Showing papers in "Journal of Statistical Physics in 1973"
TL;DR: In this article, exact generalized Langevin equations are derived for arbitrarily nonlinear systems interacting with specially chosen heat baths, and an example is presented in which the Langevin equation is nonlinear but approximately Markovian.
Abstract: Exact generalized Langevin equations are derived for arbitrarily nonlinear systems interacting with specially chosen heat baths. An example is displayed in which the Langevin equation is nonlinear but approximately Markovian.
1,187 citations
TL;DR: In this article, the evolution of the variable along the most probable path and fluctuation properties around the path are considered from a general point of view with an emphasis on the relation of nonlinearity of evolution and the associated fluctuation.
Abstract: Assuming that a macrovariable follows a Markovian process, the extensive property of its probability distribution is proved to propagate. This is a generalization of the Gaussian properties of the equilibrium distribution to nonequilibrium nonstationary processes. It is basically a WKB-like asymptotic evaluation in the inverse of the size of the macrosystem. Evolution of the variable along the most probable path and fluctuation properties around the path are considered from a general point of view with an emphasis on the relation of nonlinearity of evolution and the associated fluctuation. Anomalous behavior of the fluctuation is discussed in connection with unstable, critical, or marginal states. A general treatment is given for the asymptotic properties of relaxation eigenmodes.
621 citations
TL;DR: In this paper, the existence of equilibrium points in TV-person non-cooperative games in normal form is generalized to the case when there is a continuum of players endowed with a nonatomic measure.
Abstract: The Nash theorem on the existence of equilibrium points inTV-person non-cooperative games in normal form is generalized to the case when there is a continuum of players endowed with a nonatomic measure. The mathematical tools are those used in mathematical economics, in particular, markets with a continuum of traders. The main result shows that under a restriction on the payoff functions there exists an equilibrium in pure strategies.
573 citations
TL;DR: In this paper, an equivalence between generalized master equations and continuous-time random walks is established by means of an explicit relationship between the pausing time distribution in the theory of continuous time random walks and the memory in the kernel of a generalized master equation.
Abstract: An equivalence is established between generalized master equations and continuous-time random walks by means of an explicit relationship betweenψ(t), which is the pausing time distribution in the theory of continuous-time random walks, andφ(t), which represents the memory in the kernel of a generalized master equation. The result of Bedeaux, Lakatos-Lindenburg, and Shuler concerning the equivalence of the Markovian master equation and a continuous-time random walk with an exponential distribution forψ(t) is recovered immediately. Some explicit examples ofφ(t) andψ(t) are also presented, including one which leads to the equation of telegraphy.
351 citations
TL;DR: In this paper, the general study of random walks on a lattice is developed further with emphasis on continuous-time walks with an asymmetric bias, characterized by random pauses between jumps, with a common pausing time distributionψ(t).
Abstract: The general study of random walks on a lattice is developed further with emphasis on continuous-time walks with an asymmetric bias. Continuous time walks are characterized by random pauses between jumps, with a common pausing time distributionψ(t). An analytic solution in the form of an inverse Laplace transform for P(l, t), the probability of a walker being atl at timet if it started atlo att=0, is obtained in the presence of completely absorbing boundaries. Numerical results for P(l, t) are presented for characteristically different ψ(t), including one which leads to a non-Gaussian behavior for P(l, t) even for larget. Asymptotic results are obtained for the number of surviving walkers and the mean 〈l〉 showing the effect of the absorption at the boundary.
313 citations
TL;DR: In this paper, the Langevin equation describing Brownian motion is considered as a contraction from the more fundamental, but still phenomenological, description of an incompressible fluid governed by fluctuating hydrodynamics in which a Brownian particle with stick boundary condition is immersed.
