Showing papers in "Journal of Statistical Physics in 2001"

Lattice-Boltzmann Simulations of Particle-Fluid Suspensions

Abstract: This paper reviews applications of the lattice-Boltzmann method to simulations of particle-fluid suspensions. We first summarize the available simulation methods for colloidal suspensions together with some of the important applications of these methods, and then describe results from lattice-gas and lattice-Boltzmann simulations in more detail. The remainder of the paper is an update of previously published work,(69, 70) taking into account recent research by ourselves and other groups. We describe a lattice-Boltzmann model that can take proper account of density fluctuations in the fluid, which may be important in describing the short-time dynamics of colloidal particles. We then derive macro-dynamical equations for a collision operator with separate shear and bulk viscosities, via the usual multi-time-scale expansion. A careful examination of the second-order equations shows that inclusion of an external force, such as a pressure gradient, requires terms that depend on the eigenvalues of the collision operator. Alternatively, the momentum density must be redefined to include a contribution from the external force. Next, we summarize recent innovations and give a few numerical examples to illustrate critical issues. Finally, we derive the equations for a lattice-Boltzmann model that includes transverse and longitudinal fluctuations in momentum. The model leads to a discrete version of the Green–Kubo relations for the shear and bulk viscosity, which agree with the viscosities obtained from the macro-dynamical analysis. We believe that inclusion of longitudinal fluctuations will improve the equipartition of energy in lattice-Boltzmann simulations of colloidal suspensions.

Computational Mechanics: Pattern and Prediction, Structure and Simplicity

Abstract: Computational mechanics, an approach to structural complexity, defines a process's causal states and gives a procedure for finding them. We show that the causal-state representation—an ∈-machine—is the minimal one consistent with accurate prediction. We establish several results on ∈-machine optimality and uniqueness and on how ∈-machines compare to alternative representations. Further results relate measures of randomness and structural complexity obtained from ∈-machines to those from ergodic and information theories.

Spectral Determinants for Schrödinger Equation and Q-Operators of Conformal Field Theory

Abstract: Relation between the vacuum eigenvalues of CFT Q-operators and spectral determinants of one-dimensional Schrodinger operator with homogeneous potential, recently conjectured by Dorey and Tateo for special value of Virasoro vacuum parameter p, is proven to hold, with suitable modification of the Schrodinger operator, for all values of p.

Statistics of the Occupation Time of Renewal Processes

Abstract: We present a systematic study of the statistics of the occupation time and related random variables for stochastic processes with independent intervals of time. According to the nature of the distribution of time intervals, the probability density functions of these random variables have very different scalings in time. We analyze successively the cases where this distribution is narrow, where it is broad with index θ<1, and finally where it is broad with index 1<θ<2. The methods introduced in this work provide a basis for the investigation of the statistics of the occupation time of more complex stochastic processes (see joint paper by G. De Smedt, C. Godreche, and J. M. Luck(26)).

Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models

Abstract: We introduce a class of one-dimensional discrete space-discrete time stochastic growth models described by a height function ht(x) with corner initialization. We prove, with one exception, that the limiting distribution function of ht(x) (suitably centered and normalized) equals a Fredholm determinant previously encountered in random matrix theory. In particular, in the universal regime of large x and large t the limiting distribution is the Fredholm determinant with Airy kernel. In the exceptional case, called the critical regime, the limiting distribution seems not to have previously occurred. The proofs use the dual RSK algorithm, Gessel's theorem, the Borodin–Okounkov identity and a novel, rigorous saddle point analysis. In the fixed x, large t regime, we find a Brownian motion representation. This model is equilvalent to the Seppalainen–Johansson model. Hence some of our results are not new, but the proofs are.

A consistent BGK-type model for gas mixtures

Abstract: In this paper we introduce a collision operator for a mixture of gases which satisfies several fundamental properties. Different BGK type collision operators for gas mixtures have been introduced earlier but none of them could satisfy all the basic physical properties: positivity, correct exchange coefficients, entropy inequality, indifferentiability principle. We show that all those properties are verified for our model, and we derive its Chapman-Enskog expansion. In this paper we introduce a collision operator for a mixture of gases which satisfies several fundamental properties. Different BGK type collision operators for gas mixtures have been introduced earlier but none of them could satisfy all the basic physical properties: positivity, correct exchange coefficients, entropy inequality, indifferentiabi- lity principle. We show that all those properties are verified for our model, and we derive its Chapman-Enskog expansion.

