Showing papers in "Journal of Statistical Physics in 1977"
TL;DR: In this paper, the interrelation between the well-known non-Markovian master equation and the new memoryless one used in the previous paper is clarified on the basis of damping theory, and the latter equation is generalized to include cases in which the Hamiltonian or the Liouvillian is a random function of time.
Abstract: The interrelation between the well-known non-Markovian master equation and the new memoryless one used in the previous paper is clarified on the basis of damping theory. The latter equation is generalized to include cases in which the Hamiltonian or the Liouvillian is a random function of time, and is written in a form feasible for perturbational analysis. Thus, the existing stochastic theory in which those cases mentioned above are discussed is equipped with a more tractable basic equation. Two problems discussed in the previous paper, i.e., the random frequency modulation of a quantal oscillator and the Brownian motion of a spin, are treated from the viewpoint of the stochastic theory without such explicit consideration of external reservoirs as was taken in the previous paper.
398 citations
TL;DR: In this article, an explicit solution is obtained for a special form of the potential, where the diffusing particle is initially at an arbitrary point near the potential maximum and various suggested approximation schemes are tested, with the following conclusions, (i) in linear approximation around the maximum the probability distribution is Gaussian.
Abstract: After a general discussion of diffusion in a potential field having two minima, an explicit solution is obtained for a special form of the potential. The potential is symmetric, but the diffusing particle is initially at an arbitrary point near the potential maximum. Various suggested approximation schemes are tested, with the following conclusions, (i) In linear approximation around the maximum the probability distribution is Gaussian. A two-peaked distribution emerges only through nonlinear terms, (ii) The chances for the particle to go to the right or the left valley cannot be found from the linear approximation. Matching the linear approximation with a macroscopic description far away from the maximum is therefore wrong, (iii) Kramers' treatment of the escape across a potential barrier yields a practically exact result for this model.
220 citations
TL;DR: In this paper, a single (nonrelativistic, spinless) electron subject to a constant external electric field interacts with impurities located on an infinitely extended lattice by a potential of random strength, given by a field of Gaussian random variables.
Abstract: A single (nonrelativistic, spinless) electron subject to a constant external electric field interacts with impurities located on an infinitely extended lattice by a potential of random strength The random strength is given by a field of Gaussian random variables We show the existence of the averaged dynamics and prove that in the weak coupling limit, λ → 0, λ2
t=τ fixed, one obtains the usual transport equation for the velocity distribution
197 citations
TL;DR: In this paper, a new memoryless expression for the equation of motion for the reduced density matrix is derived, which is equivalent to that proposed by Tokuyama and Mori, but has a more convenient form for the application of the perturbational expansion method.
Abstract: A new memoryless expression for the equation of motion for the reduced density matrix is derived. It is equivalent to that proposed by Tokuyama and Mori, but has a more convenient form for the application of the perturbational expansion method. The master equation derived from this form of equation in the first Born approximation is applied to two examples, the Brownian motion of a quantal oscillator and that of a spin. In both examples the master equation is rewritten into the coherent-state representation. A comparison is made with the stochastic theory of the spectral line shape given by Kubo, and it is shown that this theory of the line shape can be incorporated into the framework of the present theory.
194 citations
TL;DR: In this article, an application of Kubo's linear response theory is used to study the nonequilibrium situation that results from placing a cluster, of vortices in a weak external velocity field, such as that produced by a distant vortex cluster.
Abstract: Equilibrium statistics of a cluster of a large number of positive two-dimensional point vortices in an infinite region and the associated thermodynamic functions, exhibiting negative temperatures, are evaluated analytically and numerically from a microcanonical ensemble. Extensive numerical simulations of vortex motion are performed to verify the predicted equilibrium configurations. An application of Kubo's linear response theory is used to study the nonequilibrium situation that results from placing a cluster, of vortices in a weak external velocity field, such as that produced by a distant vortex cluster. The weak field causes the cluster to grow in size as if there were an effective positive eddy viscosity. When a number of clusters interact, the effect is for each to grow while the distances between them decrease with time. The latter effect is an exhibit of negative viscosity. The application of this to the motion of the atmosphere is discussed.
