Showing papers in "Journal of Statistical Physics in 1974"
TL;DR: The continuous-time random walk of Montroll and Weiss has a complete separation of time (how long a walker will remain at a site) and space (how far a walkers will jump when it leaves a site). The time part is completely described by a pausing time distributionψ(t) as mentioned in this paper.
Abstract: The continuous-time random walk of Montroll and Weiss has a complete separation of time (how long a walker will remain at a site) and space (how far a walker will jump when it leaves a site). The time part is completely described by a pausing time distributionψ(t). This paper relates the asymptotic time behavior of the probability of being at sitel at timet to the asymptotic behavior ofψ(t). Two classes of behavior are discussed in detail. The first is the familiar Gaussian diffusion packet which occurs, in general, when at least the first two moments ofψ(t) exist; the other occurs whenψ(t) falls off so slowly that all of its moments are infinite. Other types of possible behavior are mentioned. The relationship of this work to solutions of a generalized master equation and to transient photocurrents in certain amorphous semiconductors and organic materials is discussed.
353 citations
TL;DR: Two simple models of random systems made up out of a finite collection of elements are considered: one generalizing the notion of “Gibbs ensemble” abstracted from statistical physics; the other, “Markov fields” derived from the idea of a Markov chain.
Abstract: This paper concerns random systems made up out of a finite collection of elements. We are interested in how a fixed structure of interactions reflects on the assignment of probabilities to overall states. In particular, we consider two simple models of random systems: one generalizing the notion of “Gibbs ensemble” abstracted from statistical physics; the other, “Markov fields” derived from the idea of a Markov chain. We give background for these two types, review proofs that they are in fact identical for systems with nonzero probabilities, and explore the new behavior that arises with constraints. Finally, we discuss unsolved problems and make suggestions for further work.
192 citations
TL;DR: In this article, the authors analyzed the far from equilibrium steady states of a simple nonlinear chemical system and showed that the nonlinearity introduces an instability which causes a transition analogous to a thermodynamic second-order phase transition.
Abstract: The far from equilibrium steady states of a simple nonlinear chemical system are analyzed. A standard macroscopic analysis shows that the nonlinearity introduces an instability which causes a transition analogous to a thermodynamic second-order phase transition. Fluctuations are introduced into this model through a stochastic master equation approach. The solution of this master equation in the steady state reveals that the system goes into a more ordered state above the transition point. An analogy is drawn with the nonequilibrium phase transition occurring in the laser at threshold.
71 citations
TL;DR: In this article, Monte Carlo calculations were carried out for a two-dimensional Ising model of a binary alloy with nearest-neighbor attractive interactions between like atoms, and the pair correlation observed had the form of an exponentially damped cosine function with parameters varying as the one-sixth power of the time.
Abstract: Monte Carlo calculations were carried out for a two-dimensional Ising model of a binary alloy with nearest-neighbor attractive interactions between like atoms. The pair correlation observed had the form of an exponentially damped cosine function with parameters varying as the one-sixth power of the time.
61 citations
TL;DR: In this article, it was shown that in the case of quantum mechanical systems associated with finite-dimensional Hilbert spaces, the proposition is completely determined by the premises of the quantum theory itself.
Abstract: Many physicists take it for granted that their theories can be either refuted or verified by comparison with experimental data. In order to evaluate such data, however, one must employ statistical estimation and inference methods which, unfortunately, always involve an ad hoc proposition. The nature of the latter depends upon the statistical method adopted; in the Bayesian approach, for example, one must usesome Lebesgue measure in the “set of all possible distributions.” The ad hoc proposition has usually nothing in common with the physical theory in question, thus subjecting its verification (or refutation) to further doubt. This paper points out one notable exception to this rule. It turns out that in the case of the quantum mechanical systems associated with finite-dimensional Hilbert spaces the proposition is completely determined by the premises of the quantum theory itself.
54 citations
TL;DR: In this paper, asymptotic distributions of the Montroll-Weiss equation for the continuous-time random walk are investigated for long times and it is shown that, for a certain subclass of the hopping waiting time distributions belonging to the domain of attraction of stable distributions, these distributions are of stable form.
Abstract: Asymptotic distributions of the Montroll-Weiss equation for the continuous-time random walk are investigated for long times. It is shown that, for a certain subclass of the hopping waiting time distributions belonging to the domain of attraction of stable distributions, these asymptotic distributions are of stable form. This indicates that the realm of applicability of the diffusion equation is limited. The Montroll-Weiss equation is rederived to include the influence of the initial waiting interval and the role of the stable distributions in physical problems is briefly discussed.
