Showing papers in "Journal of Statistical Physics in 1994"

Schrodinger Invariance and Strongly Anisotropic Critical Systems

Abstract: The extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated. For the special case of an anisotropy or dynamical exponent θ=z=2, the group of local scale transformation considered is the Schrodinger group, which can be obtained as the nonrelativistic limit of the conformal group. The requirement of Schrodinger invariance determines the two-point function in the bulk and reduces the three-point function to a scaling form of a single variable. Scaling forms are also derived for the two-point function close to a free surface which can be either spacelike or timelike. These results are reproduced in several exactly solvable statistical systems, namely the kinetic Ising model with Glauber dynamics, lattice diffusion, Lifshitz points in the spherical model, and critical dynamics of the spherical model with a nonconserved order parameter. For generic values of θ, evidence from higher-order Lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two-point function.

On the Hamiltonian Interpolation of Near-to-the-Identity Symplectic Mappings with Application to Symplectic Integration Algorithms

Abstract: We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Moser's scheme, we prove that for any mapping ψe, analytic and e-close to the identity, there exists an analytic autonomous Hamiltonian system, He such that its time-one mapping ΦHe differs from ψe by a quantity exponentially small in 1/e. This result is applied, in particular, to the problem of numerical integration of Hamiltonian systems by symplectic algorithms; it turns out that, when using an analytic symplectic algorithm of orders to integrate a Hamiltonian systemK, one actually follows “exactly,” namely within the computer roundoff error, the trajectories of the interpolating Hamiltonian He, or equivalently of the rescaled Hamiltonian Ke=e-1He, which differs fromK, but turns out to be e5 close to it. Special attention is devoted to numerical integration for scattering problems.

The story of coulombic critiality

Abstract: Recent experiments on phase separation and criticality in ionic fluids are reviewed briefly. The data suggest a sharp distinction betweensolvophobic criticality, displayed by nonionic fluids and some electrolytes, that is associated with Ising-like exponents, β≅0.325, γ≅1.239, and ν≅0.631, andCoulombic (orionic)criticality characterized by classical, van der Waals exponents, β=0.5, γ=1, and ν=0.5. Only experiments on the sodium-ammonia system seem to straddle this dichotomy: they show crossover from classical to Ising behavior close toTc at a characteristic crossover scaletx=|Tx−Tc|/Tc. A range of theoretical issues thus raised is discussed, including other conceivable options (spherical model, tricriticality, etc.). Attention is drawn to Nabutovskii's work and various scenarios are illustrated with the aid of schematic phase diagrams containing multicritical points that could, in principle, separate two distinct universality classes of electrolyte criticality. The advantages of examining a basic four-state lattice model that allows for ionic association-dissociation, etec., are reviewed. The issue of the existence, location, and nature of the long-heralded but still elusive gas-liquid transition and critical point in the continuum restricted primitive model (hard spheres carrying charges +q and −q) is taken up in further detail. Earlier theoretical work and recent Monte Carlo simulations are summarized. In an effort to obtain a physically transparent, semiquantitative description, the work of Debye and Huckel and its subsequent elaboration via Bjerrum's concept of bound ion paris is revisited and seen to predict phase separation and criticality. Recent work by Levin and the author is described which repairs serious defects of the earlier theories by including the interaction of the ion-pair dipoles with the screening ionic fluid, following Debye-Huckel methods. The resulting mean field theory agrees quite well with the simulations and appears to embody the most crucial physical effects. However, the role of critical fluctuations, the related interplay of the charge and density correlation functions, the likelihood of Ising-like behavior, and the associated crossover scaletx remain important unsettled questions. An Appendix presents a critique of arguments by Stell to the effect that the restricted primitive model should display Ising behavior and that 1/r4 effective interactions might be significant.

Generalized Hartree-Fock theory and the Hubbard model

Abstract: The familiar unrestricted Hartree-Fock variational principles is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices and these, in turn, provide a convenient formulation of a generalized Hartree-Fock variational principle, which includes the BCS theory as a special case. While this generalization is not new, it is not well known and we begin by elucidating it. The Hubbard model, with its particle-hole symmetry, is well suited to exploring this theory because BCS states for the attractive model turn into usual HF states for the repulsive model. We rigorously determine the true, unrestricted minimizers for zero and for nonzero temperature in several cases, notably the half-filled band. For the cases treated here, we can exactly determine all broken and unbroken spatial and gauge symmetries of the Hamiltonian.

