# Showing papers in "Journal of Statistical Physics in 1986"

••

TL;DR: In this paper, the authors derived a reformulation of statistical thermodynamics for fluids of molecules which interact by highly directional attraction, which consists of a repulsive core and several sites of very short-ranged attraction.

Abstract: We derive a reformulation of statistical thermodynamics for fluids of molecules which interact by highly directional attraction. The molecular model consists of a repulsive core and several sites of very short-ranged attraction. We explore the relationship between graph cancelation in the fugacity expansion and three types of steric incompatibility between repulsive and attractive interactions involving several molecules. The steric effects are used to best advantage in a limited regrouping of bonds. This controls the density parameters which appear when articulation points are eliminated in the graphical representation. Each density parameter is a singlet density for a species consisting of molecules with a specified set of sites bonded. The densities satisfy subsidiary conditions of internal consistency. These conditions are equivalent to a minimization of the Helmholtz free energyA. Graphical expressions forA and for the pressurep are derived. Analogs of thes-particle direct correlation functions and of the Ornstein-Zernike equation are found.

1,394 citations

••

TL;DR: In this paper, the authors used a previously derived reformulation of statistical thermodynamics, in which the particle species are monomeric units with a specified set of attraction sites bonded.

Abstract: We investigate approximation methods for systems of molecules interacting by core repulsion and highly directional attraction due to several attraction sites. The force model chosen imitates a chemical bond by providing for bond saturation when binding occurs. The dense fluid is an equilibrium mixture ofs-mers with mutual repulsion. We use a previously derived reformulation of statistical thermodynamics, in which the particle species are monomeric units with a specified set of attraction sites bonded. Thermodynamic perturbation theory (TPT) and integral equations of two types are derived. The use of TPT is illustrated by explicit calculation for a molecular model with two attraction sites, capable of forming chain and ring polymers. Successes and defects of TPT are discussed. The integral equations for pair correlations between particles of specified bonding include calculation of self-consistent densities of species. Methods of calculating thermodynamic properties from the solutions of integral equations are given.

1,335 citations

••

TL;DR: In this article, a model of cellular automata is presented, in which particles with discrete velocities populate the links of a fixed array of sites, and equations for microscopic particle distributions are derived.

Abstract: Continuum equations are derived for the large-scale behavior of a class of cellular automaton models for fluids. The cellular automata are discrete analogues of molecular dynamics, in which particles with discrete velocities populate the links of a fixed array of sites. Kinetic equations for microscopic particle distributions are constructed. Hydrodynamic equations are then derived using the Chapman-Enskog expansion. Slightly modified Navier-Stokes equations are obtained in two and three dimensions with certain lattices. Viscosities and other transport coefficients are calculated using the Boltzmann transport equation approximation. Some corrections to the equations of motion for cellular automaton fluids beyond the Navier-Stokes order are given.

621 citations

••

TL;DR: P pores in the RSA configuration are introduced and there is a direct correspondence between vertices of the VD network and these holes, and also between direct/indirect geometrical neighbors and theseholes, and the hole size distribution is found to be a parabola.

Abstract: By sequentially adding line segments to a line or disks to a surface at random positions without overlaps, we obtain configurations of the one- and two-dimensional random sequential adsorption (RSA) problem. We have simulated the one- and two-dimensional problem with periodic boundary condition. The one-dimensional simulations are compared with the exact analytical solutions to give an estimate of the accuracy of the simulation. In two dimensions the geometrical properties of the RSA configuration are discussed and in addition known results of the RSA process are reproduced. Various statistical distributions of the Voronoi-Dirichlet (VD) network corresponding to the RSA disk configuration are analyzed. In order to characterize pores in the RSA configuration, we introduce circular holes. There is a direct correspondence between vertices of the VD network and these holes, and also between direct/indirect geometrical neighbors and these holes. The hole size distribution is found to be a parabola. We also find general relations that connect the asymptotic behavior of the surface coverage, the correlation function, and the hole size distribution.

