Showing papers in "Journal of Statistical Physics in 1981"
TL;DR: A survey of the facts and fancies concerning the nonlinear Langevin or Ito equation can be found in this paper, where it is shown that it is merely a pre-equation, which becomes an equation when an interpretation rule is added.
Abstract: A survey is given of the facts and fancies concerning the nonlinear Langevin or Ito equation. Actually, it is merely a pre-equation, which becomes an equation when an interpretation rule is added. The rules of Ito and Stratonovich differ, but both are mathematically consistent and therefore equally admissible conventions. The reason why they seem to lead to physical differences is that the Langevin approach used to arrive at the equation involves a tacit assumption. For systems with external noise this assumption can be justified, and it is then clear that the Stratonovich rule applies. Systems with internal noise, however, can only be properly described by a master equation and the Ito-Stratonovich controversy never enters. Afterward one is free to model the resulting fluctuations either with an Ito or a Stratonovich scheme, but that does not lead to any new information.
372 citations
TL;DR: In this paper, the qualitative nature of infinite clusters in percolation models is investigated and the results apply to both independent and correlated percolations in any dimension, concern the number and density of infinite cluster, the size of their external surface, the value of their (total) surface-to-volume ratio, and the fluctuations in their density.
Abstract: The qualitative nature of infinite clusters in percolation models is investigated. The results, which apply to both independent and correlated percolation in any dimension, concern the number and density of infinite clusters, the size of their external surface, the value of their (total) surface-to-volume ratio, and the fluctuations in their density. In particular it is shown thatN0, the number of distinct infinite clusters, is either 0, 1, or ∞ and the caseN0=∞ (which might occur in sufficiently high dimension) is analyzed.
177 citations
TL;DR: In this paper, an approximate renormalization procedure is derived for the HamiltonianH(v,x,t)=v2/2−M cosx−P cosk(x−t).
Abstract: An approximate renormalization procedure is derived for the HamiltonianH(v,x,t)=v2/2−M cosx−P cosk(x−t). It gives an estimate of the large scale stochastic instability threshold which agrees within 5–10% with the results obtained from direct numerical integration of the canonical equations. It shows that this instability is related to the destruction of KAM tori between the two resonances and makes the connection with KAM theory. Possible improvements of the method are proposed. The results obtained forH allow us to estimate the threshold for a large class of Hamiltonian systems with two degrees of freedom.
173 citations
TL;DR: In this article, it was shown that the same is true for 1-pf analytic maps from ℂ n ≥ 2 to ℆ n ≥ 3, whose restriction to n is real, with the asymptotic geometric ratio 1 /4.6692.
Abstract: Infinite sequences of period doubling bifurcations in one-parameter families (1-pf) of maps enjoy very strong universality properties: This is known numerically in a multitude of cases and has been shown rigorously for certain 1-pf of maps on the interval. These bifurcations occur in 1-pf of analytic maps at values of the parameter tending to a limit with the asymptotically geometric ratio 1 /4.6692 ....In this paper we indicate the main steps of a proof that the same is true for 1-pf of analytic maps from ℂ
n
to ℂ
n
, whose restriction to ℝn is real.
147 citations
TL;DR: The behavior of the logistic system which is generated by the function f(x =ax (1−x) changes in an interesting way if it is perturbed by external noise as mentioned in this paper.
Abstract: The behavior of the logistic system which is generated by the functionf(x =ax (1−x) changes in an interesting way if it is perturbed by external noise. It turns out that the chaotic behavior which was predicted by Li and Yorke for orbits of period 3, becomes visible and that a sequence of mergence transitions occurs at the critical parameter. The change of the invariant probability density and the Lyapunov exponents are examined numerically. The power spectrum for the period 3 orbit for different fluctuations is calculated and a recursion formula for the time evolution of the probability density is presented as a discrete-time analog of a Chapman-Kolmogorov equation.
126 citations
TL;DR: In this paper, the authors consider two-dimensional models of point particles interacting through short-range two-body potentials and prove that their zero temperature, zero pressure states are crystalline.
Abstract: We consider some two-dimensional models of point particles interacting through short-range two-body potentials and prove that their zero temperature, zero pressure states are crystalline.
120 citations
TL;DR: In this article, the q-state Potts model on the square lattice is studied by Monte Carlo simulation for q = 3, 4, 5, 6, and very good agreement is obtained with exact results of Kiharaet al. and Baxter for energy and free energy at the critical point.