Linking anisotropic sharp and diffuse surface motion laws via gradient flows

Abstract: We compare four surface motion laws for sharp surfaces with their diffuse interface counterparts by means of gradient flows on corresponding energy functionals. The energy functionals can be defined to give the same dependence on normal direction for the energy of sharp plane surfaces as for their diffuse counterparts. The anisotropy of the kinetics can be incorporated into the inner product without affecting the energy functional.

Brownian dynamics of polydisperse colloidal hard spheres: Equilibrium structures and random close packings

Abstract: Recently we presented a new technique for numerical simulations of colloidal hard-sphere systems and showed its high efficiency. Here, we extend our calculations to the treatment of both 2- and 3-dimensional monodisperse and 3-dimensional polydisperse systems (with sampled finite Gaussian size distribution of particle radii), focusing on equilibrium pair distribution functions and structure factors as well as volume fractions of random close packing (RCP). The latter were determined using in principle the same technique as Woodcock or Stillinger had used. Results for the monodisperse 3-dimensional system show very good agreement compared to both pair distribution and structure factor predicted by the Percus-Yevick approximation for the fluid state (volume fractions up to 0.50). We were not able to find crystalline 3d systems at volume fractions 0.50–0.58 as shown by former simulations of Reeet al. or experiments of Pusey and van Megen, due to the fact that we used random start configurations and no constraints of particle positions as in the cell model of Hoover and Ree, and effects of the overall entropy of the system, responsible for the melting and freezing phase transitions, are neglected in our calculations. Nevertheless, we obtained reasonable results concerning concentration-dependent long-time selfdiffusion coefficients (as shown before) and equilibrium structure of samples in the fluid state, and the determination of the volume fraction of random close packing (RCP, glassy state). As expected, polydispersity increases the respective volume fraction of RCP due to the decrease in free volume by the fraction of the smaller spheres which fill gaps between the larger particles.

Amplitude expansions for instabilities in populations of globally-coupled oscillators

Abstract: We analyze the nonlinear dynamics near the incoherent state in a mean-field model of coupled oscillators. The population is described by a Fokker-Planck equation for the distribution of phases, and we apply center-manifold reduction to obtain the amplitude equations for steady-state and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is described by a native frequency distribution that is reflection-symmetric about zero, the problem has circular symmetry. In the limit of zero extrinsic noise, although the critical eigenvalues are embedded in the continuous spectrum, the nonlinear coefficients in the amplitude equation remain finite, in contrast to the singular behavior found in similar instabilities described by the Vlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both types of bifurcation are possible and they coincide at a codimension-two Takens-Bogdanov point. The steady-state bifurcation may be supercritical or subcritical and produces a time-independent synchronized state. The Hopf bifurcation produces both supercritical stable standing waves and supercritical unstable traveling waves. Previous work on the Hopf bifurcation in a bimodal population by Bonilla, Neu, and Spigler and by Okuda and Kuramoto predicted stable traveling waves and stable standing waves, respectively. A comparison to these previous calculations shows that the prediction of stable traveling waves results from a failure to include all unstable modes.

Exact results for the asymmetric simple exclusion process with a blockage

Abstract: We present new results for the current as a function of transmission rate in the one-dimensional totally asymmetric simple exclusion process (TASEP) with a blockage that lowers the jump rate at one site from one tor<1. Exact finitevolume results serve to bound the allowed values for the current in the infinite system. This proves the existence of a nonequilibrium “phase transition,” corresponding to an “immiscibility” gap in the allowed values of the asymptotic densities which the infinite system can have in a stationary state. A series expansion inr, derived from the finite systems, is proven to be asymptotic for all sufficiently large systems. Pade approximants based on this series, which make specific assumptions about the nature of the singularity atr=1, match numerical data for the “infinite” system to 1 part in 104.