431 citations

••

TL;DR: In this article, the authors report, extend, and interpret various theories of noise-activated escape, and discuss the connection between many-body transition state theory and Kramers' original diffusive Brownian motion approach (both in one-and multidimensional potential fields) and emphasize the physical situation inherent in Kramer's rate for weak friction.

Abstract: Many important processes in science involve the escape of a particle over a barrier. In this review, we report, extend, and interpret various theories of noise-activated escape. We discuss the connection between many-body transition state theory and Kramers' original diffusive Brownian motion approach (both in one-and multidimensional potential fields) and emphasize the physical situation inherent in Kramers' rate for weak friction. A rate theory accounting for memory friction is presented together with a set of criteria which test its validity. The complications and peculiarities of noise-activated escape in driven systems exhibiting multiple, locally stable stationary nonequilibrium states are identified and illustrated. At lower temperatures, quantum tunneling effects begin to play an increasingly important role. Early approaches and more recent developments of the quantum version of Kramers approach are discussed, thereby providing a description for dissipative escape at all temperatures.

273 citations

••

TL;DR: In this article, the authors show that the diffusion-limited reaction rate is proportional to the effective spectral dimension, where h = 1- d��s/2, X = 1+2/d��s)=(h-2)(h-1), and d====== s>>\s = 4/3.

Abstract: Heterogeneous kinetics are shown to differ drastically from homogeneous kinetics. For the elementary reaction A + A → products we show that the diffusion-limited reaction rate is proportional tot
− h[A]2 or to [A]x, whereh=1- d
s/2, X=1+2/d
s
=(h-2)(h-1), andd
s
is the effective spectral dimension. We note that ford = d
s
=1, h =1/2 andX = 3, for percolating clustersd
s = 4/3,h = 1/3 andX = 5/2, while for “dust” ds h > 1/2 and ∞ >X > 3. Scaling arguments, supercomputer simulations and experiments give a consistent picture. The interplay of energetic and geometric heterogeneity results in fractal-like kinetics and is relevant to excitation fusion experiments in porous membranes, films, and polymeric glasses. However, in isotopic mixed crystals, the geometric fractal nature (percolation clusters) dominates.

268 citations

••

TL;DR: In this article, it was shown that when the particle-conserving exchanges (stirrings) occur on a fast time scale of order ǫ−2 the macroscopic density evolves according to an autonomous nonlinear diffusion-reaction equation.

Abstract: We study interacting spin (particle) systems on a lattice under the combined influence of spin flip (Glauber) and simple exchange (Kawasaki) dynamics. We prove that when the particle-conserving exchanges (stirrings) occur on a fast time scale of order ɛ−2 the macroscopic density, defined on spatial scale ɛ−1, evolves according to an autonomous nonlinear diffusion-reaction equation. Microscopic fluctuations about the deterministic macroscopic evolution are found explicitly. They grow, with time, to become infinite when the deterministic solution is unstable.

180 citations

••

TL;DR: In this article, the authors provide an introduction to several of the problems to be discussed in greater depth by other speakers at a symposium held at the National Institutes of Health on May 6-8, 1985.

Abstract: There is enormous recent interest in the development of models for rate processes because rates are an almost universal characterization in the physical and biological sciences. In this paper we provide an introduction to several of the problems to be discussed in greater depth by other speakers at a symposium held at the National Institutes of Health on May 6–8, 1985. This review will focus on (1) the Smoluchowski model for reaction rates together with its extension by Onsager, (2) first passage time formalism for discrete and continuous master equations and Fokker-Planck equations, (3) the Kramers model and its extensions, (4) diffusion in the presence of trapping centers.

180 citations

••

TL;DR: In this paper, it was shown that the magnetization is continuous at T c and its critical exponents take the classical values δ=3 and β=1/2, with possible logarithmic corrections atd=4.