Abstract: The Langevin equation describing Brownian motion is considered as a contraction from the more fundamental, but still phenomenological, description of an incompressible fluid governed by fluctuating hydrodynamics in which a Brownian particle with stick boundary condition is immersed. First, the derivation of fluctuating hydrodynamics is reconsidered to clarify certain ambiguities as to the treatment of boundaries. Subsequently the contraction is carried out. Since Brownian particles of arbitrary shape are considered, rotations and translations are in general coupled. The symmetry of the 6×6 friction tensorγ
ij
(t) is proved for arbitrary shape without appeal to microscopic arguments. This symmetry is then used to prove that the fluctuation-dissipation theorem on the contracted level (nonwhite noise in general) follows from the corresponding statement on the level of fluctuating hydrodynamics (white noise). The condition under which the contracted description reduces to the classical Langevin equation is given, and the connection between our theory and related work is discussed.
265 citations
TL;DR: In this paper, the Monte Carlo sampling technique is used to calculate the equilibrium thermodynamics of fluids and magnets, and the questions of convergence and accuracy of this method can be understood in terms of the dynamics of the appropriate stochastic model.
Abstract: By means of the Monte Carlo sampling technique the equilibrium thermodynamics of fluids and magnets can be calculated numerically. We show that the questions of convergence and accuracy of this method can be understood in terms of the dynamics of the appropriate stochastic model. Also, we discuss to what extent various choices of transition probabilities lead to different dynamic properties of the system. As examples of applications, we consider Ising and Heisenberg spin systems. The numerical results about the dynamic correlation functions are compared to simple approximations taken from the theory of the kinetic Ising model.
214 citations
TL;DR: It is shown that the Kullback measure, unlike the Shannon measure, provides the basis for a consistent theory of information which extends to continuous sample spaces and to nonconstant prior distributions.
Abstract: Two widely used information measures are compared. It is shown that the Kullback measure, unlike the Shannon measure, provides the basis for a consistent theory of information which extends to continuous sample spaces and to nonconstant prior distributions. It is shown that the Kullback measure is a generalization of the Shannon measure, and that the Kullback measure has more reasonable additivity properties than does the Shannon measure. The results lend support to Jaynes's entropy maximization procedure.
91 citations
TL;DR: In this article, the spontaneous staggered polarization of the two-dimensional F-model has been derived exactly and the order parameter of the F model is derived exactly at the critical temperature.
Abstract: The “order parameter” of the two-dimensionalF-model, namely the spontaneous staggered polarizationP
0, is derived exactly. At the critical temperatureP
0 has an essential singularity, bothP
0 and all its derivatives with respect to temperature vanishing.
87 citations
TL;DR: In this article, the authors obtained complex nonlinear integral equations for the two asymptotically degenerate maximum eigenvalues of the transfer matrix of the eight-vertex model.
Abstract: We obtain complex nonlinear integral equations for the two asymptotically degenerate maximum eigenvalues of the transfer matrix of the eight-vertex model. These are exact for a lattice of a finite numberN of columns. Solving the equations recursively gives an expansion of the eigenvalues aboutN = ∞. Thus we can obtain the interfacial tension of the model, as well as rederiving our previous result for the free energy.
74 citations
TL;DR: In this article, Thorne's method for obtaining transport coefficients in a binary rigid-sphere mixture is reexamined and a close look is taken at the way in which the point where the Enskog functionsx ij are evaluated is introduced.
Abstract: Thorne's method for obtaining transport coefficients in a binary rigid-sphere mixture is reexamined. First, a close look is taken at the way in which the point where the Enskog functionsx ij are evaluated is introduced. Second, the calculation of the fluxes in the system and the transport coefficients is given. Thorne's results are found to be correct and independent of the choice of the point where the transfer plane is located. This does not hold true for the diffusion flux. It is shown that a different diffusion force is obtained for each selection and that only those diffusional effects which are of first order in the density are consistent with irreversible thermodynamics.
TL;DR: In this paper, an exact representation for the density-density response function is presented, which is a generalization of the result obtained in the mean field approximation and amounts to replacing the static, effective potential by one which is both wavenumber and frequency-dependent.
