Showing papers in "Journal of Statistical Physics in 1991"

Fluid dynamic limits of kinetic equations. I. Formal derivations

Abstract: The connection between kinetic theory and the macroscopic equations of fluid dynamics is described. In particular, our results concerning the incompressible Navier-Stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of Hilbert or Chapman-Enskog. The moment formalism shows that the limit leading to the incompressible Navier-Stokes equations, like that leading to the compressible Euler equations, is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible Navier-Stokes equations. Some indications of the validity of these limits are given. More specifically, the connection between the DiPerna-Lions renormalized solution of the classical Boltzmann equation and the Leray solution of the Navier-Stokes equations is discussed.

Stability of incoherence in a population of coupled oscillators

Abstract: We analyze a mean-field model of coupled oscillators with randomly distributed frequencies. This system is known to exhibit a transition to collective oscillations: for small coupling, the system is incoherent, with all the oscillators running at their natural frequencies, but when the coupling exceeds a certain threshold, the system spontaneously synchronizes. We obtain the first rigorous stability results for this model by linearizing the Fokker-Planck equation about the incoherent state. An unexpected result is that the system has pathological stability properties: the incoherent state is unstable above threshold, butneutrally stable below threshold. We also show that the system is singular in the sense that its stability properties are radically altered by infinitesimal noise.

Spatiotemporal analysis of complex signals: Theory and applications

Abstract: We present a space-time description of regular and complex phenomena which consists of a decomposition of a spatiotemporal signal into orthogonal temporal modes that we call chronos and orthogonal spatial modes that we call topos. This permits the introduction of several characteristics of the signal, three characteristic energies and entropies (one temporal, one spatial, and one global), and a characteristic dimension. Although the technique is general, we concentrate on its applications to hydrodynamic problems, specifically the transition to turbulence. We consider two cases of application: a coupled map lattice as a dynamical system model for spatiotemporal complexity and the open flow instability on a rotating disk. In the latter, we show a direct relation between the global entropy and the different instabilities that the flow undergoes as Reynolds number increases.

Implementing the fast multipole method in three dimensions

Abstract: The Rokhlin-Greengard fast multipole algorithm for evaluating Coulomb and multipole potentials has been implemented and analyzed in three dimensions. The implementation is presented for bounded charged systems and systems with periodic boundary conditions. The results include timings and error characterizations.

Disks vs. Spheres: Contrasting Properties of Random Packings

Abstract: Collections of random packings of rigid disks and spheres have been generated by computer using a previously described concurrent algorithm. Particles begin as infinitesimal moving points, grow in size at a uniform rate, undergo energy-onconserving collisions, and eventually jam up. Periodic boundary conditions apply, and various numbers of particles have been considered (N⩽2000 for disks,N⩽8000 for spheres). The irregular disk packings thus formed are clearly polycrystalline with mean grain size dependent upon particle growth rate. By contrast, the sphere packings show a homogeneously amorphous texture substantially devoid of crystalline grains. This distinction strongly influences the respective results for packing pair correlation functions and for the distributions of particles by contact number. Rapidly grown disk packings display occasional vacancies within the crystalline grains; no comparable voids of such distinctive size have been found in the random sphere packings. “Rattler” particles free to move locally but imprisoned by jammed neighbors occur in both the disk and sphere packings.

A maximum-entropy principle for two-dimensional perfect fluid dynamics

Abstract: We use Kullback entropy for Young measures to define statistical equilibrium states for a two-dimensional incompressible flow of a perfect fluid. This approach is justified, as it gives a concentration property about the equilibrium state in the phase space. It might give a statistical understanding of the appearance of coherent structures in two-dimensional turbulence.

Strict monotonicity for critical points in percolation and ferromagnetic models

Abstract: When is the numerical value of the critical point changed by an enhancement of the process or of the interaction? Ferromagnetic spin models, independent percolation, and the contact process are known to be endowed with monotonicity properties in that certain enhancements are capable of shifting the corresponding phase transition in only an obvious direction, e. g., the addition of ferromagnetic couplings can only increase the transition temperature. The question explored here is whether enhancements do indeed change the value of the critical point. We present a generally applicable approach to this issue. For ferromagnetic Ising spin systems, with pair interactions of finite range ind⩾2 dimensions, it is shown that the critical temperatureTc is strictly monotone increasing in each coupling, with the first-order derivatives bounded by positive functions which are continuous on the set of fullyd-dimensional interactions. For independent percolation, with 0

Two results concerning asymptotic behavior of solutions of the Burgers equation with force

Abstract: We consider the Burgers equation with an external force. For the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast → ∞. If the force is the product of a periodic function ofx and white noise in time, we prove the existence of an invariant distribution concentrated on the space of space-periodic functions which is the limit of a wide class of distributions ast → ∞.