Abstract: We derive rigorously general results on the critical behavior of the magnetization in Ising models, as a function of the temperature and the external field. For the nearest-neighbor models it is shown that ind⩾4 dimensions the magnetization is continuous atT
c and its critical exponents take the classical valuesδ=3 andβ=1/2, with possible logarithmic corrections atd=4. The continuity, and other explicit bounds, formally extend tod>3 1/2. Other systems to which the results apply include long-range models ind=1 dimension, with 1/|x−y|
λ
couplings, for which 2/(λ−1) replacesd in the above summary. The results are obtained by means of differential inequalities derived here using the random current representation, which is discussed in detail for the case of a nonvanishing magnetic field.

153 citations

••

TL;DR: In this paper, the authors analyze the commensurate-in-commensurate transition in uniaxial systems in terms of a gas of interacting domain walls and show that the transition can be modeled as a rounding of facets in the equilibrium shape of crystals.

Abstract: In this paper we study several problems in statistical mechanics involving systems of fluctuating extended objects, such as interacting steps and domain walls. We reconsider the roughening transition and relate it to the free energy of a gas of steps and to the rounding of facets in the equilibrium shape of crystals, defined via the Wulff construction. Using an idealized description due to Fisher and Fisher we analyze the commensurate-incommensurate transition in uniaxial systems in terms of a gas of interacting domain walls. We also study the interactions between a domain wall and a rigid wall and between two interfaces, a problem which is central for the understanding of wetting. Among our results are a quantitative analysis of entropic repulsion between extended objects and a calculation of deviations from mean-field theory in the commensurate-incommensurate transition in dimension 2 ⩽d ⩽ 3.

148 citations

••

TL;DR: In this paper, the authors revisited Nekhoroshev's theorem on the stability of motions in quasi-integrable Hamiltonian systems and proved the confinement of orbits in resonant regions by using the elementary idea of energy conservation instead of more complicated mechanisms.

Abstract: Nekhoroshev's theorem on the stability of motions in quasi-integrable Hamiltonian systems is revisited. At variance with the proofs already available in the literature, we explicitly consider the case of weakly perturbed harmonic oscillators; furthermore we prove the confinement of orbits in resonant regions, in the general case of nonisochronous systems, by using the elementary idea of energy conservation instead of more complicated mechanisms. An application of Nekhoroshev's theorem to the study of perturbed motions inside resonances is also provided.

••

TL;DR: In this paper, it was shown that, with large probability, there are quasiperiodic lattice vibrations of finite total energy which lie on some infinite-dimensional, compact invariant tori in phase space.

Abstract: We study localization and wave trapping in disordered, nonlinear dynamical systems. For some models of classical, disordered anharmonic crystal lattices, we prove that, with large probability, there are quasiperiodic lattice vibrations of finite total energy which lie on some infinite-dimensional, compact invariant tori in phase space. Such vibrations remain localized, for all times, and there is no transport of energy through the lattice. Our general concepts and techniques extend to other systems, such as disordered, nonlinear Schrodinger equations, or randomly coupled rotors.

••

TL;DR: In this article, the invariant measures of a dynamical system can be answered by computations of regular functionals or by ranking methods based on a set of observations, including symmetry tests and the determination of dimension coefficients.

Abstract: Various questions about the invariant measures of a dynamical system can be answered by computations of regular functionals or by ranking methods based on a set of observations. This includes symmetry tests and the determination of dimension coefficients. The paper contains the theoretical results and several simulations explain the methods.

••

TL;DR: In this article, a simple model where the position of resonances can be estimated is presented, and the effect of noise is also discussed, where the poles (or resonances) are difficult to locate.

Abstract: For a class of differentiable dynamical systems (called Axiom A systems) it has been shown by Pollicott and the author that correlation functions have Fourier transforms which are meromorphic in a strip. The poles (or resonances) are, however, not easy to locate. This note reviews the results which are known and discusses a simple model where the position of resonances can be estimated. The effect of noise is also discussed.