Abstract: An exact representation for the density-density response function is presented. This representation is a generalization of the result obtained in the mean field approximation and amounts to replacing the static, effective potential by one which is both wavenumber- and frequency-dependent. This potential possesses both a real and an imaginary part; the latter describes the collisional damping of collective modes. Analyticity and sum rule arguments are used to describe the basic properties of this complex potential. The formalism allows us to write an exact formula for the scattering functionS(k, ω) in which the basic unknown is the collisional damping function. Using a small portion of the recent experimental data on coherent neutron scattering in liquid argon, we are able to calculateS(k, ω) and other quantities of interest and to make comparisons with the rest of the data.
TL;DR: In this article, the form of the coefficientsg1(x) andg2(x), as well as the boundary values [a, b] of the Fokker-Planck equation are tabulated.
Abstract: We have tabulated the form of the coefficientsg1(x) andg2(x) as well as the boundary values [a, b] of the Fokker-Planck equation
$$\frac{{\partial P(x, t)}}{{\partial t}} = - \frac{\partial }{{\partial x}}[g_1 (x)P(x, t)] + \frac{{\partial ^2 }}{{\partial x^2 }}[g_2 (x)P(x, t)],a \leqslant x \leqslant b$$
for which the solution can be written as an eigenfunction expansion in the classical orthogonal polynomials. We also discuss the problem of finding solutions in terms of the discrete classical polynomials for the differential difference equations of stochastic processes.
TL;DR: In this paper, the existence of a phase transition for all sizes of hard hexagons on a triangular lattice is proved by means of the Peierls' argument, which does not imply a phase-transition in the continuous limit.
Abstract: Systems of hard hexagons on a triangular lattice are investigated. The orientation of the hexagons is kept fixed, while the size of the hexagons is varied. The existence of a phase transition is proved for all sizes by means of the Peierls'argument. The proof does not imply a phase transition in the continuous limit.
TL;DR: The Schultz-Mattis-Lieb fermion formulation of the two-dimensional Ising model is simplified by means of longwavelength approximations which become exact in the critical region as discussed by the authors.
Abstract: The Schultz-Mattis-Lieb fermion formulation of the two-dimensional Ising model is simplified by means of long-wavelength approximations which become exact in the critical region. The resulting continuum theory has a Hamiltonian density which is shown to be identical, to within a perfect derivative, to that of free, spinless particles satisfying the one-dimensional Dirac equation. Filling the negative-energy single-particle states of momentumq and massκ gives an integral over the single-particle energies -(θ2+k2)1/2. Becauseκ varies linearly with the temperature, differentiating twice gives Onsager's logarithmic singularity in the specific heat.
TL;DR: In this article, the short-time kinetic equation for hard spheres derived by Lebowitz, Percus, and Sykes is compared with the Enskog equation, and it is shown that, to leading order in the density, they are identical and equivalent to the memory function equation used by Mazenko, Wei, and Yip.
Abstract: The short-time kinetic equation for hard spheres derived by Lebowitz, Percus, and Sykes is compared with the Enskog equation. It is shown that, to leading order in the density, the short-time equation and the Enskog equation are identical and equivalent to the memory function equation used by Mazenko, Wei, and Yip. By using simple properties of the collision integrals, the scattering function calculated from the short-time equation can be related to the scattering function obtained from the Enskog equation: This relationship is exact for all values of the density. We examine the relationship in the short-time limit and in the hydrodynamic limit and argue that the short-time kinetic equation gives a better description of the scattering function than does the Enskog equation.
TL;DR: In this paper, a detailed discussion of the thermodynamic properties of the magnetoelastic Ising model at zero-field conditions is given. But the authors focus on the properties of magnetic phase transition at zero field, which is found to be either first-order or second-order.
Abstract: We examine the soluble magnetoelastic Ising model developed by Baker and Essam and give a detailed discussion of its thermodynamic properties. Particular attention is devoted to the properties of the magnetic phase transition at zero field, which is found to be either first order or second order, depending on whether the experiment is performed at negative or positive pressure.
IBM1
TL;DR: In this paper, the dissipative energy dissipation of a system subject to a slow modulation of external parameters is analyzed and it is shown that the time-integrated dissipation consists of three terms.
