Showing papers in "Journal of Statistical Physics in 1972"

Fluctuations around nonequilibrium states in open nonlinear systems

Abstract: The theory of nonequilibrium fluctuations in open systems is extended to nonlinear situations. It is shown that the usual birth-and-death type of stochastic formulation of chemical kinetics is in general inadequate and has to be replaced by a more detailed phase-space description. As a consequence, for large classes of nonlinear systems arbitrarily far from equilibrium, the classical Einstein fluctuation formula can be extended, provided the steady reference state is asymptotically stable. The case of oscillatory or unstable systems is also discussed. It is conjectured that in such systems, the departure from the steady state is governed by large fluctuations of “macroscopic” size, while small fluctuations are still described by the extended Einstein formula. Nonequilibrium macroscopic instabilities such as chemical or hydrodynamic instabilities seem therefore to bear strong similarities to first-ordei phase transitions.

The thermodynamics of cluster formation in nucleation theory

Abstract: We have derived a precise thermodynamic definition of the standard free energy to form a cluster which is used in nucleation theory. The results [Eq. (9)] have a form differing slightly from the form usually used in nucleation theory and show that the Lothe-Pound correction factor is based on a misconception concerning the standard states involved.

Monte Carlo study of the surface area of liquid droplets

Abstract: Computer simulation of droplets containingl molecules (l ⩽ 1000) in a lattice gas shows that the average surface area is proportional tol σ′;σ′≃ 0.6 in two and σ′= 0.825 in three dimensions for small droplets. These exponents agree approximately with those in Kadanoff's modification of Fisher's droplet model near critical points [σ′= (1 + β)/βδ; ourT/T c is 0.4, 0.7, and 0.9]. For larger droplets, these exponents change to 1/2 (d = 2) and 2/3 (d = 3), the transition occurring for droplet diameters larger than the coherence length and smaller than the critical diameter in the nucleation of supersaturated vapors. This latter result rises some doubts on a recent nucleation theory of Eggingtonet al.

Nonlinear fluctuation-dissipation theorem

Abstract: Using statistical mechanical perturbation theory, the second-order average current density response is calculated for magnetic field-free classical plasmas. A dynamical fluctuation-dissipation theorem is then derived, thus establishing a connection between triplet microscopic current-current correlations and quadratic response functions; it also leads to a static fluctuation-dissipation theorem which provides a dielectric description of the equilibrium ternary correlation. A comparison of the latter with its expansion in terms of the Mayer pair correlation clusters is discussed.

On the solutions and the steady states of a master equation

Abstract: A complete characterization of the time behavior of the means and variance of a stochastic process which is generated by a finite number of independent systems is presented based on the master equation for the conditional probability. It is found that the means and variance relax to a steady state and that the steady state will be independent of the initial state if and only if a matrix related to the transition matrix is nonsingular. Finally, the result that the variance approaches its steady-state form at twice the rate of the means is shown to depend on the nonsingularity of the same matrix.

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

Abstract: 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.

The velocity autocorrelation function of a finite model system

Abstract: We investigate in detail the dependence of the velocity autocorrelation function of a one-dimensional system of hard, point particles with a simple velocity distribution function (all particles have velocities ±c) on the size of the system. In the thermodynamic limit, when both the number of particlesN and the length of the “box”L approach infinity andN/L → ρ, the velocity autocorrelation functionψ(t) is given simply by c2 exp(−2ρct@#@). For a finite system, the functionψN(t) is periodic with period 2L/c. We also show that for more general velocity distribution functions (particles can have velocities ±ci,i = 1,...),ψN(t) is an almost periodic function oft. These examples illustrate the role of the thermodynamic limit in nonequilibrium phenomena: We must keept fixed while letting the size of the system become infinite to obtain an auto-correlation function, such asψ(t), which decays for all times and can be integrated to obtain transport coefficients. For any finite system, ourψ N (t) will be “very close” toψ(t) as long ast is small compared to the effective “size” of the system, which is 2L/c for the first model.