••

TL;DR: In this article, photochemical isomerization in stilbene and diphenyl butadiene has been used as a model for activated barrier crossing, and the qualitative features predicted by Kramers theory are observed.

Abstract: Photochemical isomerization in stilbene and diphenyl butadiene has been studied as a model for activated barrier crossing. Experiments have been carried out from isolated molecule conditions up to 3000 atm pressure in solution-phase samples. The qualitative features predicted by Kramers theory are observed. The system undergoes a transition from energy-controlled to diffusion-controlled behavior in the high-pressure gas phase. The influences of multidimensionality, intramolecular vibrational relaxation, and frequency dependent friction are discussed.

••

TL;DR: In this article, the local bifurcation structure of a heteroclinic loop in the Lorenz equations has been analyzed and the existence of a particular loop at one point in a two-dimensional parameter space (a “T point”) has been shown to imply the presence of a line of heteroclineic loops and a logarithmic spiral of homoclinical orbits.

Abstract: The local bifurcation structure of a heteroclinic bifurcation which has been observed in the Lorenz equations is analyzed. The existence of a particular heteroclinic loop at one point in a two-dimensional parameter space (a “T point”) implies the existence of a line of heteroclinic loops and a logarithmic spiral of homoclinic orbits, as well as countably many other topologically more complicatedT points in a small neighborhood in parameter space.

••

TL;DR: In this article, the symmetry group of the finite lattice partition function for the general ℤn Baxter model is determined. But the symmetry groups are not known for the special case of the Belavin parametrization.

Abstract: The ℤn Baxter model is an exactly solvable lattice model in the special case of the Belavin parametrization. For this parametrization we calculate the partition function,κ, in an antiferromagnetic region and the order parameter in a ferromagnetic region. We find that the order parameter is expressible in terms of a modular function of leveln which forn = 2 is the Onsager-Yang-Baxter result. In addition we determine the symmetry group of the finite lattice partition function for the general ℤn Baxter model.

••

TL;DR: A short review of the quantum statistical Monte Carlo method based on the equivalence theorem that d-dimensional quantum systems are mapped onto (d+1)-dimensional classical systems is given in this article.

Abstract: A short review is given concerning the quantum statistical Monte Carlo method based on the equivalence theorem(1) thatd-dimensional quantum systems are mapped onto (d+1)-dimensional classical systems. The convergence property of this approximate tansformation is discussed in detail. Some applications of this geneal appoach to quantum spin systems are reviewed. A new Monte Carlo method, “thermo field Monte Carlo method,” is presented, which is an extension of the projection Monte Carlo method at zero temperature to that at finite temperatures.

••

TL;DR: In this article, the statistical error in the ground state energy as calculated by Green's Function Monte Carlo (GFMC) is analyzed and a simple approximate formula is derived which relates the error to the number of steps of the random walk, the variational energy of the trial function, and the time step of the Random Walk.

Abstract: The statistical error in the ground state energy as calculated by Green's Function Monte Carlo (GFMC) is analyzed and a simple approximate formula is derived which relates the error to the number of steps of the random walk, the variational energy of the trial function, and the time step of the random walk. Using this formula it is argued that as the thermodynamic limit is approached withN identical molecules, the computer time needed to reach a given error per molecule increases asN
h
where 0.5

**••**

TL;DR: In this article, the propagation of phonons in one-dimensional quasicrystals is investigated using the projection method which has been recently proposed to generate almost periodic tilings of the line.

Abstract: The propagation of phonons in one-dimensional quasicrystals is investigated. We use the projection method which has been recently proposed to generate almost periodic tilings of the line. We define a natural Laplace operator on these structures, which models phonon (and also tight-binding electron) propagation. The selfsimilarity properties of the spectrum are discussed, as well as some characteristic features of the eigenstates, which are neither extended nor localized. The long-wavelength limit is examined in more detail; it is argued that one is the lower critical dimension for this type of models.