174 citations
TL;DR: In this article, the solution of the Ornstein-Zernike equation with Yukawa closure was generalized for an arbitrary number of Yukawas, using the Fourier transform technique introduced by Baxter, and full equivalence to the results of Waisman, Hoye, and Stell was proved for the case of a single Yukawa.
Abstract: The solution of the Ornstein-Zernike equation with Yukawa closure [c(r)=\(\sum\limits_i {K_i e^{ - z_i (r - 1)} /r} \) forr>1] is generalized for an arbitrary number of Yukawas, using the Fourier transform technique introduced by Baxter. Full equivalence to the results of Waisman, Hoye, and Stell is proved for the case of a single Yukawa. Finally, a convenient form of the Laplace transform ofg(s) is found, which can be easily inverted to give a stepwise, rapidly converging series forg(r).
169 citations
TL;DR: In this article, a new technique for handling chemical master equations, based on an expansion of the probability distribution in Poisson distributions, is introduced, which enables chemical master equation to be transformed into Fokker-Planck and stochastic differential equations and yields very simple descriptions of chemical equilibrium states.
Abstract: We introduce a new technique for handling chemical master equations, based on an expansion of the probability distribution in Poisson distributions. This enables chemical master equations to be transformed into Fokker-Planck and stochastic differential equations and yields very simple descriptions of chemical equilibrium states. Certain nonequilibrium systems are investigated and the results are compared with those obtained previously. The Gaussian approximation is investigated and is found to be valid almost always, except near critical points. The stochastic differential equations derived have a few novel features, such as the possibility of pure imaginary noise terms and the possibility of higher order noise, which do not seem to have been previously studied by physicists. These features are allowable because the transform of the probability distribution is a quasiprobability, which may be negative or even complex.
150 citations
TL;DR: In this paper, the authors derive a new inequality for ferromagnetic Ising spin systems and then use it to obtain information about the number of phases which can coexist in such systems.
Abstract: We derive a new inequality for ferromagnetic Ising spin systems and then use it to obtain information about the number of phases which can coexist in such systems. We show in particular that for even interactions only two phases (up and down magnetization) can coexist below the critical temperature at zero magnetic field (h=0) whenever the energy is a continuous function of the temperature. We also prove that the derivatives with respect toh ath=0 of the odd correlation functions (triplet,...) diverge like the susceptibility in the vicinity of the critical temperature (at least for pair interactions). Our results also apply to higher order Ising spins (not just spin 1/2).
105 citations
TL;DR: In this article, the short-range behavior of the pair correlation function in a dense one-component plasma (jellium) is investigated, and the results are applied to the computation of the nuclear reaction rate in dense stellar matter (pycnonuclear reactions).
Abstract: The short-range behavior of the pair correlation function in a dense onecomponent plasma (jellium) is investigated. As an intermediate step, the short-range behavior of the classical pair correlation function is obtained. Actually, although the temperature and the density are assumed to be such that the thermodynamic properties are almost classical, quantum mechanics (tunnel effect) always dominates the pair correlation function at short distances. The quantum pair correlation function is calculated by treating the many-body quantum effects by a perturbation theory, and by using a semiclassical approximation based on path integrals. The results are applied to the computation of the nuclear reaction rate in dense stellar matter (pycnonuclear reactions).
102 citations
TL;DR: In this paper, a scaling theory of transient phenomena is formulated near the instability point for the moments of the relevant macrovariable, for the generating function, and for the probability distribution function.
Abstract: A general scaling theory of transient phenomena is formulated near the instability point for the moments of the relevant intensive macrovariable, for the generating function, and for the probability distribution function. This scaling theory is based on a generalized scale transformation of time. The whole range of time is divided into three regions, namely the initial, scaling, and final regions. The connection procedure between the initial region and the scaling region is studied in detail. This scaling treatment has overcome the difficulty of divergence of the variance for a large time which was encountered in the Ω-expansion, and this scaling theory yields correct values of moments to order unity for an infinite time. Some instructive examples are discussed for the purpose of clarifying the concepts of the scaling theory.