Abstract: The dissipative steady state far from equilibrium and subject to a slow modulation of external parameters is analyzed. It is shown that the time-integrated energy dissipation consists of three terms. The first of these is irreversible and consists of the time-integrated dissipation of the sequence of exact steady states defined by the externally controlled parameters traversed during the modulation. The second term is reversible and reflects the fact that the dissipation of the time-dependent modulated system, as calculated in a macroscopic way from ensemble averages, is not the same as the dissipation of a sequence of exact steady states. The third term is also reversible and relates to the ensemble dispersion in changes in stored energy during the modulation. If the system has a single degree of freedom and narrow fluctuations, then these fluctuations can be characterized by an effective temperature TN. The third term can then be shown to be equal toTN dS, whereS is the entropy calculated from the distribution function by the usual definition.
TL;DR: A recent defense of Jaynes' information-theoretical approach to statistical mechanics is rejected, and an earlier critique of this approach is extended as discussed by the authors, and a recent defence of the approach is rejected.
Abstract: A recent defense of Jaynes' information-theoretical approach to statistical mechanics is rejected, and an earlier critique of this approach is extended.
TL;DR: It is concluded that the Shannon information measure should be retained as the basis of information theory.
Abstract: Hobson and Chang recommend that the Kullback information measure replace the Shannon information measure as a basis for information theory. They cite several items in support of their proposal. The items are considered individually and it is shown that they do not in fact constitute sufficient reasons for accepting the Hobson/Chang proposal. It is concluded that the Shannon information measure should be retained as the basis of information theory.
TL;DR: In this paper, the Heisenberg paramagnet in one, two, and three dimensions is analyzed by a second-order Green's function theory similar to that used by Knapp and ter Haar.
Abstract: The Heisenberg paramagnet in one, two, and three dimensions is analyzed by a second-order Green's function theory similar to that used by Knapp and ter Haar. This theory, which incorporates the exact values for the zero, first, and second moments of the relaxation function as boundary conditions, yields results satisfying the rotational symmetry of the paramagnetic region as well as the principle of detailed balance. We find that our predictions for equal time properties in the classical limit are identical with the RPA Green's function theory of Liu as well as the spherical model results of Lax. The quantum limit is analyzed, and our predictions for the 1/T series coefficients for both internal energy and susceptibility are compared with exact results.
TL;DR: In this article, a scalar, classical, and time-independent field theory of the type proposed by Ferrell for the treatment of continuous phase transitions is developed along lines similar to those of many-body theory.
Abstract: We present a formalism of a scalar, classical, and time-independent field theory of the type proposed by Ferrell for the treatment of continuous phase transitions. The formalism is developed along lines similar to those of many-body theory. All physical quantities, e.g., susceptibility, correlation length, and free energy, are expressed as functionals of the two-point time-independent correlation function and the order parameter. This is done both in the ordered and in the disordered phase. We obtain renormalized equations and diagram expansions of all quantities and self-consistent approximation schemes arc presented. It is shown that near the transition temperature, which is defined within the theory, no weak coupling limit exists. The generalization to more complicated field symmetries is straight-forward.
TL;DR: In this paper, a two-dimensional model of polymer chain folding was studied using a grand ensemble and transfer matrix method, and it was shown that the probability densities describing the length of a segment and the correlations among segments can be derived from the probability distributions of the folded chain.
Abstract: A two-dimensional model of polymer chain folding invented by Zwanzig and Lauritzen is here studied using a grand ensemble and transfer matrix method. Due to the character of the model, there are no extensive parameters in the grand ensemble and the dispersion in system size is large, raising doubts about the validity and usefulness of the ensemble. We find it possible to define a thermodynamic limit such that it leads to near equivalence between the canonical and grand ensembles in the limit of large systems. The transfer matrix in this case is a nonlocal operator on a space of L2 functions, and the eigenvalue equation is a homogeneous Fredholm integral equation of the second kind which can be completely solved in terms of Bessel functions. The grand partition function can then be expressed as a sum of powers of the known eigenvalues. It is an easy matter to reproduce the second-order phase transition in the canonical ensemble found in the original work on the model. The investigation is extended to yield the probability densities describing the length of a segment and the correlations among segments. The concept of a local width of the folded chain is found to break down at higher temperatures, while critical correlations are characterized by infinite range, as expected. Apart from physical and methodological implications, the new solution provides striking illustrations of some basic ideas concerning phase transitions.