**••**

TL;DR: Using a new microcanonical algorithm efficiently vectorized on a Cray XMP, this work reaches a simulation speed of 1.5 nsec per update of one spin, three times faster than the best previous method known to us.

Abstract: Using a new microcanonical algorithm efficiently vectorized on a Cray XMP, we reach a simulation speed of 1.5 nsec per update of one spin, three times faster than the best previous method known to us. Data for the nonlinear relaxation with conserved energy are presented for the two-dimensional Ising model.

**••**

TL;DR: In this paper, a general formalism is developed to statistically characterize the microstructure of porous and other composite media composed of inclusions (particles) distributed throughout a matrix phase (which, in the case of porous media, is the void phase).

Abstract: A general formalism is developed to statistically characterize the microstructure of porous and other composite media composed of inclusions (particles) distributed throughout a matrix phase (which, in the case of porous media, is the void phase). This is accomplished by introducing a new and generaln-point distribution functionH
n and by deriving two series representations of it in terms of the probability density functions that characterize the configuration of particles; quantities that, in principle, are known for the ensemble under consideration. In the special case of an equilibrium ensemble, these two equivalent but topologically different series for theH
n are generalizations of the Kirkwood-Salsburg and Mayer hierarchies of liquid-state theory for a special mixture of particles described in the text. This methodology provides a means of calculating any class of correlation functions that have arisen in rigorous bounds on transport properties (e.g., conductivity and fluid permeability) and mechanical properties (e.g., elastic moduli) for nontrivial models of two-phase disordered media. Asymptotic and bounding properties of the general functionH
n are described. To illustrate the use of the formalism, some new results are presented for theH
n and it is shown how such information is employed to compute bounds on bulk properties for models of fully penetrable (i.e., randomly centered) spheres, totally impenetrable spheres, and spheres distributed with arbitrary degree of impenetrability. Among other results, bounds are computed on the fluid permeability, for assemblages of impenetrable as well as penetrable spheres, with heretofore unattained accuracy.

**••**

TL;DR: In this paper, the role of the dynamic solvent friction in influencing the rates of chemical reactions in solution is described. Features considered include (a) the bias of the reaction coordinate toward a direction of lesser friction in the diffusive limit, (b) the importance of frequency-dependent friction in atom transfers, tunneling reactions and isomerizations, and (c) the dynamic nonequilibrium solvation in charge transfers which leads to a polar solvent molecule reorientation time dependence for the rate.

Abstract: The role of the dynamic solvent friction in influencing the rates of chemical reactions in solution is described. Features considered include (a) the bias of the reaction coordinate toward a direction of lesser friction in the diffusive limit, (b) the importance of frequency-dependent friction in atom transfers, tunneling reactions and isomerizations, (c) the dynamic nonequilibrium solvation in charge transfers which leads to a polar solvent molecule reorientation time dependence for the rate, and (d)the importance of internal degrees of freedom in the location of the Kramers turnover for isomerizations.

**••**

TL;DR: The variational modified-hypernetted-chain (VMHNC) theory, based on the approximation of universality of the bridge functions, is reformulated in this article.

Abstract: The variational modified-hypernetted-chain (VMHNC) theory, based on the approximation of universality of the bridge functions, is reformulated. The new formulation includes recent calculations by Lado and by Lado, Foiles, and Ashcroft, as two stages in a systematic approach which is analyzed. A variational iterative procedure for solving the exact (diagrammatic) equations for the fluid structure which is formally identical to the VMHNC is described, featuring the theory of simple classical fluids as a one-iteration theory. An accurate method for calculating the pair structure for a given potential and for inverting structure factor data in order to obtain the potentialand the thermodynamic functions, follows from our analysis.

**••**

TL;DR: In this paper, the authors review the feasibility of quantitative trajectory studies and the fact that these studies have yet to be fully exploited in the development of approximate theories of activated processes, and the possibilities and consequences of nonadiabatic electronic transitions in affecting this dynamics is also considered.