97 citations
TL;DR: In this article, the Onsager-Yang expression for the magnetization was verified for the case when the eight-vertex model reduces to two independent and identical square-lattice Ising models.
Abstract: In a previous paper certain “corner transfer matrices” were defined. It was conjectured that for the zero-field, eight-vertex model these matrices have a very simple eigenvalue spectrum. In this paper these conjectures are verified for the case when the eight-vertex model reduces to two independent and identical square-lattice Ising models. The Onsager-Yang expression for the magnetization follows immediately.
TL;DR: In this article, a number of new results on the Ising ferromagnet are obtained as a consequence of correlation inequalities, which concern the monotonicity properties of the correlation functions, the study of equilibrium states for certain boundary conditions, and the uniqueness of the state in a semi-infinite lattice.
Abstract: A number of new results on the Ising ferromagnet are obtained as a consequence of correlation inequalities. These results concern the monotonicity properties of the correlation functions, the study of equilibrium states for certain boundary conditions, and the uniqueness of the state in a semiinfinite lattice.
TL;DR: In this article, a detailed analysis is presented of a closely related but mathematically simpler problem: the calculation of the collision probability per unit time for a thermally equilibrized one-dimensional gas of point particles.
Abstract: A simple argument advanced recently in support of the legitimacy of the stochastic formulation of chemical kinetics has been criticized because it seems to require the imminent collision of widely separated molecules. It is argued here that this criticism is unwarranted because it is based on an incorrect use of probabilities. To illustrate the various probabilistic considerations involved, a detailed analysis is presented of a closely related but mathematically simpler problem: the calculation of the collision probability per unit time for a thermally equilibrized one-dimensional gas of point particles.
TL;DR: In this article, the existence of a time evolution for infinite anharmonic crystals for a large class of initial configurations was proved for strong forces tying particles to their equilibrium positions, and the uniqueness of the time evolution was shown under suitable assumptions on the solutions of the equations of motion.
Abstract: We prove the existence of a time evolution for infinite anharmonic crystals for a large class of initial configurations. When there are strong forces tying particles to their equilibrium positions then the class of permissible initial conditions can be specified explicitly; otherwise it can only be shown to have full measure with respect to the appropriate Gibbs state. Uniqueness of the time evolution is also proven under suitable assumptions on the solutions of the equations of motion.
TL;DR: In this article, the authors present a rigorous derivation of the stochastic Glauber dynamics of Ising models from Hamiltonian quantum mechanics and obtain an example of a nonequilibrium statistical mechanical system for which a link without mathematical gap can be established from microscopic quantum mechanics to a macroscopic irreversible thermodynamic process.
Abstract: With the help of recent results in the mathematical theory of master equations, we present a rigorous derivation of the stochastic Glauber dynamics of Ising models from Hamiltonian quantum mechanics. A thermal bath is explicitly constructed and, as an illustration, the dynamics of the Ising-Weiss model is analyzed in the thermodynamic limit. We thus obtain an example of a nonequilibrium statistical mechanical system for which a link without mathematical gap can be established from microscopic quantum mechanics to a macroscopic irreversible thermodynamic process.
TL;DR: In this article, the scaling theory of transient phenomena proposed by the author for a single macrovariable near the instability point is extended to multi-macrovariables in nonequilibrium systems.
Abstract: The essential ideas of the scaling theory of transient phenomena proposed by the author for a single macrovariable near the instability point are extended to multi-macrovariables in nonequilibrium systems. The time region is divided into three regimes according to the scaling behavior of the fluctuating parts of the macrovariables. In the first regime, the fluctuation is Gaussian and it is described by the linearized stochastic equation (or linear Fokker-Planck equation). In the second regime, the fluctuation is non-Gaussian, but it is probabilistic or stochastic (not dynamical) in the sense that the stochastic nature comes from the probability distribution in the initial regime and that each representative motion is deterministic, namely a random force can be neglected asymptotically in the second regime. In the final regime, the fluctuation is again Gaussian. A fluctuation-enhancement theorem for multi-macrovariables is given, which states that the fluctuation becomes enhanced by the order of the system size Ω in the second regime, which is of order log Ω, if the initial system is located just at the unstable point. An anomalous fluctuation theorem for multi-macrovariables is also proven, which states that the fluctuation is anomalously enhanced in proportion to δ−2 at times of order log δ if the initial system deviates by δ from the unstable point.