45 citations
TL;DR: In this paper, a dynamical coupling between the time-dependent averages and the fluctuations must be accounted for by a procedure which leads to a renormalization of the nonlinear transport equation.
Abstract: We are concerned here with the problems encountered in the derivation of nonlinear transport equations from a correspondingly nonlinear Langevin equation. A dynamical coupling between the time-dependent averages and the fluctuations must be accounted for by a procedure which leads to a renormalization of the nonlinear transport equation. Generalizing the familiar phenomenological approach to Brownian motion to nonlinear dynamics, we illustrate how the problem arises and show how the fluctuation renormalization can be obtained exactly by a formal procedure or approximately by more tractable methods.
38 citations
TL;DR: In this article, the authors studied the surface specific heat and surface susceptibilities of ind-dimensional ferromagnetic spherical models on hypercubic lattices with free surfaces and derived a scaling form for the surface free energy.
Abstract: Critical phenomena ind-dimensional ferromagnetic spherical models on hypercubic lattices with free surfaces are studied. The surface specific heat and surface susceptibilities are obtained. The exponents characterizing the divergence of these surface quantities at the bulk critical temperature are found to satisfy recently proposed scaling relations. The variation of the susceptibility with distance from the surface is also discussed. The author's recent scaling theory for surface properties is investigated in detail, and found to give an exact representation for the free energy of a three-dimensional spherical model of finite thickness in finite bulk and surface magnetic fields. A scaling form for the surface free energy is derived.
26 citations
TL;DR: In this paper, the first two moments of a planar walk were calculated in which the step lengths depend on the direction of motion, and the model was suggested by experiments on the locomotion of biological cells.
Abstract: We calculate asymptotic values of the first two moments of a planar walk in which the step lengths depend on the direction of motion. The model is suggested by experiments on the locomotion of biological cells. Internally induced persistence due to nonuniform turn angle distributions is also accounted for.
25 citations
TL;DR: In this paper, the free energy correct to second order in the interaction strength is utilized for calculation of other thermodynamic properties of the lattice-gas model, which can be brought into agreement with those of real water.
Abstract: The adaptation of the lattice-gas model to embody features possessed by water is further explored. On the basis of Martin's functional derivative formulation of Ising problems, a perturbation scheme is developed which allows calculation of the free energy to any desired order in the interaction potential at fixed density. The free energy correct to second order in the interaction strength is utilized here for calculation of other thermodynamic properties of the model. With reasonable choices of values of the interaction parameters these thermodynamic properties of the model can be brought into agreement with those of real water.
23 citations
TL;DR: In this paper, it was shown that the mean square fluctuation computed from a master equation in the space of internal states of the reacting species is identical to that calculated from Einstein's fluctuation formula.
Abstract: We study fluctuations around nonequilibrium steady states of some model nonlinear chemical systems. A previous result of Nicolis and Prigogine states that the mean square fluctuation computed from a master equation in the space of internal states of the reacting species is identical to that calculated from Einstein's fluctuation formula. Our analysis of fluctuations based on that master equation leads with the assumption of local equilibrium to a result identical to that obtained from a master equation for the total concentration of the reacting species, which is different from the equilibrium (Einstein relation) result. Nicolis and Prigogine approximated one term in their master equation, and a discussion of this approximation is presented. The master equation without this approximation yields at equilibrium the result expected on the basis of Einstein's formula.
TL;DR: In this article, the lattice-gas approach is generalized to incorporate features of the configurational problem posed by the randomly hydrogen-bonded "gel" model for liquid water.
Abstract: The lattice-gas approach is generalized to incorporate features of the configurational problem posed by the randomly hydrogen-bonded “gel” model for liquid water. Because it possesses sublattices characterized by tetrahedral angles associated with triads of sites, a body-centered cubic (bcc) lattice is used. Each water molecule is allowed 12 orientations with respect to the bcc lattice. When two nearest neighbors have relative orientations which permit hydrogen bonding, they are assigned a hydrogen bond energy. When hydrogen bonding is not permitted the pair is assigned one of two weaker interaction energies. Like the simple lattice gas, this model displays a “vapor-liquid” phase transition. The critical site density proves to be less than 1/2. The model should also exhibit a transition to a solid phase as a result of the possibility of complete hydrogen bonding associated with exclusive occupation of one sublattice. Excellent agreement is obtained with the observed temperature dependence of the second virial coefficient. The agreement in the case of the third virial coefficient is poor, however. The mean field approximation is shown to be inadequate for quantitative description of the vapor-liquid transition and the properties of the liquid phase.