TL;DR: In this paper, a time-dependent distribution for an infinite system of hard rods is presented, which is simpler than previous derivations and provides a direct physical interpretation of the individual terms in the final expressions.
Abstract: A new derivation of two important one-dimensional time-dependent distributions for an infinite system of hard rods is presented. This derivation is simpler than previous derivations and it provides a direct physical interpretation of the individual terms in the final expressions. A new, more unusual distribution is also presented and discussed. Finally, an exact expression for the diffusion of a Brownian particle is obtained and compared with the exact expression for the self-diffusion coefficient.
TL;DR: In this paper, a new kinetic equation is developed which incorporates the desirable features of the Enskog, the Rice-Allnatt, and the Prigogine-Nicolis-Misguich kinetic theories of dense fluids.
Abstract: A new kinetic equation is developed which incorporates the desirable features of the Enskog, the Rice-Allnatt, and the Prigogine-Nicolis-Misguich kinetic theories of dense fluids. Advantages of the present theory over the latter three theories are (1) it yields the correct local equilibrium hydrodynamic equations, (2) unlike the Rice-Allnatt theory, it determines the singlet and doublet distribution functions from the same equation, and (3) unlike the Prigogine-Nicolis-Misguich theory, it predicts the kinetic and kinetic-potential transport coefficients. The kinetic equation is solved by the Chapman-Enskog method and the coefficients of shear viscosity, bulk viscosity, thermal conductivity, and self-diffusion are obtained. The predicted bulk viscosity and thermal conductivity coefficients are singular at the critical point, while the shear viscosity and self-diffusion coefficients are not.
TL;DR: In this paper, the authors show that equilibrium properties of one-component liquids can be obtained from the coding procedure in terms of distances between neighboring molecules, and the connection between the probability ψ(R) of an intermolecular distanceR and the usual pair distribution function is considered.
Abstract: Equilibrium properties of one-component liquids are obtainable—as suggested by Collins-from the coding procedure in terms of distances between neighboring molecules. The monatomic case is dealt with first, and consequences of some simplifying assumptions are explored. The connection between the probabilityψ(R) of an intermolecular distanceR and the usual pair distribution function is considered. The treatment is then generalized to the case of heterogeneous multiatomic molecules.
TL;DR: In this paper, a two-component linear Ising chain with nearest-neighbor interaction is formulated by using the transfer matrix method, with emphasis placed on the case in which the two components are randomly distributed along the chain.
Abstract: The one-dimensional, two-component linear Ising chain with nearest-neighbor interaction is formulated by using the transfer matrix method, with emphasis placed on the case in which the two components are randomly distributed along the chain. Certain recurrence formulas appear such that themth-order partition function of one of the components is dependent on the lower-order ones. The algorithm provides a working basis for discussing the thermodynamic and magnetic functions with various concentrations of one of the components. An exact expression for the partition function is derived for a linear chain which is composed of a periodic distribution of the two components. The construction of a periodic sequence which would approximate a random distribution of the two components is briefly discussed.
TL;DR: In this paper, the diffusion constant for two-state brownons was calculated in terms of the Laplace transforms of the sojourn time densities, where the change of state is not, as usually assumed, Markovian.
Abstract: We calculate the diffusion constant for two-state brownons when the change of state is not, as usually assumed, Markovian. The correction to the non-interchanging species result is found to be exactly expressible in terms of the Laplace transforms of the sojourn time densities.
TL;DR: In this article, an earlier criticism of Jaynes's maximum-entropy prescription is vindicated with respect to two recent replies to that criticism, which are both written by the same authors.
Abstract: An earlier criticism of Jaynes's maximum-entropy prescription is vindicated with respect to two recent replies to that criticism.