TL;DR: In this article, a stochastic analysis of the spatial and temporal structures in the Prigogine-Lefever-Nicolis model (the Brusselator) is presented, carried out through a Langevin equation derived from a multivariate master equation using the Poisson representation method.
Abstract: A stochastic analysis of the spatial and temporal structures in the Prigogine-Lefever-Nicolis model (the Brusselator) is presented. The analysis is carried out through a Langevin equation derived from a multivariate master equation using the Poisson representation method, which is used to calculate the spatial correlation functions and the fluctuation spectra in the Gaussian approximation. The case of an infinite three-dimensional system is considered in detail. The calculations for the spatial correlation functions and the fluctuation spectra for a finite system subject to different kinds of boundary conditions are also given.
TL;DR: Using the methods of multiplicative stochastic processes, a thorough analysis of “non-Markovian,” generalized Langevin equations is presented and shows that the methods already used in the Gaussian case lead directly to results for the non-Gaussian case.
Abstract: Using the methods of multiplicative stochastic processes, a thorough analysis of “non-Markovian,” generalized Langevin equations is presented. For the Gaussian case, these methods are used to show that the nonstationary Fokker-Planck equation already found by Adelman and others is also obtainable from van Kampen's lemma for stochastic probability flows. Here, results applicable to an arbitraryn-component process are obtained and the specific two-component case of the Brownian harmonic oscillator is presented in detail in order to explicitly exhibit the matrix algebraic methods. The non-Gaussian case is presented at the end of the paper and shows that the methods already used in the Gaussian case lead directly to results for the non-Gaussian case. In order to use the methods of multiplicative stochastic processes analysis, it is necessary to transform the “non-Markovian,” generalized Langevin equation using a stochastic extension of a transformation discussed by Adelman. This transformation removes the “memory kernel” term in the usual generalized Langevin equation and in the Gaussian case leads to the result that the original process was in fact not “non-Markovian” but actually nonstationary,Markovian.
TL;DR: In this paper, the Stillinger-lovett second-moment condition of electrolyte solutions is derived rigorously and simply from only some reasonable (but apparently never proven rigorously) assumptions concerning the asymptotic form of the direct correlation function and the Ornstein-Zernike equation.
Abstract: The Stillinger-Lovett second-moment condition of electrolyte solutions is derived rigorously and simply from only some reasonable (but apparently never proven rigorously) assumptions concerning the asymptotic form of the direct correlation function and the Ornstein-Zernike equation. The derivation suggests that this condition is not the first member of a hierarchy of moment conditions and that there exists no simple result for a fourth-moment condition.
TL;DR: In this paper, the ergodic and stability properties of certain stochastic models are studied, where each model is described by a finite-dimensional processxλ(t) satisfyingdxλ=ℱλ(xλ,t)dt+ λdz(t).
Abstract: The ergodic and stability properties of certain stochastic models are studied. Each model is described by a finite-dimensional stochastic processxλ(t) satisfyingdxλ=ℱλ(xλ,t)dt+ λdz(t), where ℱλ represents a “secular force” andz(t) is a stochastic process with given statistical properties. Such a model may represent a reduced description of an infinite-particle system. Thenxλ(t) may be either a set of macrovariables fluctuating about thermal equilibrium or the macrostate of a system maintained through pumping in a nonequilibrium state. Two Markovian models for whichz(t) is Wiener and ℱλ(y, t) = G(λ,y(t)) for someG nonlinear iny(t) are shown to possess a unique stationary probability density which is approached by any other density ast → ∞. For one of these models, which is of Hamiltonian type, the stationary state is given by the Maxwell-Boltzmann distribution. A particular form of non-Markovian model is also proved to have the above mixing property with respect to the Maxwell-Boltzmann distribution. Finally, the behavior of the sample paths ofxλ(t) for small values of the parameter A is investigated. In the case whenz(t) is Wiener and ℱλ(y, t) = G(y(t), it is shown thatxλ(t) will remain close to the deterministic trajectoryx0(t) (corresponding to λ = 0) for allt ⩾= 0 if and only ifx0(t) is highly stable with respect to small perturbations of the initial conditions.