Analytic Calculation of Scaling Dimensions: Tricritical Hard Squares and Critical Hard Hexagons

Abstract: The finite-size corrections, central chargesc, and scaling dimensionsx of tricritical hard squares and critical hard hexagons are calculated analytically. This is achieved by solving the special functional equation or inversion identity satisfied by the commuting row transfer matrices of these lattice models at criticality. The results are expressed in terms of Rogers dilogarithms. For tricritical hard squares we obtainc=7/10,x=3/40, 1/5, 7/8, 6/5 and for hard hexagons we obtainc=4/5,x=2/15, 4/5, 17/15, 4/3, 9/5, in accord with the predictions of conformal and modular invariance.

Asymptotic Behavior of Densities for Two-Particle Annihilating Random Walks

Abstract: Consider the system of particles onℤ d where particles are of two types—A andB—and execute simple random walks in continuous time. Particles do not interact with their own type, but when anA-particle meets aB-particle, both disappear, i.e., are annihilated. This system serves as a model for the chemical reactionA+B→ inert. We analyze the limiting behavior of the densitiesρ A (t) andρ B (t) when the initial state is given by homogeneous Poisson random fields. We prove that for equal initial densitiesρ A (0)=ρ B (0) there is a change in behavior fromd⩽4, whereρ A (t)=ρ B (t)∼C/t d /4, tod⩾4, whereρ A (t)=ρ B (t)∼C/tast→∞. For unequal initial densitiesρ A (0)<ρ B (0),ρ A (t)∼e−c√l ind=1,ρ A (t)∼e−Ct/logt ind=2, andρ A (t)∼e−Ct ind⩾3. The termC depends on the initial densities and changes withd. Techniques are from interacting particle systems. The behavior for this two-particle annihilation process has similarities to those for coalescing random walks (A+A→A) and annihilating random walks (A+A→inert). The analysis of the present process is made considerably more difficult by the lack of comparison with an attractive particle system.

Sticky spheres and related systems

Abstract: Some properties of a system of hard-core particles with attractive wells in the Baxter sticky-sphere limit and a related limit are considered, as is the approach to these limits. A demonstration of the result of Stell and Williams that sticky spheres of equal diameter in the Baxter limit are not thermodynamically stable is given, and the way in which size polydispersity can be expected to restore thermodynamic stability is discussed. The implications of these results for the PY sticky-sphere approximation and recent sticky-sphere computer simulations are then examined. It is concluded that the Baxter PY sticky-sphere approximation for a monodisperse system may well be a reasonable one for a slightly polydisperse system of sticky spheres and that existing simulation results may also be relevant to such a system. How polydisperse a system must be in quantitative terms in order for the PY approximation to be useful remains to be seen, however. The question of whether the PY sticky-sphere approximation may prove to be useful and appropriate in describing monodisperse systems with pair potentials for which the attractive wells arenot extremely narrow is also considered; it is noted that firm evidence concerning this question also appears to be lacking. Implications for systems near, but not in, the limit of zero attractive-well width are also considered, especially in terms of the relative size of the well width and the degree of size polydispersity in the repulsive cores. The possible pertinence of such considerations to colloidal systems is observed. The importance of taking into consideration the extremely long equilibration times that can be expected for systems with very narrow attractive wells is also pointed out, in connection both with real colloidal systems and in computer simulations. It is further observed that in the Baxter limit sticky spheres described quantum mechanically are indistinguishable from hard spheres so described; near the zero-well-width limit, the quantal behavior hinges on the number of bound states and thus the well depth as well as the relative size of the de Broglie thermal wavelength and the well width. Related results and investigations relevant to the issues described above are cited.

Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model

Abstract: We prove that ifĤ N is the Sherrington-Kirkpatrick (SK) Hamiltonian and the quantity $$\bar q_N = N^{ - 1} \sum \left\langle {S_l } \right\rangle _H^2 $$ converges in the variance to a nonrandom limit asN→∞, then the mean free energy of the model converges to the expression obtained by SK. Since this expression is known not to be correct in the low-temperature region, our result implies the “non-self-averaging” of the order parameter of the SK model. This fact is an important ingredient of the Parisi theory, which is widely believed to be exact. We also prove that the variance of the free energy of the SK model converges to zero asN→∞, i.e., the free energy has the self-averaging property.

