ferromagnetic bonds J1 and J2 is shown to obey the same duality relation as the ordered rectangular model with the same two bond strengths. If the random.system has a single, sharp critical point, the critical temperature Tc must be identical to that of the ordered system, i.e., sinh(2J 1/kT c) sinh(2J 2/kT c) = 1. Since c (B) = 1/2, we can takeJ 2 → 0 and use Bergstresser-type inequalities to obtain(ρ/ρdp) exp(−2J 1/kTc¦p=pc + = 1, in agreement with Bergstresser's rigorous result for the diluted ferromagnet near the percolation threshold.

The Poisson Representation. II. Two-Time Correlation Functions

Abstract: Basic formulas for the two-time correlation functions are derived using the Poisson representation method The formulas for the chemical system in thermodynamic equilibrium are shown to relate directly to the fluctuationdissipation theorems, which may be derived from equilibrium statistical mechanical considerations For nonequilibrium systems, the formulas are shown to be generalizations of these fluctuation-dissipation theorems, but containing an extra term which arises entirely from the nonequilibrium nature of the system These formulas are applied to two representative examples of equilibrium reactions (without spatial diffusion) and to a nonequilibrium chemical reaction model (including the process of spatial diffusion) for which the first two terms in a systematic expansion for the two-time correlation functions are calculated The relation between the Poisson representation method and Glauber-SudarshanP-representation used in quantum optics is discussed

A model of heat conduction

Abstract: We construct a model of a chain of atoms coupled at its ends to two reservoirs at different temperatures. In a weak coupling limit the atoms obey a stochastic evolution law and have an equilibrium state with a uniform temperature gradient along the chain.

Clusters, metastability, and nucleation: Kinetics of first-order phase transitions

Abstract: We describe and interpret computer simulations of the time evolution of a binary alloy on a cubic lattice, with nearest neighbor interactions favoring like pairs of atoms. Initially the atoms are arranged at random; the time evolution proceeds by random interchanges of nearest neighbor pairs, using probabilities compatible with the equilibrium Gibbs distribution at temperatureT. For temperatures 0.59Tc, 0.81 Tc, and 0.89T c, with densityρ of A atoms equal to that in the B-rich phase at coexistence, the density C1 of clusters ofl A atoms approximately satisfies the following empirical formulas: C1 ≈w(1 −ρ)3 andC 1, ≈ (1 −ρ)4Q1w1 (2 ⩽l ⩽ 10). Herew is a parameter and we defineQ l =∑ K e −βE(K) , where the sum goes over all translationally nonequivalentl-particle clusters andE(K) is the energy of formation of the clusterK. Forl > 10,Q 1 is not known exactly; so we use an extrapolation formulaQ l ≈Aw −l l −α exp(−bl σ), wherew s is the value ofw at coexistence. The same formula (withw > w s) also fits the observed values of C, (for small values ofl) at densities greater than the coexistence density (forT=0.59Tc): When the supersaturation is small, the simulations show apparently metastable states, a theoretical estimate of whose lifetime is compatible with the observations. For higher supersaturation the system is observed to undergo a slow process of segregation into two coexisting phases (andw therefore changes slowly with time). These results may be interpreted as a more quantitative formulation (and confirmation) of ideas used in standard nucleation theory. No evidence for a “spinodal” transition is found.

Growth of clusters in a first-order phase transition

Abstract: The results of computer simulations of phase separation kinetics in a binary alloy quenched from a high temperature are analyzed in detail, using the ideas of Lifshitz and Slyozov. The alloy was modeled by a three-dimensional Ising model with Kawasaki dynamics. The temperature after quenching was 0.59Tc, whereTc is the critical temperature, and the concentration of minority atoms wasρ=0.075, which is about five times their largest possible single-phase equilibrium concentration at that temperature. The time interval covered by our analysis goes from about 1000 to 6000 attempted interchanges per site. The size distribution of small clusters of minority atoms is fitted approximately byc1≈(1-ρ)3w(t),c1≈ (1−ρ)4Qlw(t)l(2≤l≤10); wherecl is the concentration of clusters of sizel;Q2,...,Q10 are known constants, the “cluster partition functions”;t is the time; andw(t)=0.015(1+7.17t−1/3). The distribution of large clusters (l≥20) is fitted approximately by the type of distribution proposed by Lifshitz and Slyozov,cl,(t)=−(d/dl)ψ[lnt+pϕ(l/t)], whereϕ is a function given by those authors andψ is defined byψ(x)=Coe−x-C1e−4x/3-C2e−5x/3;C0,C1,C2 are constants determined by considering how the total number of particles in large clusters changes with time.

The Yang-Lee edge singularity in spherical models

Abstract: The density of Yang-Lee zeros in the thermodynamic limit is discussed for ferromagnetic spherical models of general dimensionalities and arbitrary range of interaction. In all cases the zeros lie on the imaginary axis in the complex magnetic field planeH=H′+iH″ with a densityℊ (H″) that exhibits a square root singularityℊ(H″) ∼(H″-H 0)σ, withσ=1/2, as the edge of the gap atH″=H 0(T) is approached forT>T c. WhenT→T c one hasH 0(T)∼(T∼T c )Δ with critical exponentΔ=β+γ.

High-density percolation: Exact solution on a Bethe lattice

Abstract: A new percolation problem is posed where the sites on a lattice are randomly occupied but where only those occupied sites with at least a given numberm of occupied neighbors are included in the clusters. This problem, which has applications in magnetic and other systems, is solved exactly on a Bethe lattice. The classical percolation critical exponentsβ=gg=1 are found. The percolation thresholds vary between the ordinary percolation thresholdp c (m=1)=l/(z − 1) andp c(m=z) =[l/(z − 1)]1/(z−1). The cluster size distribution asymptotically decays exponentially withn, for largen, p ≠ p c .