Ising Field Theory in a Magnetic Field: Analytic Properties of the Free Energy

Abstract: We study the analytic properties of the scaling function associated with the 2D Ising model free energy in the critical domain T→T c , H→0 The analysis is based on numerical data obtained through the Truncated Free Fermion Space Approach We determine the discontinuities across the Yang–Lee and Langer branch cuts We confirm the standard analyticity assumptions and propose “extended analyticity;” roughly speaking, the latter states that the Yang–Lee branching point is the nearest singularity under Langer's branch cut We support the extended analyticity by evaluating numerically the associated “extended dispersion relation”

Spectral Analysis of Fractional Kinetic Equations with Random Data

Abstract: We present a spectral representation of the mean-square solution of the fractional kinetic equation (also known as fractional diffusion equation) with random initial condition. Gaussian and non-Gaussian limiting distributions of the renormalized solution of the fractional-in-time and in-space kinetic equation are described in terms of multiple stochastic integral representations.

The sl2 Loop Algebra Symmetry of the Six-Vertex Model at Roots of Unity

Abstract: We demonstrate that the six vertex model (XXZ spin chain) with Δ=(q+q-1)/2 and q2N=1 has an invariance under the loop algebra of sl2 which produces a special set of degenerate eigenvalues. For Δ=0 we compute the multiplicity of the degeneracies using Jordan–Wigner techniques.

Bethe's equation is incomplete for the XXZ model at roots of unity

Abstract: We demonstrate for the six vertex and XXZ model parameterized by Δ= −(q+q-1)/2≠±1 that when q2N=1 for integer N≥2 the Bethe's ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl2 loop algebra symmetry group of the model. Therefore in this case the Bethe's ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function. We discuss how the evaluation parameters of the finite dimensional representations of the sl2 loop algebra can be used to complete this specification.

Regularization of the Burnett Equations via Relaxation

Abstract: The classical Chapman–Enskog expansions for the pressure deviator P and heat flux q provide a natural bridge between the kinetic description of gas dynamics as given by the Boltzmann equation and continuum mechanics as given by the balance laws of mass, momentum, energy supplemented by the expansions for P and q. Truncation of these expansions beyond first (Navier–Stokes) order yields instability of the rest state and is inconsistent with thermodynamics. In this paper we propose a visco-elastic relaxation approximation that eliminates the instability paradox. This system is weakly parabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman–Enskog expansion.

Effect of Microscopic Noise on Front Propagation

Abstract: We study the effect of the noise due to microscopic fluctuations on the position of a one dimensional front propagating from a stable to an unstable region in the “linearly marginal stability case.” By simulating a very simple system for which the effective number N of particles can be as large as N=10150, we measure the N dependence of the diffusion constant DN of the front and the shift of its velocity vN. Our results indicate that DN∼(log N)−3. They also confirm our recent claim that the shift of velocity scales like vmin−vN≃K(log N)−2 and indicate that the numerical value of K is very close to the analytical expression Kapprox obtained in our previous work using a simple cut-off approximation.

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. I. General Theory and Square-Lattice Chromatic Polynomial

Abstract: We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P G (q) for m×n rectangular subsets of the square lattice, with m≤8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin–Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n→∞, which arise from the crossing in modulus of dominant eigenvalues (Beraha–Kahane–Weiss theorem). We also provide evidence that the Beraha numbers B 2,B 3,B 4,B 5 are limiting points of partition-function zeros as n→∞ whenever the strip width m is ≥7 (periodic transverse b.c.) or ≥8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B 10) cannot be a chromatic root of any graph.

Random Incidence Matrices: Moments of the Spectral Density

Abstract: We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices: any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semicircle of “small” eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit) we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix.

The Susceptibility of the Square Lattice Ising Model: New Developments

Abstract: We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N6) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant scaling fields, arising first at order |T−Tc|9/4, though high-low temperature symmetry is still preserved. At terms of order |T−Tc|17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T−Tc)p (log |T−Tc|)q with p≥q2. Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.

The Ground State Energy of a Dilute Two-Dimensional Bose Gas

Abstract: The ground state energy per particle of a dilute, homogeneous, two-dimensional Bose gas, in the thermodynamic limit is shown rigorously to be E0/N=(2πℏ2ρ/m)|ln(ρa2)|−1, to leading order, with a relative error at most O(|ln(ρa2)|−1/5). Here N is the number of particles, ρ=N/V is the particle density and a is the scattering length of the two-body potential. We assume that the two-body potential is short range and nonnegative. The amusing feature of this result is that, in contrast to the three-dimensional case, the energy, E0 is not simply N(N−1)/2 times the energy of two particles in a large box of volume (area, really) V. It is much larger.