Superposition approximations from a variation principle

Abstract: A variation principle is introduced involving then-particle molecular distribution function (where 1 ⩽n ⩽N) for a fluid containingN molecules. An integral involving any approximaten-particle distribution function proves to define aleast upper bound to the true system free energy. This integral can, therefore, be minimized with respect to the form of a trial distribution function to provide a best estimate to the exact distribution function. When no other constraints, save the requirement of normalization, are applied to the trial function, the extremum corresponds to the exact function. Using this variation principle, it is possible to demonstrate that the optimum triplet superposition approximation is the Krikwood approximation, and that the optimum quadruplet approximation is the form suggested by Fisher and Kopeliovich. Furthermore, all higher-order optimum superposition approximations are specified.

Lattice models for hydrogen-bonded solvents

Abstract: A class of lattice models for a binary mixture is defined by assuming that one of the components may form bonds to neighboring molecules of the same species. It is assumed that the fugacity of a molecule depends on the number of bonds which connect the molecule to other molecules. If no molecule is allowed to be connected by more than two bonds to other molecules, then no phase transition occurs, while phase transition can occur if more than two bonds are allowed. If only two or no bonds are allowed, then the model can be solved rigorously for certain planar lattices by transforming it to a dimer covering problem; this model shows behavior similar to the Ising model in zero magnetic field.

A theory of 1/f noise

Abstract: Letu(θ) be an absolutely integrable function and define the random process where thet i are Poisson arrivals and thes i, are identically distributed nonnegative random variables. Under routine independence assumptions, one may then calculate a formula for the spectrum ofn(t), S n(ω), in terms of the probability density ofs, ps(α). If any probability density ps(α) having the property ps(α) ∼ I for small α is substituted into this formula, the calculated Sn(ω) is such that Sn(ω)∼ 1 ω for small ω. However, this is not a spectrum of a well-defined random process; here, it is termed alimit spectrum. If a probability density having the property ps(α) ∼α δ for small α, where δ > 0, is substituted into the formula instead, a spectrum is calculated which is indeed the spectrum of a well-defined random process. Also, if the latter ps is suitably close to the former ps, then the spectrum in the second case approximates, to an arbitrary, degree of accuracy, the limit spectrum. It is shown how one may thereby have 1/f noise with low-frequency turnover, and also strict 1/f 1−δ noise (the latter spectrum being integrable for δ > 0). Suitable examples are given. Actually, u(θ) may be itself a random process, and the theory is developed on this basis.

Models in nonequilibrium quantum statistical mechanics

Abstract: In this paper, quantum versions of statistical models are constructed. All aspects of the systems can be explicitly solved. It is possible to give magnetic realizations of these models. The most interesting conclusions are: (1) the state for time going to infinity is approached in an oscillatory manner in the quantum case; (2) in both classical and quantum cases, the exact description gives limiting states which remember the initial specifications; and (3) in these models, the time evolution generally cannot be described. even approximately, by a master equation.

Computer-Oriented Approach to the Ensemble Theory

Abstract: Computer simulation of the ensemble behavior of large interacting systems is hampered by the relative paucity of low-energy configurations. The stochastic sampling must, therefore, be guided toward the capture of configurations which, though rare, aretypical, in the sense of contributing the maximum term of the partition function, Max log ζ. In a previous Monte Carlo study by the author, such a guidance was applied to anab initio construction of lattice configurations with a stochastic chain of steps, the transition probability at each step depending of both near and far neighbors. This study is now put on a more systematic and broader basis. It is argued that any computer study of the emergence of long-range order in large systems must be guided by Max log ζ. Thus, lattice configurations are often constructed by a cyclic variation, which, guided by nearest-neighbor interaction, tries to relax an arbitrary initial configuration toward equilibrium (Metropolis). Such a relaxation may be also guided by Max log ζ, with transition probabilities dependent on both near and far neighbors. Of main interest is the relevance of such a calculation for the actual relaxation behavior of the system, and this is discussed at some length.