Abstract: Theq-state Potts model on the square lattice is studied by Monte Carlo simulation forq=3, 4, 5, 6. Very good agreement is obtained with exact results of Kiharaet al. and Baxter for energy and free energy at the critical point. Critical exponent estimates forq=3 areα≈0.4,β≈0.1,γ≈1.45, in rough agreement with high-temperature series extrapolation and real space renormalization-group methods. The transition forq=5, 6 is found to be a very weakly first-order transition, i.e., pronounced “pseudocritical” phenomena occur, specific heat, susceptibility, etc. (nearly) diverge at the first-order transition temperature. Dynamics is associated to the model in the same way as for the kinetic Ising model, and the nonlinear slowing down of the order parameter and of the energy is studied. The dynamic exponent is estimated to be Δ (=zv)≈1.9. Within our accuracy noq dependence is detected. The relaxation is found to be consistent with dynamic scaling predictions, and dynamic scaling functions associated with the nonlinear relaxation are estimated.
112 citations
TL;DR: The functional integral method for the statistical solution of stochastic differential equations is extended to a broad class of nonlinear dynamical equations with random coefficients and initial conditions as mentioned in this paper, which have applications in the calculation of particle motion in stochastically magnetic fields, in the solution of Stochastic wave equations, and in the description of electromagnetic plasma turbulence.
Abstract: The functional integral method for the statistical solution of stochastic differential equations is extended to a broad, new class of nonlinear dynamical equations with random coefficients and initial conditions. This work encompasses previous results for classical systems with random forces and initial conditions with arbitrary statistics and provides new results for systems with nonlinear interactions which are nonlocal in time. Closed equations of motion for the correlation and response functions are derived which have applications in the calculation of particle motion in stochastic magnetic fields, in the solution of stochastic wave equations, and in the description of electromagnetic plasma turbulence. As an illustration of the new results for nonlocal interactions, the electromagnetic dispersion tensor is calculated to first order in renormalized theory.
109 citations
TL;DR: In this paper, the authors discuss the question of determining the entropy given the phase space trajectory which describes the detailed history of a many-body system over a period of observation, and propose an illustrative program based on the kinetic Ising model.
Abstract: We discuss the question of determining the entropy given the phase space trajectory which describes the detailed history of a many-body system over a period of observation. Our viewpoint is that the determination of entropy, as well as all other thermodynamic properties, should require no concepts or information other than those given and defined by the trajectory. The counting of coincidence (or repetition) of states along the trajectory is presented as a way to determine entropy given the trajectory. An illustrative program based on the kinetic Ising model is described in detail.
108 citations
TL;DR: In this article, it was shown that the Hausdorff dimension of the attractor is universally equal to D = 0.538 for the same class of mappings for which the Feigenbaum scaling laws hold.
Abstract: We consider such mappingsx
n+1=F(xn) of an interval into itself for which the attractor is a Cantor set. For the same class of mappings for which the Feigenbaum scaling laws hold, we show that the Hausdorff dimension of the attractor is universally equal toD=0.538 ...
107 citations
TL;DR: The hard hexagon model in statistical mechanics is a special case of a solvable class of hard-square-type models, in which certain special diagonal interactions are added as mentioned in this paper.
Abstract: The hard hexagon model in statistical mechanics is a special case of a solvable class of hard-square-type models, in which certain special diagonal interactions are added. The sublattice densities and order parameters of this class are obtained, and it is shown that many Rogers-Ramanujan-type identities naturally enter the working.
TL;DR: In this article, it was shown that non-unitary equivalence between stochastic Markov processes and non-isomorphic K flows is intrinsically random, and the connection of intrinsic randomness with local instability of motion is briefly discussed.
Abstract: We discuss the problem of nonunitary equivalence, via positivity-preserving similarity transformations, between the unitary groups associated with deterministic dynamical evolution and semigroups associated with stochastic processes. Dynamical systems admitting such nonunitary equivalence with stochastic Markov processes are said to be intrinsically random. In a previous work, it was found that the so-called Bernoulli systems (discrete time) are intrinsically random in this sense. This result is extended here by showing that a more general class of dynamical systems---the so-called K systems and K flows---are intrinsically random. The connection of intrinsic randomness with local instability of motion is briefly discussed. We also show that Markov processes associated through nonunitary equivalence to nonisomorphic K flows are necessarily nonisomorphic.