Abstract: This paper reviews the reactive flux correlation function approach to studying the classical dynamics of activated processes in liquids. The possibilities and consequences of nonadiabatic electronic transitions in affecting this dynamics is also considered. We emphasize the feasibility of quantitative trajectory studies and the fact that these studies have yet to be fully exploited in the development of approximate theories of activated processes.

**••**

TL;DR: In this article, the critical properties of a diffusive system with a single conserved density subject to a constant uniform external field were studied. And a fixed point stable below d ≥ 5 was found to govern the critical behavior.

Abstract: We use a field theoretic renormalization group method to study the critical properties of a diffusive system with a single conserved density subject to a constant uniform external field. A fixed point stable belowd
c=5 is found to govern the critical behavior. Scaling forms of density correlation functions are derived and critical exponents are obtained to all orders in ɛ=5−d. Spatial correlations are found to be very anisotropic with elongated correlations along the external field. Long wavelength transverse fluctuations are suppressed completely to yield mean field transverse exponents.

**••**

TL;DR: In this article, the relation between the underlying dynamics of randomly evolv ing systems and the extrema statistics for such systems is investigated, and independent processes, Fokker-Planck processes and Levy processes are considered.

Abstract: We investigate the relation between the underlying dynamics of randomly evolv ing systems and the extrema statistics for such systems. Independent processes, Fokker-Planck processes and Levy processes are considered.

**••**

TL;DR: In this article, the authors studied the transmission problem associated with the non-linear Schrodinger equation with a random potential and showed that for almost every realization of the medium the rate of transmission vanishes when increasing the size of medium; however, whereas the rate decays exponentially in the linear regime, it decays polynomially in the nonlinear one.

Abstract: This is the first study of one of the transmission problems associate to the non-linear Schrodinger equation with a random potential. We show that for almost every realization of the medium the rate of transmission vanishes when increasing the size of the medium; however, whereas it decays exponentially in the linear regime, it decays polynomially in the nonlinear one.

**••**

TL;DR: In this article, a high-dimensional chaotic attractor in an infinite-dimensional phase space is examined for the purpose of studying the relationships between the physical processes occurring in the real space and the characteristics of highdimensional attractors in the phase space.

Abstract: The nature of a very high-dimensional chaotic attractor in an infinite-dimensional phase space is examined for the purpose of studying the relationships between the physical processes occurring in the real space and the characteristics of high-dimensional attractor in the phase space. We introduce two complementary bases from which the attractor is observed, one the Lyapunov basis composed of the Lyapunov vectors and the another the Fourier basis composed of the Fourier modes. We introduce the “exterior” subspaces on the basis of the Lyapunov vectors and observe the chaotic motion projected onto these exteriors. It is shown that a certain statistical property of the projected motion changes markedly as the exterior subspace “goes out” of the attractor. The origin of such a phenomenon is attributed to more fundamental features of our attractor, which become manifest when the attractor is observed from the Lyapunov basis. A counterpart of the phenomenon can be observed also on the Fourier basis because there is a statistical one-to-one correspondence between the Lyapunov vectors and the Fourier modes. In particular, a statistical property of the high-pass filtered time series reflects clearly the difference between the interior and the exterior of the attractor.

**••**

TL;DR: It is proved that the integrated density of states ρ(λ) for a potential Wω =Vper +Vω has Lifshitz tails where Vper is a periodic potential with reflection symmetry and Vω is a random potential.

Abstract: W. Kirsch I and B. Simon 2 Received November 27, 1984 We prove that the integrated density of states p(2) for a potential W,~ = Vper + V,o has Lifshitz tails where Vp~ r is a periodic potential with reflec- tion symmetry and V,o is a random potential, e.g., of the form V~ = Z q,((.o) f(x - i). KEY WORDS: Lifshitz tails; Anderson model; Dirichlet-Neumann bracketing.