Euler characteristic and related measures for random geometric sets

Abstract: By an elementary calculation we obtain the exact mean values of Minkowksi functionals for a standard model of percolating sets. In particular, a recurrence theorem for the mean Euler characteristic recently put forward is shown to be incorrect. Related previous mathematical work is mentioned. We also conjecture bounds for the threshold density of continuum percolation, which are associated with the Euler characteristic.

Finite-size scaling for Potts models

Abstract: Recently, Borgs and Kotecký developed a rigorous theory of finite-size effects near first-order phase transitions. Here we apply this theory to the ferromagneticq-state Potts model, which (forq large andd⩾2) undergoes a first-order phase transition as the inverse temperatureβ is varied. We prove a formula for the internal energy in a periodic cube of side lengthL which describes the rounding of the infinite-volume jumpΔE in terms of a hyperbolic tangent, and show that the position of the maximum of the specific heat is shifted byΔβ m (L)=(Inq/ΔE)L −d +O(L −2d ) with respect to the infinite-volume transition pointβ t . We also propose an alternative definition of the finite-volume transition temperatureβ t (L) which might be useful for numerical calculations because it differs only by exponentially small corrections fromβ t .

Cellular Automata Model for the Diffusion Equation

Abstract: We consider a new cellular automata rule for a synchronous random walk on a two-dimensional square lattice, subject to an exclusion principle. It is found that the macroscopic behavior of our model obeys the telegraphists's equation, with an adjustable diffusion constant. By construction, the dynamics of our model is exactly described by a linear discrete Boltzmann equation which is solved analytically for some boundary conditions. Consequently, the connection between the microscopic and the macroscopic descriptions is obtained exactly and the continuous limit studied rigorously. The typical system size for which a true diffusive behavior is observed may be deduced as a function of the parameters entering into the rule. It is shown that a suitable choice of these parameters allows us to consider quite small systems. In particular, our cellular automata model can simulate the Laplace equation to a precision of the order (λ/L)6, whereL is the size of the system andλ the lattice spacing. Implementation of this algorithm on special-purpose machines leads to the fastest way to simulate diffusion on a lattice.

Some results on the behavior and estimation of the fractal dimensions of distributions on attractors

Abstract: The strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation. In this paper we first provide a rigorous mathematical framework for the study of dimension, focusing on pointwise dimensionσ(x) and the generalized Renyi dimensionsD(q), and give a rigorous proof of inequalities first derived by Grassberger and Procaccia and Hentschel and Procaccia. We then specialize to the problem of statistical estimation of the correlation dimension ν and information dimensionσ. It has been recognized for some time that the error estimates accompanying the usual procedures (which generally involve least squares methods and nearest neighbor calculations) grossly underestimate the true statistical error involved. In least squares analyses of ν andσ we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates. We then develop an estimation procedure forσ which corrects for an important bias term (the local measure density) and provides confidence intervals forσ. The general applicability of this method is illustrated with various numerical examples.

Space- and Time-Resolved Diffusion-Limited Binary Reaction Kinetics in Capillaries: Experimental Observation of Segregation, Anomalous Exponents, and Depletion Zone

Abstract: An experimental investigation of one-dimensional, diffusion-limited A+B→C chemical reactions is reported. The persistence of reactant segregation and the formation of a depletion zone is observed and expressed in terms of the universal time exponents:α (motion of the boundary zone),β (width of instantaneous product formation zone),γ (rate of instantaneous local formation of product),δ (rate of instantaneous global formation of product), etc. There is good agreement with the recently predicted and/or simulated values:α=1/2,β=1/6,γ=2/3,δ=1/2, in contrast to classical predictions (α=0,β=1/2,γ=0,δ=−1/2). Furthermore, classically the segregation would not be preserved and there would be no formation of a depletion zone and no motion (just dissipation) of the reaction zone. We also discuss the relations to electrode oxidation-reduction reactions, i.e., A+C→C where C is a catalyst, electrode, or “trap.”