On interesting walks in a graph

Abstract: Notions of interesting walks and of their equivalence are introduced. A general formula for the numberΓ l, of equivalence classes of interesting walks of lengthl in a given graphG is derived and applied forl⩽ 5 so as to expressΓ l in terms of the adjacency matrix ofG.

The interpretation of inductive probabilities

Abstract: An apparent inconsistency in the inductive logic interpretation of probabilities is examined and resolved.

On the validity of stochastic rate equations in finite systems with finite-strength interactions

Abstract: Starting with the Hamiltonian for a linear harmonic chain of 2N particles of massm and one of massM, we have carried out numerical calculations for the momentum autocorrelation function of the mass defect particle for chains with finite numberN of mass points and for nonzero values of the mass ratioμ=m/M. These results have been compared with the well-known exponential relaxation of the momentum autocorrelation function which is found to be the rigorous result when passing to the thermodynamic and weak-coupling limit. In these limits, the dynamics of the mass defect particle is exactly described by a Fokker-Planck equation, i.e., a stochastic equation of motion. We have shown that, to an excellent approximation, an exponential relaxation of the momentum autocorrelation function is obtained for mass ratios as high asμ=0.1 and for chains with only 50 particles. Thus, for the harmonic chain considered here, the stochastic equations of motion can be applied to a very good approximation far outside the usually imposed thermodynamic and weak-coupling limits.

Kinetic equations without “memory” for the time-displaced correlation functions

Abstract: The relations between the kinetic equations with and without convolution in time are discussed on the basis of the kinetic equation for the Van Hove self-correlation function. Formal equivalence of both the equations is shown, and approximate scattering operators for the dilute-gas case and for the Brownian particle are considered.

Random integral equation formulation of a generalized Langevin equation

Abstract: In this paper, the generalized Langevin equation introduced by Kubo and Mori is formulated as a random integral equation. We consider (1) the existence and uniqueness of the solution, (2) moments of the solution process, (3) a comparison theorem for solution processes, and (4) the Cauchy polygonal approximation to the solution.

Some approximate solutions to the discrete master equation

Abstract: Recent mathematical developments on approximate diffusionlike solutions to the master equation are summarized. The technique is applied to two master equations of physical interest-one that describes the phenomenon of superradiance and a second that characterizes generation-recombination noise in semiconductors. For this second case, some previously obtained equilibrium results are found and the extension of these results to finite times is given.

Time-dependent statistics of binary linear lattices

Abstract: The time-dependent statistics of binary linear lattices is investigated on the basis of a master equation at the microscopic level. It is assumed that the kinetics may be formulated as transformations of specified sequences of clusters ofA units andB units into other specified sequences. On the basis of aStosszahlansatz, a master equation at the macroscopic level is derived. In the limit of a large system, the densities of clusters of all types satisfy rate equations similar to the equations of chemical kinetics. AnH-theorem is proven and the nonequilibrium thermodynamics of the system is studied. The theory has application to the kinetics of the helix-coil phase transition in biopolymers.

Inversion of the Classical Second Virial Coefficient

Abstract: It is shown that, on the basis of some weak assumptions regarding the nature of the intermolecular pair potential, the classical second virial coefficient determines the potential uniquely.

Explicit Relations for the Radial Distribution Functions for One-Dimensional Lattice (Arbitrary Spacing) Fluids and Solutions

Abstract: It is pointed out that the size of the matrix required to formulate the grand partition function for a one-dimensional lattice fluid for a fixed and finite range of the interatomic potential varies linearly with the density of lattice points used and hence is much smaller and more manageable than the expected size (which varies exponentially with the same quantity) and thus allows very fine grids to be examined. Using the matrix treatment of the grand partition function, it is shown that the radial distribution function for a one-dimensional fluid or solution can be formulated as an explicit matrix product which is simply performed by computer. The resulting distribution functions (which can be extrapolated to the continuum by varying the lattice spacing) are useful as starting solutions for the iterative solution of integral equations for three-dimensional fluids.