Scaling, optimality, and landscape evolution

Abstract: A nonlinear model is studied which describes the evolution of a landscape under the effects of erosion and regeneration by geologic uplift by mean of a simple differential equation. The equation, already in wide use among geomorphologists and in that context obtained phenomenologically, is here derived by reparametrization invariance arguments and exactly solved in dimension d=1. Results of numerical simulations in d=2 show that the model is able to reproduce the critical scaling characterizing landscapes associated with natural river basins. We show that configurations minimizing the rate of energy dissipation (optimal channel networks) are stationary solutions of the equation describing the landscape evolution. Numerical simulations show that a careful annealing of the equation in the presence of additive noise leads to configurations very close to the global minimum of the dissipated energy, characterized by mean field exponents. We further show that if one considers generalized river network configurations in which splitting of the flow (i.e., braiding) and loops are allowed, the minimization of the dissipated energy results in spanning loopless configurations, under the constraints imposed by the continuity equations. This is stated in the form of a general theorem applicable to generic networks, suggesting that other branching structures occurring in nature may possibly arise as optimal structures minimizing a cost function.

The Ergodic Theory of Traffic Jams

Abstract: We introduce and analyze a simple probabilistic cellular automaton which emulates the flow of cars along a highway. Our Traffic CA captures the essential features of several more complicated algorithms, studied numerically by K. Nagel and others over the past decade as prototypes for the emergence of traffic jams. By simplifying the dynamics, we are able to identify and precisely formulate the self-organized critical evolution of our system. We focus here on the Cruise Control case, in which well-spaced cars move deterministically at maximal speed, and we obtain rigorous results for several special cases. Then we introduce a symmetry assumption that leads to a two-parameter model, described in terms of acceleration (α) and braking (β) probabilities. Based on the results of simulations, we map out the (α, β) phase diagram, identifying three qualitatively distinct varieties of traffic which arise, and we derive rigorous bounds to establish the existence of a phase transition from free flow to jams. Many other results and conjectures are presented. From a mathematical perspective, Traffic CA provides local, particle-conserving, one-dimensional dynamics which cluster, and converge to a mixture of two distinct equilibria.

Generalized Clausius–Mossotti Formula for Random Composite with Circular Fibers

Abstract: An important area of materials science is the study of effective dielectric, thermal and electrical properties of two phase composite materials with very different properties of the constituents. The case of small concentration is well studied and analytical formulas such as Clausius–Mossotti (Maxwell–Garnett) are successfully used by physicists and engineers. We investigate analytically the case of an arbitrary number of unidirectional circular fibers in the periodicity cell when the concentration of the fibers is not small, i.e., we account for interactions of all orders (pair, triplet, etc.). We next consider transversely-random unidirectional composite of the parallel fibers and obtain a closed form representation for the effective conductivity (as a power series in the concentration v). We express the coefficients in this expansion in terms of integrals of the elliptic Eisenstein functions. These integrals are evaluated and the explicit dependence of the parameter d, which characterizes random position of the fibers centers, is obtained. Thus we have extended the Clausius–Mossotti formula for the non dilute mixtures by adding the higher order terms in concentration and qualitatively evaluated the effect of randomness in the fibers locations. In particular, we have proven that the periodic array provides extremum for the effective conductivity in our class of random arrays (“shaking” geometries). Our approach is based on complex analysis techniques and functional equations, which are solved by the successive approximations method.

Universality in Quantum Hall Systems: Coset Construction of Incompressible States

Abstract: Incompressible Quantum Hall fluids (QHF's) can be described in the scaling limit by three-dimensional topological field theories. Thanks to the correspondence between three-dimensional topological field theories and two dimensional chiral conformal field theories (CCFT's), we propose to study QHF's from the point of view of CCFT's. We derive consistency conditions and stability criteria for those CCFT's that can be expected to describe a QHF. A general algorithm is presented which uses simple currents to construct interesting examples of such CCFT's. It generalizes the description of QHF's in terms of Quantum Hall lattices. Explicit examples, based on the coset construction, provide candidates for the description of Quantum Hall fluids with Hall conductivity σH=1/2(e2/h), 1/4(e2/h), 3/5(e2/h), (e2/h),... .