TL;DR: The renormalization group approach to critical phenomena is seen as a theory of the maximum of the renormalized coupling-constant which may, or may not, be a theory for the Ising-model critical point as discussed by the authors.
Abstract: This paper sketches briefly the ideas of the ferromagnetic critical point, or bifurcation point, as exemplified in the Ising model. Historically, the only reliable, general methods available to attack this problem have been series expansion methods. The development of scaling ideas and the realization that Euclidean, Boson, quantum field-theory is the same problem as the scaling limit of critical phenomena has led to the development of the renormalization group approach to critical phenomena. For clarity, attention is focused on that continuous- spin Ising model which is equivalent to a g0:ɸ4:d field theory. A review of the ideas of the renormalization group approach in statistical mechanical language shows that the key assumption is that the limits as the bare coupling-constant goes to infinity and the ultra-violet cut-off is removed are independent of order. Calculations show that this assumption is satisfied in one and two dimensions, but fails in three and four dimensions where the renormalized coupling-constant is not a monotonic function of the bare coupling constant. The renormalization group approach to critical phenomena is seen as a theory of the maximum of the renormalized coupling-constant which may, or may not, be a theory of the Ising-model critical-point.
TL;DR: In this article, a parametrically forced pendulum is studied numerically both with and without friction, and period-doubling sequences of bifurcation are found.
Abstract: A parametrically forced pendulum is studied numerically both with and without friction. In both cases, period-doubling sequences of bifurcation are found. In the dissipative case, the period-doubling sequence leads to strange attractors, while in the conservative case, the sequence is responsible for the destruction of stable zones.
TL;DR: In this article, a detailed study of the statistical mechanics of two-dimensional Coulomb monopole and dipole gases in two or more dimensions is presented and the Kosterlitz-Thouless transition is analyzed in that perspective.
Abstract: A detailed, rigorous study of the statistical mechanics-screening- and critical properties, phase diagrams, etc., of classical Coulomb monopole and dipole gases in two or more dimensions is presented. The statistical mechanics of the two-dimensionalXY and Villain models is reconsidered and related to the one of two-dimensional lattice Coulomb gases. At low temperatures and moderate densities those gases behave like dipole gases. The Kosterlitz-Thouless transition is analyzed in that perspective and characterized by an order parameter. Techniques useful for a proof of existence of such a transition in a two-dimensional hard-core Coulomb gas are developed and applied to the study of dipole gases.
TL;DR: The third law of thermodynamics for quantum and classical lattice models is investigated in this article, and it is shown that the question of whether the third law is satisfied can be decided completely in terms of ground state degeneracies alone, provided these are computed for all possible "boundary conditions".
Abstract: The third law of thermodynamics, in the sense that the entropy per unit volume goes to zero as the temperature goes to zero, is investigated within the framework of statistical mechanics for quantum and classical lattice models. We present two main results: (i) For all models the question of whether the third law is satisfied can be decided completely in terms of ground-state degeneracies alone, provided these are computed for all possible “boundary conditions.” In principle, there is no need to investigate possible entropy contributions from low-lying excited states. (ii) The third law is shown to hold for ferromagnetic models by an analysis of the ground states.
TL;DR: In this article, the authors evaluated the time-correlation function for hard spheres at volumes of 1.6 and 3 times the close-packed volume by a Monte Carlomolecular dynamics technique and showed that at high density the asymptotic behavior is not established until times substantially longer than those attainable in the present work.
Abstract: The time-correlation function for shear viscosity is evaluated for hard spheres at volumes of 1.6 and 3 times the close-packed volume by a Monte Carlomolecular dynamics technique. At both densities, the kinetic part of the timecorrelation function is consistent, within its rather large statistical uncertainty, with the long-timet−3/2 tail predicted by the mode-coupling theory. However, at the higher density, the time-correlation function is dominated by the cross and potential terms out to 25 mean free times, whereas the mode-coupling theory predicts that these are asymptotically negligible compared to the kinetic part. The total time-correlation function decays roughly asαt−3/2, withα much larger than the mode-coupling value, similar to the recent observations by Evans in his nonequilibrium simulations of argon and methane. The exact value of the exponent is, however, not very precisely determined. By analogy with the case of the velocity autocorrelation function, for which results are also presented at these densities, it is argued that it is quite possible that at high density the asymptotic behavior is not established until times substantially longer than those attainable in the present work. At the lower density, the cross and potential terms are of the same magnitude as the kinetic part, and all are consistent with the mode-coupling predictions within the relatively large statistical uncertainties.