Phenomenological equations for multicomponent fluids

Abstract: Ahydrodynamic equation of motion for each component of a multicomponent fluid is derived on the basis of nonequilibrium thermodynamics. Special care has been directed to the choice of state variables. In some limiting cases, this equation leads to customary phenomenological equations, such as the equation for diffusion and the Navier-Stokes equation. The viscosity is a consequence of nonlocal coupling of forces and fluxes. The reciprocity between the linear coefficients is examined closely.

Comments on universality of critical exponents for fluid systems

Abstract: Critical exponents (γ, ν, β, andμ) of one-component fluid systems from recent experiments show agreement with the universality concept and the critical exponent relations. Variations in the magnitude of exponents from different systems are well within the limits of error of present-day techniques. Thus, it is within reason to expect that observable deviations from universality could come from more complex fluid mixtures, where such a concept may break down, even though, based on published literature values, the prospects remain very small.

Action-angle variables in quantum statistical mechanics

Abstract: Utilizing the facts (i) that the number of particles in the many-boson system is conserved and (ii) that the Hamiltonian is Hermitian, a new set of variables comprising “action” and “angle” variables has been introduced. These variables are conjugate in the “mean” and provide a rigorous approach to introducing phase variables for “total-number-conserving many-boson systems.”

Statistical Mechanics and the Gauss Principle

Abstract: In an essentially statistical approach to statistical mechanics, it is seen that the Gauss principle of the arithmetic mean may be taken as the starting point. The equations from which the subject can be built up are deduced from the Gauss principle of the arithmetic mean.

Nonequilibrium statistical mechanics of finite classical systems-II

Abstract: The work of the previous paper is applied to the study of weakly interacting systems. Either by “quasilinear” techniques or by analyzing the perturbation series for the smoothed probability density, it is possible to derive a master equation equivalent to that of Brout and Prigogine without requiring the size of the system to become infinite. The properties of this equation are discussed. The equation is self-consistent provided the interactions are weak enough; however, examination of higher terms in the perturbation series shows that their effect might make the master equation invalid for times longer than that taken by a typical particle to cross the containing vessel. In many physical cases, the relaxation time will be shorter than this; also, further studies may show the higher terms to be less important than they seem.

Time Partition Function Analysis of a Neural Network

Abstract: The time characteristics of a linear network in the brain are obtained by the method of the “time partition function,” which is analogous to a grand partition function or a distribution function in statistical mechanics. The analogy between the average density in a many-particle system and the reciprocal of the frequency in a network is shown. By this method, the frequency distribution functions are obtained with respect to a network composed of two layers, the network used in information retrieval and the network generating a brain wave.

A constant-magnetization ensemble for the classical anisotropic Heisenberg model

Abstract: A constant-magnetization ensemble is introduced in order to study classical, anisotropic Heisenberg systems. Existence, uniform convergence, and convexity properties are proved for an appropriate thermodynamic potential. The thermodynamic equivalence of this ensemble with the more common canonical ensemble is also established. In a subsequent paper, this formulation is used to obtain an exact statistical mechanical solution of classical Heisenberg systems with long-range Kac interactions.

Direct-Interaction Approximation by the Modal-Interaction Perturbation Technique,

Abstract: An alternate method is presented of obtaining the direct-interaction equations by combining the heuristic and rigorous derivations of Kraichnan. Within the framework of the model dynamic representation of Kraichnan's rigorous theory, we have developed the irreducible diagram expansion systematically * by formalizing the perturbation argument of his heuristic derivation. It is hoped that the present work will provide a further insight into the analytical structure of the irreducible diagram expansion and bridge the gap apparent in the two original derivations of the direct-interaction equations given by Kraichnan.