On the correlation for Kac-like models in the convex case

Abstract: The aim of this paper is to stu the behavior asm tends to ∞ of a family of measures exp[-Φ(m)(x)]dx(m) on ℝ m , whereΦ(m) is a potential on ℝ m which is a perturbation “in a suitable sense” of the harmonic potential Σ j x j 2 .

Structure of two-dimensional sandpile. I. Height probabilities

Abstract: The height probabilities of the two-dimensional Abelian sandpile model are the fractionial numbers of lattice sites having heights 1, 2, 3, 4. A combinatorial method for evaluation of these quantities is proposed. The method is based on mapping the set of allowed sandpile configurations onto the set of spanning trees covering a given lattice. Exact analytical expressions for all probabilities are obtained.

Nonequilibrium Statistical Mechanics of Preasymptotic Dispersion.

Abstract: Turbulent transport in bulk-phase fluids and transport in porous media with fractal character involve fluctuations on all space and time scales. Consequently one anticipates constitutive theories should be nonlocal in character and involve constitutive parameters with arbitrary wavevector and frequency dependence. We provide here a nonequilibrium statistical mechanical theory of transport which involves both diffusive and convective mixing (dispersion) at all scales. The theory is based on a generalization of classical approaches used in molecular hydrodynamics and on time-correlation functions defined in terms of nonequilibrium expectations. The resulting constitutive laws are nonlocal and constitutive parameters are wavevector and frequency dependent. All results reduce to their convolution-Fickian quasi-Fickian, or Fickian counterparts in the appropriate limits.

Critical percolation on the torus

Abstract: We compute the various crossing probabilities defined by R. Langlands, P. Pouliot, and Y. Saint-Aubin for the critical percolation on the torus.

Large deviations for the 2D ising model: A lower bound without cluster expansions

Abstract: We show that a lower large-deviation bound for the block-spin magnetization in the 2D Ising model can be pushed all the way forward toward its correct “Wulff” value for all β>βc.

Symmetry breaking and finite-size effects in quantum many-body systems

Abstract: We consider a quantum many-body system on a lattice which exhibits a spontaneous symmetry breaking in its infinite-volume ground states, but in which the corresponding order operator does not commute with the Hamiltonian. Typical examples are the Heisenberg antiferromagnet with a Neel order and the Hubbard model with a (superconducting) off-diagonal long-range order. In the corresponding finite system, the symmetry breaking is usually “obscured” by “quantum fluctuation” and one gets a symmetric ground state with a long-range order. In such a situation, Horsch and von der Linden proved that the finite system has a low-lying eigenstate whose excitation energy is not more than of orderN −1, whereN denotes the number of sites in the lattice. Here we study the situation where the broken symmetry is a continuous one. For a particular set of states (which are orthogonal to the ground state and with each other), we prove bounds for their energy expectation values. The bounds establish that there exist ever-increasing numbers of low-lying eigenstates whose excitation energies are bounded by a constant timesN −1. A crucial feature of the particular low-lying states we consider is that they can be regarded as finite-volume counterparts of the infinite-volume ground states. By forming linear combinations of these low-lying states and the (finite-volume) ground state and by taking infinite-volume limits, we construct infinite-volume ground states with explicit symmetry breaking. We conjecture that these infinite-volume ground states are ergodic, i.e., physically natural. Our general theorems not only shed light on the nature of symmetry breaking in quantum many-body systems, but also provide indispensable information for numerical approaches to these systems. We also discuss applications of our general results to a variety of interesting examples. The present paper is intended to be accessible to readers without background in mathematical approaches to quantum many-body systems.

Asymptotic behavior of solutions to the Coagulation-Fragmentation Equations. II. Weak Fragmentation

Abstract: The discrete coagulation-fragmentation equations are a model for the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones. The assumptions made on the fragmentation coefficients have the physical interpretation that surface effects are important. Our results on the asymptotic behavior of solutions generalize the corresponding results of Ball, Carr, and Penrose for the Becker-Doring equation.