TL;DR: In this article, the Fokker-Planck equation for the distribution of position and velocity of a Brownian particle is determined explicitly and an apparently complete set of stationary boundary layer solutions can be determined explicitly.
Abstract: The Fokker-Planck equation for the distribution of position and velocity of a Brownian particle is a particularly simple linear transport equation. Its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly. By a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall, and that approach a linearly increasing position space density far from the wall. Various aspects of these approximate solutions are discussed. In particular we find that the extrapolated asymptotic density reaches zero at a distance xM beyond the wall. We find xM=1.46 in units of the velocity persistence length of the Brownian particle. This study was motivated by certain problems in the theory of diffusion-controlled reactions, and the results might be used to test approximate theories employed in that field.
TL;DR: In this article, with the help of the Onsager-Machlup functional integral approach, the distributionP of a single stochastic variable, the evolution of which is described by a Fokker-Planck equation with a first moment deriving from a bistable potential, is studied.
Abstract: We study, with the help of the Onsager-Machlup functional integral approach, the distributionP of a single stochastic variable, the evolution of which is described by a Fokker-Planck equation with a first moment deriving from a bistable potential. We set up the approximation scheme appropriate, in this approach, to the limit of constant and small diffusion coefficient. Two regimes are to be distinguished: Very long times (Kramers regime) are treated within the frame of a free-instanton-molecule gas approximation, and at intermediate times (Suzuki regime) a standard semiclassical calculation is legitimate. We thus rederive exactly the results obtained from the mode expansion and WKB method.
TL;DR: In this article, the authors consider percolation on the sites of a graphG, e.g., a regulard-dimensional lattice, where all sites of G are occupied (vacant) with probabilityp (respectively,q=1−p), independently of each other.
Abstract: We consider percolation on the sites of a graphG, e.g., a regulard-dimensional lattice. All sites ofG are occupied (vacant) with probabilityp (respectively,q=1−p), independently of each other.W denotes the cluster of occupied sites containing a fixed site (which will usually be taken to be the origin) andW the cardinality ofW. The percolation probabilityθ is the probability that #W=∞, i.e.,θ(p)=Pp{# W=∞}. Some critical values ofp,pH andpT, are defined, respectively, as the smallest value ofp for whichθ(p)> 0, and for which the expectation of #W is infinite. Formally,pH=inf {p∶θ(p)>0} andpT=inf{p∶ Ep{#W}=∞}. We show for fairly general graphsGthat ifp
TL;DR: In this article, the authors present a method for making rigorous various arguments which predict that certain situations are unstable because of a balance of energy vs. entropy, and prove that the two-dimensional plane rotor has no spontaneous magnetization and make rigorous Thouless' arguments on the one-dimensional Ising model with couplingJ/n2.
Abstract: We present a method for making rigorous various arguments which predict that certain situations are unstable because of a balance of energy vs. entropy. As applications, we give yet another proof that the two-dimensional plane rotor has no spontaneous magnetization and we make rigorous Thouless' arguments on the one-dimensional Ising model with couplingJ/n2.
TL;DR: In this paper, the authors present GR-PF-ARTICLE-1981-002, which is the first publication in the Web of Science Record created on 2008-11-27, modified on 2017-05-12.
Abstract: Reference GR-PF-ARTICLE-1981-002doi:10.1007/BF01013174View record in Web of Science Record created on 2008-11-27, modified on 2017-05-12
TL;DR: In this paper, the properties of fluctuations inμ space in or outside thermal equilibrium are obtained by solving hierarchies of equations derived either from the Liouville or the Master equation, and the results are compared with those obtained in the extensive literature, which is reviewed in some detail.
Abstract: The properties of fluctuations inμ space in or outside thermal equilibrium are obtained by solving hierarchies of equations derived either from the Liouville or the Master equation. In particular we study the one-, two-, etc., time correlation functions that describe the spatial and temporal behavior of the fluctuations inμ space. Explicit solutions are obtained for a dilute gas. The Langevin approach is briefly discussed. Our results are compared with those obtained in the extensive literature, which is reviewed in some detail.