TL;DR: In this paper, a general method for evaluating collision integrals in the linearized Boltzmann transport equation is described. But the method is not dependent on scattering cross-section variables.
Abstract: We develop and describe a general method for evaluating collision integrals in the linearized Boltzmann transport equation which eliminates the necessity to repeat similar integration steps for each force law. Integrations not dependent on scattering cross-section variables have been carried out once and for all. The two mathematical innovations which facilitate these general integrations are (i) the development of an expansion of the Burnett functionXNML(x+y) into products of Burnett functions of argument x with other functions; and (ii) the use of representations of the full rotation group to transform from space-fixed axes to axes aligned with the relative velocity vector of colliding atoms. The relations so derived allow rapid evaluation of the collision integral from a knowledge of the scattering cross section.
TL;DR: The first ten terms of the high-temperature expansion of the susceptibility of the single-band Hubbard model in the strong correlation limit are obtained for arbitrary electron density as discussed by the authors, and the series is analyzed by ratio methods and Pade approximants.
Abstract: The first ten terms of the high-temperature expansion of the susceptibility of the single-band Hubbard model in the strong correlation limit are obtained for arbitrary electron density. The series is analyzed by ratio methods and Pade approximants. A critical temperature is found for 0.2 ⩽ρ ⩽ 0.8; forρ > 1 further terms in the series are required.
TL;DR: The physical aspects of the statistical theory of the energy levels of complex physical systems and their relation to the mathematical theory of random matrices are discussed in this article, where different kinds of ensembles are discussed.
Abstract: In this paper the physical aspects of the statistical theory of the energy levels of complex physical systems and their relation to the mathematical theory of random matrices are discussed. After a preliminary introduction we summarize the symmetry properties of physical systems. Different kinds of ensembles are then discussed. This includes the Gaussian, orthogonal, and unitary ensembles. The problem of eigenvalue-eigenvector distributions of the Gaussian ensemble is then discussed, followed by a discussion on the distribution of the widths. In the appendices we discuss the symplectic group and quaternions, and the Gaussian ensemble in detail.
TL;DR: In this article, a simulation of a one-dimensional binary alloy system is presented, where fluctuations can be observed on an atomic time scale, and it is shown that thermal fluctuations may play an important role in coarsening in real alloys.
Abstract: Recently Cahn's generalized diffusion equation theory of spinodal decomposition in binary alloys has been modified to include the effects of thermal fluctuations. This paper reports studies of a one-dimensional binary alloy system in which fluctuations can be observed on an atomic time scale. The system, a computer-simulated linear chain binary alloy which evolves from an initially random atomic arrangement through interchange of unlike nearest neighbors via the Monte Carlo technique, rapidly develops grains of two different concentrations and then slowly experiences coarsening. A numerical solution of the diffusion equation successfully predicts the development of grain structure, but only predicts coarsening to the extent present as fluctuations in the initial atomic arrangement. The simulated alloy coarsens further than the prediction of the diffusion equation because of thermal fluctuations which develop naturally during its evolution. This suggests that thermal fluctuations may play an important role in coarsening in real alloys.
TL;DR: In this paper, the Ising model for an alloy with an arbitrary number of components was studied and an approximation to the Bethe and Peierls model was developed for the case when the concentration of one of the components is unity.
Abstract: We study the Ising model for an alloy with an arbitrary number of components. We develop an approximation which reduces to that of Bethe and Peierls when the concentration of one of the components is unity. We investigate within this approximation the dependence of the various thermodynamic quantities, in particularTc, on the composition of the alloy and the magnetic properties of its constituents. Comparison with the only exact calculation available, that of F. T. Leeet al., for a linear chain, shows extremely satisfactory agreement.
TL;DR: The relation between information and entropy for measurements of thermo-dynamic parameters is considered in this paper, where the reduction in entropy that an observer can obtain in a system described by a fluctuating thermodynamic parameter is shown to be smaller than the information possessed by the observer.
Abstract: The relation between information and entropy for measurements of thermo-dynamic parameters is considered. The reduction in entropy that an observer can obtain in a system described by a fluctuating thermodynamic parameter is shown to be smaller than the information possessed by the observer. The information transfer and the entropy production due to the irreversible interaction of the observed system with the measuring instrument are compared.