Trace maps as 3D reversible dynamical systems with an invariant

Abstract: One link between the theory of quasicrystals and the theory of nonlinear dynamics is provided by the study of so-called trace maps. A subclass of them are mappings on a one-parameter family of 2D surfaces that foliate ℝ3 (and also ℂ3). They are derived from transfer matrix approaches to properties of 1D quasicrystals. In this article, we consider various dynamical properties of trace maps. We first discuss the Fibonacci trace map and give new results concerning boundedness of orbits on certain subfamilies of its invariant 2D surfaces. We highlight a particular surface where the motion is integrable and semiconjugate to an Anosov system (i.e., the mapping acts as a pseudo-Anosov map). We identify properties of symmetry and reversibility (time-reversal symmetry) in the Fibonacci trace map dynamics and discuss the consequences for the structure of periodic orbits. We show that a conservative period-boubling sequence can be identified when moving through the one-parameter family of 2D surfaces. By using generator trace maps, in terms of which all trace maps obtained from invertible two-letter substitution rules can be expressed, we show that many features of the Fibonacci trace map hold in general. The role of the Fricke character\(\hat I(x,y,z) = x^2 + y^2 + z^2 - 2xyz - 1\), its symmetry group, and reversibility for the Nielsen trace maps are described algebraically. Finally, we outline possible higher-dimensional generalizations.

Entropy production estimates for Boltzmann equations with physically realistic collision kernels

Abstract: We establish strict entropy production bounds for the Boltzmann equation with the hard-sphere collision kernel. Using these entropy production bounds, we prove results asserting that the rate at which strongL 1 convergence to equilibrium occurs is uniform in wide classes of initial data. This extends our previous results in this direction, which applied only to a very special collision kernel. Moreover, the present results provide computable lower bounds; compactness arguments are entirely avoided. The uniformity is an important ingredient in our study of scaling limits of solutions of the non-spatially homogeneous Boltzmann equation, and is the main focus of this paper. However, the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous Boltzmann equation with initial data far from equilibrium to reach any given small strongL 1 neighborhood of equilibrium.

Anderson localization and the space-time characteristic of continuum states

Abstract: A proof of Anderson localization is obtained by ruling out any continuous spectrum on the basis of the space-time characteristic of its states.

On the two-dimensional dynamical Ising model in the phase coexistence region

Abstract: We consider a Glauber dynamics reversible with respect to the two-dimensional Ising model in a finite square of sideL, in the absence of an external field and at large inverse temperature β. We first consider the gap in the spectrum of the generator of the dynamics in two different cases: with plus and open boundary conditions. We prove that, when the symmetry under global spin flip is broken by the boundary conditions, the gap is much larger than the case in which the symmetry is present. For this latter we compute exactly the asymptotics of −(1/βL) log(gap) asL→∞ and show that it coincides with the surface tension along one of the coordinate axes. As a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin-spin time correlation in the infinite-volume plus phase. Our results establish a connection between the dynamical large deviations and those of the equilibrium Gibbs measure studied by Shlosman in the framework of the rigorous description of the Wulff shape for the Ising model. Finally we show that, in the case of open boundary conditions, it is possible to rescale the time withL in such a way that, asL→∞, the finite-dimensional distributions of the time-rescaled magnetization converge to those of a symmetric continuous-time Markov chain on the two-state space {−m*(β),m*(β)},m*(β) being the spontaneous magnetization. Our methods rely upon a novel combination of techniques for bounding from below the gap of symmetric Markov chains on complicated graphs, developed by Jerrum and Sinclair in their Markov chain approach to hard computational problems, and the idea of introducing “block Glauber dynamics” instead of the standard single-site dynamics, in order to put in evidence more effectively the effect of the boundary conditions in the approach to equilibrium.