Random sequential addition : a distribution function approach

Abstract: Random sequential addition (RSA) of hard objects is an irreversible process defined by three rules: objects are introduced on a surface (or ad-dimensional volume) randomly and sequentially, two objects cannot overlap, and, once inserted, an object is clamped in its position. The configurations generated by an RSA can be characterized, in the macroscopic limit, by a unique set of distribution functions and a density. We show that these “nonequilibrium” RSA configurations can be described in a manner which, in many respects, parallels the usual statistical mechanical treatment of equilibrium configurations: Kirkwood-Salsburg-like hierarchies for the distribution functions, zero-separation theorems, diagrammatic expansions, and approximate equations for the pair distribution function. Approximate descriptions valid for low to intermediate densities can be combined with exact results already derived for higher densities close to the jamming limit of the process. Similarities and differences between the equilibrium and the RSAconfigurations are emphasized. Finally, the potential application of RSA processes to the study of glassy phases is discussed.

Rheology of suspensions of rodlike particles

Abstract: Suspensions of rigid rodlike particles in Newtonian suspending fluids are considered. We discuss the dependence of the relative viscosityμr upon the volume fraction of particlesϕ, their aspect ratioar, and the particle orientation distribution when the particles are sufficiently large that hydrodynamic forces are dominant. Theoretical results are reviewed for a variety of long slender particles. Experimental results obtained using classical rheometrical techniques are discussed. It is shown that whenar⩽25, data from several laboratories agree and they indicate thatμr depends more strongly uponϕ thanar. Previous experimental results using falling ball rheometry are discussed as well as some more recent findings. These are shown to provide insights heretofore unavailable into the macroscopic rheology of suspensions of randomly oriented and oriented rods.

Phase diagrams of Ising models on Husimi trees. I. Pure multisite interaction systems

Abstract: Lattice spin systems with multisite interactions have rich and interesting phase diagrams. We present some results for such systems involving Ising spins (σ=±1) using a generalization of the Bethe lattice approximation. First, we show that our approach yields good approximations for the phase diagrams of some recently studied multisite interaction systems. Second, a multisite interaction system with competing interactions is investigated and a strong connection with results from the theory of dynamical systems is made. We exhibit a full bifurcation diagram, chaos, period-3 windows, etc., for the magnetization of the base site of this system.

Characteristics and mechanisms of electrorheological fluids

Abstract: Electrorheological (ER) fluids consist of suspensions of fine polarizable particles in a dielectric medium, which upon application of an electric field take on the characteristics of a solid in times of the order of milliseconds and reversibly return to liquid behavior upon removal of the field. The rheology, electrical characteristics, and structure of typical ER fluids are here reviewed. The proposed mechanisms and their accord with experimental data are discussed. Some directions for future research are mentioned.

Power Law Growth for the Resistance in the Fibonacci Model

Abstract: Many one-dimensional quasiperiodic systems based on the Fibonacci rule, such as the tight-binding HamiltonianHψ(n)=ψ(n+1)+ψ(n−1)+λv(n)ψ(n),neℤ,ψel2(ℤ),λeℝ, wherev(n)=[(n+1)α]−[nα],[x] denoting the integer part ofx and α the golden mean\((\sqrt 5 --1)/2\), give rise to the same recursion relation for the transfer matrices. It is proved that the wave functions and the norm of transfer matrices are polynomially bounded (critical regime) if and only if the energy is in the spectrum of the Hamiltonian. This solves a conjecture of Kohmoto and Sutherland on the power-law growth of the resistance in a one-dimensional quasicrystal.

On a forest fire model with supposed self-organized criticality

Abstract: We study a stochastic forest fire model introduced by P. Baket al. as a model showing self-organized criticality. This model involves a growth parameterp, and the criticality is supposed to show up in the limitp→0. By simulating the model on much larger lattices, and with much smaller values ofp, we find that the correlations with longest range do not show a nontrivial critical phenomenon in this limit, though we cannot rule out percolation-like critical behavior on a smaller but still divergent length scale. In contrast, the model shows nontrivialdeterministic evolution over time scales ≫1/p in the limitp→0.

Some properties of the A + B → C reaction-diffusion system with initially separated components

Abstract: We study some properties of the A+B→C reaction-diffusion system with initially separated components, first analyzed by means of an asymptotic scaling argument by Galfi and Racz. We show that, in contrast to the asymptotic result that predicts that the rate of production of C goes liket−1, at early times it is shown to increase ast1/2. Deviations from this behavior appear at times inversely proportional to the reaction constant. Analogous crossover properties appear in the kinetic behavior of the reaction front. A second part of the study is concerned with the same chemical reaction on a fractal surface. When the substrate is a percolation cluster at criticality, both the maximum production rate and the width of the reaction zone differ considerably from those for the homogeneous space.