TL;DR: In this paper, the memory function and derivation of a kinetic equation for one-body phase space correlation functions are discussed. But the analysis is restricted to the case of a single-body additive projection operator and the Liouville operator.
Abstract: This paper is concerned with the calculation of the memory function and derivation of a kinetic equation for one-body phase space correlation functions. The theory uses a one-body additive projection operator and a division of the Liouville operator with an unperturbed part that describes dressed particles. Binary collisions are neglected, for the theory aims at describing the screening and backflow effects of a type contained in the plasma kinetic theory of Balescu and Lenard. We obtain an explicit kinetic equation which is an improvement of these theories for the plasma case, and involves the exact equilibrium pair and triplet distributions. The equation also describes systems with strong short-range forces and shows how the screening effects occur in this case as well. The unifying function is the direct correlation function. The theory is meant to provide understanding for a more complete theory of fluids where a proper account is given of close collisions.
TL;DR: In this paper, the influence of initial deviations from bath equilibrium on the motion of a Brownian particle in a harmonic chain is investigated by exact calculation, and it is shown that these initial condition effects, which are excluded by convention in standard projection operator treatments of relaxation processes, are relatively long-lived.
Abstract: The influence of initial deviations from bath equilibrium on the motion of a Brownian particle in a harmonic chain is investigated by exact calculation. These initial condition effects, which are excluded by convention in standard projection operator treatments of relaxation processes, are found to be relatively long-lived, contrary to usual assumption. For weak, localized initial deviations from bath equilibrium these effects on the motion are small in magnitude and may be accounted for by a modified initial condition on the particle velocity. For initial deviations involving many bath particles these effects are more substantial and retention of their time dependence in the particle equation of motion is generally required.
TL;DR: An axiomatic characterization of an information-theoretic quantity associated with a pair of probability distributions having the same number of elements has been given, which leads to Kullback's information and Kerridge's inaccuracy concepts.
Abstract: An axiomatic characterization of an information-theoretic quantity associated with a pair of probability distributions having the same number of elements has been given. This quantity, under additional suitable conditions, leads to Kullback's information and Kerridge's inaccuracy concepts. By modifying one of the axioms, the two-parameter generalization of these is obtained.
TL;DR: In this paper, the screening approximation of Ferrell and Scalapino @-1 expansion is tested in the exactly soluble zero-dimensional case and the expansion is carried to fifth order in n- 1, where, for n = 2, it appears to start diverging.
Abstract: The screening approximation of Ferrell and Scalapino @-1 expansion) is tested in the exactly soluble zero-dimensional case. The expansion is carried to fifth order in n- 1, where, for n = 2, it appears to start diverging. For n = 1 divergence sets in at the second-order term. The "self-consistent" screening approximation of Bray and Rickayzen converges more rapidly but is more difficult to apply in higher dimensionalities. The usefulness of the zero-dimensional case for checking the enumeration of the Feynman graphs which appear in third and higher order is emphasized.
TL;DR: In this paper, a probabilistic model for tracer transport in multiphase spatially inhomogeneous transport (plug-flow) systems is presented and the properties of the trajectories are completely described by a two-component Markov process with absorbing boundaries.
Abstract: A probabilistic model describing tracer transport in multiphase spatially inhomogeneous transport (plug-flow) systems is presented. The properties of the trajectories are completely described by a two-component Markov process with absorbing boundaries. The first component is continuous, the second discrete. Infinitesimal conditions are given. Probabilities associated with the process are derived.
TL;DR: In this article, the transition matrix method based on the theory of Markov chains is used to classify the eigenvalues of a transition matrix according to the various representations of the appropriate group.
Abstract: A restricted walk of orderr on a lattice is defined as a random walk in which polygons withr vertices or less are excluded. A study of restricted walks for increasingr provides an understanding of how the transition in properties is effected from random to self-avoiding walks which is important in our understanding of the excluded volume effect in polymers and in the study of many other problems. Here the properties of restricted walks are studied by the transition matrix method based on the theory of Markov chains. A group theoretical method is used to reduce the transition matrix governing the walk in a systematic manner and to classify the eigenvalues of the transition matrix according to the various representations of the appropriate group. It is shown that only those eigenvalues corresponding to two particular representations of the group contribute to the correlations among the steps of the walk. The distributions of eigenvalues for walks of various ordersr on the two-dimensional triangular lattice and the three-dimensional face-centered cubic lattice are presented, and they are shown to have some remarkable features.