TL;DR: In this paper, the Langevin equations with different random sources are constructed for the description of fluctuations to varying degrees of accuracy in inverse powers of the system size, and the relaxation frequency for the decay of correlations in a critical equilibrium and the scaling law for anomalous fluctuations are determined and compared to those obtained by Kubo et al.
Abstract: Fluctuations in nonlinear Markovian systems are studied by the Langevin equation method using system-size expansion. Langevin equations with different random sources are constructed for the description of fluctuations to varying degrees of accuracy in inverse powers of the system size ~. Evolution equations for the deterministic path, deviation of the mean from the deterministic path, and the variance are obtained in a nonstationary state in the lowest order of e. The power spectral density for fluctuations about a stable equilibrium is calculated correct to order ~2 and is compared to the exact expression for the Alkemade diode. The relaxation frequency for the decay of correlations in a critical equilibrium and the scaling law for the anomalous fluctuations are determined and compared to those obtained by Kubo et al.
TL;DR: In this article, the external potential needed to produce an arbitrary equilibrium density profile for a one-dimensional lattice gas with nearest neighbor interactions is solved exactly, and the resulting sequence of direct correlation functions is shown to be of short range.
Abstract: The external potential needed to produce an arbitrary equilibrium density profile for a one-dimensional lattice gas with nearest neighbor interactions is solved exactly. The resulting sequence of direct correlation functions is shown to be of short range, and in the ferromagnetic case the even members alternate in sign at zero spin. The even Ursell distributions in this case likewise alternate in sign.
TL;DR: In this article, a unified method for deriving exact kinetic equations for dynamical quantities of a many-body system is presented, and the well-known results of Mori and Zwanzig are recovered as special cases.
Abstract: A unified method for deriving exact kinetic equations for dynamical quantities of a many-body system is presented. The well-known results of Mori and Zwanzig are recovered as special cases. Furthermore, it is shown that they differ only by the way in which the system is prepared at the initial time. Connections between this method and others recently developed are also discussed.
TL;DR: In this article, the Gaussian inequality for multicomponent rotators with negative correlations between two spin components was shown to be a consequence of the Lebowitz inequality, which implies that the decay rate of the truncated correlation functions is dominated by that of the two-point function.
Abstract: The Gaussian inequality is proven for multicomponent rotators with negative correlations between two spin components. In the case of one-component systems, the Gaussian inequality is shown to be a consequence of the Lebowitz inequality. For multicomponent models, the Gaussian inequality implies that the decay rate of the truncated correlation (or Schwinger) functions is dominated by that of the two-point function. Applied to field theory these inequalities give information on the absence of bound states in the lambda ( phi /sub 1 //sup 2/+ phi /sub 2//sup 2/)/sup 2/ model.
TL;DR: In this paper, the general master equations describing nonlinear birth and death processes with one variable are analyzed in terms of their eigenmodes and eigenvalues using the method of a WKB approximation.
Abstract: Nonlinear birth and death processes with one variable are considered. The general master equations describing these processes are analyzed in terms of their eigenmodes and eigenvalues using the method of a WKB approximation. Formulas for the density of eigenstates are obtained. The lower lying eigenmodes are calculated to investigate long-time relaxation, such as relaxations of metastable and unstable states. Anomalous accumulation of the lower lying eigenvalues is shown to exist when the system is infinitesimally close to a critical or marginal state. The general results obtained are applied to some instructive examples, such as the kinetic Weiss-Ising model and a stochastic model of nonlinear chemical reactions.