Evolution equations for phase separation and ordering in binary alloys

Abstract: We explore two phenomenological approaches leading to systems of coupled Cahn-Hilliard and Cahn-Allen equations for describing the dynamics of systems which can undergo first-order phase separation and order-disorder transitions simultaneously, starting from the same discrete lattice free energy function. In the first approach, a quasicontinuum limit is taken for this discrete energy and the evolution of the system is then assumed to be given by gradient flow. In the second approach, a discrete set of gradient flow evolution equations is derived for the lattice dynamics and a quasicontinuum limit is then taken. We demonstrate in the context of BCC Fe−Al binary alloys that it is important that variables be chosen that accommodate the variations in the average concentration as well as the underlying ordered structure of the possible coexistent phases. Only then will the two approaches lead to roughly the same continuum descriptions. We conjecture that in general the number of variables necessary to describe the dynamics of such systems is equal toN 1 +N 2−1, whereN 1 is given by the dimension of the span of the bases of the irreducible representations needed to describe the symmetry groups of the possible equilibrium phases andN 2 is the number of chemical components.N 1 of these variables are nonconserved, and the remaining are conserved and represent the average concentrations. For the Fe−Al alloys this implies a description of one conserved order parameter and one nonconserved order parameter. The resultant description is given by a Cahn-Hilliard equation coupled to a Cahn-Allen equation via the lower-order nonlinear terms. The rough equivalence of the two phenomenological methods adds credibility to the validity of the resulting evolution equations. A similar description should also be valid for alloy systems in which the structure of the competing phases is more complicated.

Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel

Abstract: Smoluchowski's coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version Open image in new window wherect=c1(t) is the concentration ofl-particle clusters at timet. We prove that for initial data satisfyingc1(0)>0 and the condition 0 ⩽cl(0) 0), the solutions behave asymptotically likec1(t)∼t−2≈c(lt−1) ast→∞ withlt−1 kept fixed. The scaling function ≈c(ξ) is (1/gr)ξ, where\(\rho = \sum _l lc_l (0)\), a conserved quantity, is the initial number of particles per unit volume. An analous result is obtained for the continuous version of Smoluchowski's coagulation equation\(\frac{\partial }{{\partial t}}c(v,{\text{ }}t) = \int_0^v {du{\text{ }}c(v - u,{\text{ }}t){\text{ }}c(u,{\text{ }}t) - 2c(v,{\text{ }}t)} \int_0^\infty {du{\text{ }}c(u,{\text{ }}t)}\) wherec(v, t) is the oncentration of clusters of sizev.

Coulomb systems seen as critical systems: Finite-size effects in two dimensions

Abstract: It is known that the free energy at criticality of a finite two-dimensional system of characteristic sizeL has in general a term which behaves like logL asL→∞; the coefficient of this term is universal. There are solvable models of two-dimensional classical Coulomb systems which exhibit the same finite-size correction (except for its sign) although the particle correlations are short-ranged, i.e., noncritical. Actually, the electrical potential and electrical field correlationsare critical at all temperatures (as long as the Coulomb system is a conductor), as a consequence of the perfect screening property of Coulomb systems. This is why Coulomb systems have to exhibit critical finite-size effects.

Perturbation analysis of a stationary nonequilibrium flow generated by an external force

Abstract: The stationary flow of a gas in a slab under the action of a constant external force parallel to the walls is analyzed in the context of the Bhatnagar-Gross-Krook model kinetic equation. The force produces spatial gradients along the coordinate normal to the walls. By performing a perturbation expansion in powers of the force, we obtain the hydrodynamic fields up to fifth order in the force. Then the velocity distribution function and all its moments are evaluated to third order. The expansion coefficients are polynomials in the space variable of a degree increasing linearly with the expansion order. Although the series expansion is only asymptotic, it shows how the state of the system is modified by a variation of the external force beyond the linear regime.

Order parameters of the dilute A models

Abstract: The free energy and local height probabilities of the dilute A models with broken Z2 symmetry are calculated analytically using inversion and corner transfer matrix methods. These models possess four critical branches. The first two branches provide new realizations of the unitary minimal series and the other two branches give a direct product of this series with an Ising model. We identify the integrable perturbations which move the dilute A models away from the critical limit. Generalized order parameters are defined and their critical exponents extracted. The associated conformal weights are found to occur on the diagonal of the relevant Kac table. In an appropriate regime the dilute A 3 model lies in the universality class of the lsing model in a magnetic field. In this case we obtain the magnetic exponent 6 = 15 directly, without the use of scaling relations.