TL;DR: In this article, exact analytic results for symmetric, non-nearest-neighbor random walks in one-dimensional finite and semi-infinite lattices are presented.
Abstract: Exact analytic results for symmetric, nonnearest-neighbor random walks in one-dimensional finite and semiinfinite lattices are presented. Random walks with exponentially distributed step lengths are considered such that variation of a single parameter permits one to cover the whole range of step lengths from nearest-neighbor transitions to steps of aribtrary length. The generating functions for such lattices are derived and used to calculate a number of moment properties (mean first passage times, dispersion in the mean recurrence time). Since explicit expressions for the generating functions for these walks are obtained, additional moment properties can readily be calculated. The results found here for a finite system are compared to results found previously for a system with periodic boundary conditions. Two different semiinfinite systems are also considered.
TL;DR: The Gibbs neg-entropy -ηG=∫ II ln II is compared to the Shannon negentropy ηs=∑p Inp, and the coarse-grained density is II, whilep is a probability sequence.
Abstract: The Gibbs neg-entropy -ηG=∫ II ln II is compared to the Shannon negentropy ηs=∑p Inp. The coarse-grained density is II, whilep is a probability sequence. Both objects are defined over partitions of the energy shell within a set-theoretic framework. The dissimilarity of these functionals is exhibited throughηG vs.GηS curves. A positive information interpretation of ηG is given referring it to the maximum information contained in the solution to the Liouville equation. The physical relevance ofηG over ηS in classical physics is argued. In quantum mechanics, the fine-grained Shannon entropy remains relevant to the uncertainty principle, while the coarsegrained densities maintain their properties as in the classical case.
TL;DR: In this paper, the generalized Langevin equation for Brownian motion is extended to the case of a Brownian particle with arbitrary shape and withslip boundary condition, which naturally arises in a treatment of the problem from first principles.
Abstract: Previous work by Hauge and Martin-Lof discussing the generalized Langevin equation for Brownian motion as a contraction from the more fundamental but still phenomenological description of a particle immersed in an incompressible fluid governed by fluctuating hydrodynamics with stick boundary condition is extended to the case of a Brownian particle with arbitrary shape and withslip boundary condition. The motivation for this extension is the fact that the latter condition naturally arises in a treatment of the problem from first principles.
TL;DR: Two general measures associated with two distributions of a discrete random variate are characterized, one of these measures is logarithmic, while the other contains powers of variables.
Abstract: Starting with an additive property for distributions of two statistically independent random variates in terms of different sum functions, we have characterized two general measures associated with two distributions of a discrete random variate. One of these measures is logarithmic, while the other contains powers of variables. An interesting aspect is that under suitable additional boundary conditions the logarithmic measure leads to measure of information (directed divergence) studied by Kullback and measure of inaccuracy studied by Kerridge, while the other solution leads to their parametric generalizations.
TL;DR: In this paper, the canonical statistical sum for the Breit-Darwin plasma is investigated by means of a generalized van Kampen cellular method, and the pair correlation function is derived.
Abstract: The canonical statistical sum for the Breit-Darwin plasma is investigated by means of a generalized van Kampen cellular method. In particular, the pair correlation function is derived. This function agrees with that previously obtained by Trubnikov from the approximate closure of the BBGKY hierarchy. The method developed in this paper can be used for the description of other systems in which the velocity-dependent forces are pairwise.
NEC1
TL;DR: The least irreversible decay rate is a state function characteristic of macrostructure on a coarse-grained time scale as mentioned in this paper, which is consistent with the assertion for the minimum K-entropy which has been argued to apply to the nonequilibrium asymptotic state.
Abstract: Probabilistic kinetics following the Pauli master equation without microscopic reversibility determines an asymptotic structure of macroprocesses in a coarse-grained phase space of many degrees of freedom The structure, which is asymptotically realized, minimizes its irreversible decay rate among various candidates This least irreversible decay rate is consistent with the assertion for the minimum K-entropy which has been argued to apply to the nonequilibrium asymptotic state The irreversible decay rate is a state function characteristic of macrostructure on a coarse-grained time scale Macrofluctuations, which always appear around the asymptote as fluctuations of the state function, do not obey the central limit theorem, implying that fluctuations whose characteristic times are not less than some finite value are never excluded