TL;DR: In this paper, the authors used the inequalities of Fortuin, Kasteleyn, and Ginibre (FKG) to obtain fairly accurate upper and lower bounds on the equilibrium probabilityp(K) of finding the set of sitesK occupied and the adjacent sites unoccupied, i.e., on the probabilities of finding specified clusters.
Abstract: For a lattice gas with attractive potentials of finite range we use the inequalities of Fortuin, Kasteleyn, and Ginibre (FKG) to obtain fairly accurate upper and lower bounds on the equilibrium probabilityp(K) of finding the set of sitesK occupied and the adjacent sites unoccupied, i.e., on the probabilities of finding specified clusters. The probability that a given site, say the origin, is empty or belongs to a cluster of at mostl particles is shown to be a nonincreasing function of the fugacityz and the reciprocal temperatureβ=(κT)
−1; hence the percolation probability is a nondecreasing function ofz andβ. If the forces are not entirely attractive, or if the ensemble is restricted by forbidding clusters larger than a certain size, the FKG inequalities no longer apply, but useful upper and lower bounds onp(K) can still be obtained if the density of the system and the size of the clusterK are not too large. They are obtained from a generalization of the Kirkwood-Salsburg equation, derived by regarding the system as a mixture of different types of cluster, whose only interaction is that they cannot overlap or touch.
TL;DR: A site percolation approach to classical transport in disordered two-phase materials is presented in this paper, where a Monte Carlo computer experiment gives the bulk conductivity of a 30 × 30 x 30 site simple cubic resistor network consisting of two kinds of unit resistor with different conductance.
Abstract: A site percolation approach to classical transport in disordered two-phase materials is presented. A Monte Carlo computer experiment gives the bulk conductivity of a 30 x 30 x 30 site simple cubic resistor network consisting of two kinds of unit resistor with different conductance. A modified effective-medium theory predicts very accurately the bulk conductivity of the network. This theory is found to agree well with available data for the thermal conductivity of real two-phase materials: glass particle-silicon rubber; glass fiber-plastics; air-saturated porous sandstone; and air-saturated fire brick.
TL;DR: In this paper, it was shown that the dynamics of open systems interacting with a chaotic environment can be formulated in a quantum mechanical scheme by means of Nelson's stochastic quantization procedures, and a wave equation for a particle in the chaotic environment was found to be of the Schrodinger-Langevin type associated with an additional nonlinear and random potential.
Abstract: It is shown that the dynamics of open systems interacting with a chaotic environment can be formulated in a quantum mechanical scheme by means of Nelson's stochastic quantization procedures. As a result, a wave equation for a particle in the chaotic environment is found to be of the Schrodinger-Langevin type associated with an additional nonlinear and random potential.
TL;DR: In this paper, the authors derived adsorption isotherms for an adsorbate of hard-sphere particles with "sticky" interactions at any fluid density being adsorbed onto a plane, sticky surface.
Abstract: We derive adsorption isotherms for an adsorbate of hard-sphere particles with “sticky” interactions at any fluid density being adsorbed onto a plane, “sticky” surface. The theory is based on the Percus-Yevick theory for bulk fluids and explicitly includes the equilibrium between the adsorbed fluid and the bulk adsorbate. The theory predicts a surface condensation at low temperatures and low bulk densities in good agreement with surface condensations found in experimental studies of adsorption of gases onto graphite. An approximate law of corresponding states for these transitions is developed. At higher bulk densities and room temperatures, the adsorption isotherms can show a maximum, in accord with recent experimental work.
TL;DR: In this paper, the average partition function for an electron moving in a Gaussian random potential is computed, with a trial action like that in Feynman's polaron theory.
Abstract: We compute the average partition function for an electron moving in a Gaussian random potential. A path integral formulation is used, with a trial action like that in Feynman's polaron theory. We compute the variational bound as well as the first correction in a systematic cumulant expansion. The results are checked against exact formulas for the onedimensional white noise problem. The density of states in the low-energy tail has the correct exponential energy dependence, and energy-dependent prefactor to within a few percent. In addition, the partition function goes over smoothly to the perturbation theory result at high temperatures.