Relation Between the Free Energy and the Direct Correlation Function in the Mean Spherical Approximation

Abstract: It is shown that the direct correlation function of a mixture of hard ions in the mean spherical approximation (MSA) can be expressed in terms of overlap functions of charged spherical shells. In particular, if the system is a mixture of pairs of ions of equal size and opposite charge, then the MSA direct correlation function is given by the electrostatic energy of a pair of charged shells, of radius equal to the radius of the hard ion plus 1/(2Γ). This direct correlation function can be derived from a free energy functional, and a simple extension to nonuniform systems is given.

Random surfaces with two-sided constraints: An application of the theory of dominant ground states

Abstract: We consider some models of classical statistical mechanics which admit an investigation by means of the theory of dominant ground states. Our models are related to the Gibbs ensemble for the multidimensional SOS model with symmetric constraints ∣φ x ∣ ⩽m/2. The main result is that for β⩾β0, where β0 does not depend onm, the structure of thermodynamic phases in the model is determined by dominant ground states: for an evenm a Gibbs state is unique and for an oddm the number of space-periodic pure Gibbs states is two.

A Mean-Field Equation of Motion for the Dynamic Ising Model

Abstract: A mean-field type of approximation is used to derive two differential equations, one approximately representing the average behavior of the Ising model with Glauber (spin-flip) stochastic dynamics, and the other doing the same for Kawasaki (spin-exchange) dynamics. The proposed new equations are compared with the Cahn-Allen and Cahn-Hilliard equations representing the same systems and with information about the exact behavior of the microscopic models.

Fluctuating Hydrodynamics and Principal Oscillation Pattern Analysis

Abstract: Principal oscillation pattern (POP) analysis was recently introduced into climatology to analyze multivariate time series xi(t) produced by systems whose dynamics are described by a linear Markov process x=Bx + ξ. The matrixB gives the deterministic feedback and ξ is a white noise vector with covariances 〈ξ(t)ξ j (t′〉*Q ij δ(t−t′. The POP method is applied to data from a direct simulation Monte Carlo program. The system is a dilute gas with 50,000 particles in a Rayleigh-Benard configuration. The POP analysis correctly reproduces the linearized Navier-Stokes equations (in the matrixB) and the stochastic fluxes (in the matrixQ) as given by Landau-Lifschitz fluctuating hydrodynamics. Using this method, we find the Landau-Lifschitz theory to be valid both in equilibrium and near the critical point of Rayleigh-Benard convection.

Statistical and systematic errors in Monte Carlo sampling

Abstract: We have studied the statistical and systematic errors which arise in Monte Carlo simulations and how the magnitude of these errors depends on the size of the system being examined when a fixed amount of computer time is used. We find that, depending on the degree of self-averaging exhibited by the quantities measured, the statistical errors can increase, decrease, or stay the same as the system size is increased. The systematic underestimation of response functions due to the finite number of measurements made is also studied. We develop a scaling formalism to describe the size dependence of these errors, as well as their dependence on the “bin length” (size of the statistical sample), both at and away from a phase transition. The formalism is tested using simulations of thed=3 Ising model at the infinite-lattice transition temperature. We show that for a 96×96×96 system noticeable systematic errors (systematic underestimation of response functions) are still present for total run lengths of 106 Monte Carlo steps/site (MCS) with measurements taken at regular intervals of 10 MCS.

A Random Field Model for Anomalous Diffusion in Heterogeneous Porous Media

Abstract: Heterogeneity, as it occurs in porous media, is characterized in terms of a scaling exponent, or fractal dimension. A feature of primary interest for two-phase flow is the mixing length. This paper determines the relation between the scaling exponent for the heterogeneity and the scaling exponent which governs the mixing length. The analysis assumes a linear transport equation and uses random fields first in the characterization of the heterogeneity and second in the solution of the flow problem, in order to determine the mixing exponents. The scaling behavior changes from long-length-scale dominated to short-length-scale dominated at a critical value of the scaling exponent of the rock heterogeneity. The long-length-scale-dominated diffusion is anomalous.