Statistical properties of the periodic Lorentz gas. Multidimensional case

Abstract: In 1981 Bunimovich and Sinai established the statistical properties of the planar periodic Lorentz gas with finite horizon. Our aim is to extend their theory to the multidimensional Lorentz gas. In that case the Markov partitions of the Bunimovich-Sinai type, the main tool of their theory, are not available. We use a crude approximation to such partitions, which we call Markov sieves. Their construction in many dimensions is essentially different from that in two dimensions; it requires more routine calculations and intricate arguments. We try to avoid technical details and outline the construction of the Markov sieves in mostly qualitative, heuristic terms, hoping to carry out our plan in full detail elsewhere. Modulo that construction, our proofs are conclusive. In the end, we obtain a stretched-exponential bound for the decay of correlations, the central limit theorem, and Donsker's Invariance Principle for multidimensional periodic Lorentz gases with finite horizon.

Classical Diffusion in Strong Random Media

Abstract: We study classical diffusion of particles in random media. Although many of our results are general, we focus on the case of an ion in a three-dimensional medium with random, quenched charge centers obeying bulk charge neutrality. Within a functional-integral framework, we calculate the effective diffusion coefficients by first-order and second-order self-consistent perturbation theory (with a Gaussian reference in both cases). We also carry out a one-loop order momentum space renormalization group calculation. The self-consistent methods are complicated numerically and fail beyond intermediate disorder strengths. In contrast, the renormalization group calculation gives an analytical result that appears valid even to high disorder strengths. The methodology, generally applicable to a quantitative calculation of effective diffusion coefficients in disordered media, resolves deficiencies in self-consistent perturbation theory approaches to this class of problems.

The replica-symmetric solution without replica trick for the Hopfield model

Abstract: We derive the saddle-point equations for the order parameters of the Hopfield model in the case of replica symmetry without using the replica trick, but assuming that the Edwards-Anderson parameter is a self-averaging quantity.

Surface-directed spinodal decomposition in a thin-film geometry: A computer simulation

Abstract: The phase separation kinetics of a two-dimensional binary mixture at critical composition confined between (one-dimensional) straight walls which preferentially attract one component of the mixture is studied for a wide range of distancesD between the walls. Following earlier related work on semiinfinite systems, two choices of surface forces at the walls are considered, one corresponding to an incompletely wet state of the walls, the other to a completely wet state (forD→∞). The nonlinear Cahn-Hilliard-type equation, supplemented with appropriate boundary conditions which account for the presence of surfaces, is replaced by a discrete equivalent and integrated numerically. Starting from a random initial distribution of the two species (say,A andB), an oscillatory concentration profile rapidly forms across the film. This is characterized by two thin enrichment layers of the preferred component at the walls, followed by adjacent depletion layers. While in these layers phase separation is essentially complete, the further oscillations of the average composition at distanceZ from a wall get rapidly damped asZ increases toward the center of the film. This structure is relatively stable for an intermediate time scale, while the inhomogeneous structure in the center of the film coarsens. The concentration correlation function in directions parallel to the walls (integrated over allZ) and the associated structure factor (describing small-angle scattering from the film) exhibit a scaling behavior, similar to bulk spinodal decomposition, and the characteristic length scale grows with time asl ‖, wherea is close to the Lifshitz-Slyozov value 1/3, and the coefficients α, β depend on film thickness only weakly. Only when one considers the local correlation function at distances close to the walls are deviations from scaling observed due to the competing effects of the grwing surface enrichment layers. However, at very late times [whenl ‖ (t) becomes comparable toD] this bulklike description breaks down, and a concentration distribution is expected to be established which is a superposition of domains separated by interfaces perpendicular to the walls, the one type of domain being rich inA and nearly homogeneous, and the other poor inA except for two thin enrichment layers at the walls.

Shapes of growing droplets—A model of escape from a metastable phase

Abstract: Nucleation from a metastable state is studied for an Ising ferromagnet with nearest and next nearest neighbor interaction and at very low temperatures. The typical escape path is shown to follow a sequence of configurations with a growing droplet of stable phase whose shape is determined by dynamical considerations and differs significantly from the equilibrium shape corresponding to the instantaneous volume.