TL;DR: In this article, a generalized nonlinear Langevin equation (GLE) forn interacting particles in a bath is derived, and the exact form of the fluctuation-dissipation theorem is obtained.
Abstract: The main result of this paper is a derivation of a generalized nonlinear Langevin equation (GLE) forn interacting particles in a bath. A consequence of the derivation is that the exact form of the (generalized) fluctuation-dissipation theorem is obtained. We discuss also the relation between the memory kernel of the GLE and some corresponding correlation functions which can be easily obtained in a molecular dynamics computer experiment. In the same spirit it is shown that the approach applies to a Brownian particle subjected to a stationary external field. The technique presented in a previous paper to simulate generalized Brownian dynamics can be easily extended to the present case. Our derivation intends to clarify the uses and (possibly) abuses of the Langevin equation in Brownian dynamics studies.
TL;DR: In the presence of internal noise the variables describing a system are intrinsically stochastic as mentioned in this paper, and the question whether the equation has to be interpreted according to Ito or Stratonovich concerns these higher orders, for which the equation is not valid anyway.
Abstract: In the presence of internal noise the variables describing a system are intrinsically stochastic. If they constitute a Markov process the Ω expansion enables one to extract a deterministic macroscopic equation and to compute the fluctuations in successive approximations. In the lowest or linear noise approximation the fluctuations can be represented by a Langevin equation, provided it is handled appropriately. Higher orders cannot be described by any white noise Langevin equation. The question whether the equation has to be interpreted according to Ito or Stratonovich concerns these higher orders, for which the equation is not valid anyway.
TL;DR: In this article, it was shown that for some given values of the parameter (in particular in the case first investigated by Henon) the Henon mapping has a transversal homoclinic orbit.
Abstract: By using a parametric representation of the stable and unstable manifolds, we prove that for some given values of the parameter (in particular in the case first investigated by Henon) the Henon mapping has a transversal homoclinic orbit.
TL;DR: In this paper, a Goldstone-type theorem for a wide class of lattice and continuum quantum systems, both for the ground state and at nonzero temperature, was proved.
Abstract: We prove a Goldstone-type theorem for a wide class of lattice and continuum quantum systems, both for the ground state and at nonzero temperature. For the ground state (T=0) spontaneous breakdown of a continuous symmetry implies no energy gap. For nonzero temperature, spontaneous symmetry breakdown implies slow clustering (noL1 clustering). The methods apply also to nonzero-temperature classical systems.
TL;DR: In this paper, the one-parameter family of mappingsfa(x)=4ax(1−x), a, x e[0, 1] and define an infinite countable set of parameter values a for which the solutions show observable chaos.
Abstract: We consider the one-parameter family of mappingsfa(x)=4ax(1−x), a, x e[0, 1] and define an infinite countable set of parameter values a for which the solutions show observable chaos. Their properties are investigated by means of correlation functions and spectra, which can be interpreted and approximated by separating periodic and chaotic components in the solutions and introducing two simple assumptions on the statistics of the chaotic component.
TL;DR: In this article, the spin-1/2 anisotropic Heisenberg model is studied by generalizing the Migdal-Kadanoff renormalization transformations to quantum spin systems.
Abstract: The spin-1/2 anisotropic Heisenberg model is studied by generalizing the Migdal-Kadanoff renormalization transformations to quantum spin systems. An approximate one-dimensional decimation is employed besides the potential-moving approximation in this generalization. It is shown that these approximations are valid at high temperatures. The results obtained from these approximations suggest that the two-dimensional spin-1/2X-Y model shows the critical behavior similar to that expected for the classicalX-Y and planar models.
TL;DR: In this paper, it was shown that the variational principle for the grand potential of a non-uniform fluid as a functional of the singlet density yields the potential distribution theory for the equilibrium density.
Abstract: It is shown that the variational principle for the grand potential of a nonuniform fluid as a functional of the singlet density yields the potential distribution theory for the equilibrium density. We derive the explicit form that the functional takes for a system of hard rods, and propose an approximate one for hard spheres. Attractive interactions are also considered in mean-field approximation. In all cases the pair direct correlation function of the nonuniform system is obtained and the density gradient expansion of the free energy is investigated.