Enumerations of lattice animals and trees

Abstract: We have developed an improved algorithm that allows us to enumerate the number of site animals on the square lattice up to size 46. We also calculate the number of lattice trees up to size 44 and the radius of gyration of both lattice animals and trees up to size 42. Analysis of the resulting series yields an improved estimate, λ=4.062570(8), for the growth constant of lattice animals, and, λ0=3.795254(8), for the growth constant of trees, and confirms to a very high degree of certainty that both the animal and tree generating functions have a logarithmic divergence. Analysis of the radius of gyration series yields the estimate, ν=0.64115(5), for the size exponent.

Completing Bethe's Equations at Roots of Unity

Abstract: In a previous paper we demonstrated that Bethe's equations are not sufficient to specify the eigenvectors of the XXZ model at roots of unity for states where the Hamiltonian has degenerate eigenvalues. We here find the equations which will complete the specification of the eigenvectors in these degenerate cases and present evidence that the sl 2 loop algebra symmetry is sufficiently powerful to determine that the highest weight of each irreducible representation is given by Bethe's ansatz.

On Stable Oscillations and Equilibriums Induced by Small Noise

Abstract: We consider the motion of a light particle when the force field is perturbed by a small nose. If a certain relation between the mass of the particle and the noise intensity holds, the motion of the particle will be close to periodic oscillations or to a stable equilibrium which do not exist without the noise. We study various classes of random perturbations. In particular, we consider the question of computer simulation of these effects and calculate the correction term which appears when the Gaussian perturbations are replaced by the simple random walk. These are the stochastic-resonance-type effects, and their mathematical description is based on the large deviation theory.

The Strength of First and Second Order Phase Transitions from Partition Function Zeroes

Abstract: We present a numerical technique employing the density of partition function zeroes (i) to distinguish between phase transitions of first and higher order, (ii) to examine the crossover between such phase transitions and (iii) to measure the strength of first and second order phase transitions in the form of latent heat and critical exponents. These techniques are demonstrated in applications to a number of models for which zeroes are available.

A Reader's Guide to Gacs's “Positive Rates” Paper

Abstract: Peter Gacs's monograph, which follows this article, provides a counterexample to the important Positive Rates Conjecture. This conjecture, which arose in the late 1960's, was based on very plausible arguments, some of which come from statistical mechanics. During the long gestation period of the Gacs example, there has been a great deal of skepticism about the validity of his work. The construction and verification of Gacs's counterexample are unavoidably complex, and as a consequence, his paper is quite lengthy. But because of the novelty of the techniques and the significance of the result, his work deserves to become widely known. This reader's guide is intended both as a cheap substitute for reading the whole thing, as well as a warm-up for those who want to plumb its depths.

Shear Flow of a Granular Material

Abstract: The shear flow of a granular material between parallel plates is treated by means of the Boltzmann equation with pseudo-Maxwellian grains. The moments for reverse reflection boundary conditions are found explicitly. The shearing stress is found to depend quadratically on the shear rate.

Frogs in Random Environment

Abstract: We study the so-called frog model: Initially there are some “sleeping” particles and one “active” particle. A sleeping particle is activated when an active particle hits it, after that the activated particle starts to walk independently of everything and can activate other sleeping particles as well. The initial configuration of sleeping particles is random with density p(x). We identify the critical rate of decay of p(x) separating transience from recurrence, and study some other properties of the model.

On Spatially Homogeneous Solutions of a Modified Boltzmann Equation for Fermi–Dirac Particles

Abstract: The paper considers a modified spatially homogeneous Boltzmann equation for Fermi–Dirac particles (BFD). We prove that for the BFD equation there are only two classes of equilibria: the first ones are Fermi–Dirac distributions, the second ones are characteristic functions of the Euclidean balls, and they can be simply classified in terms of temperatures: T>2/5TF and T=2/5TF, where TF denotes the Fermi temperature. In general we show that the L∞-bound 0≤f≤ 1/e derived from the equation for solutions implies the temperature inequality T≥2/5TF, and if T>2/5TF, then f trend towards Fermi–Dirac distributions; if T=2/5TF, then f are the second equilibria. In order to study the long-time behavior, we also prove the conservation of energy and the entropy identity, and establish the moment production estimates for hard potentials.

A survey of one-dimensional random polymers

Abstract: In the last decade there has been an enormous progress in the mathematical understanding of one-dimensional polymer measures, which are path measures that suppress self-intersections. We are currently in the situation that many interesting questions have either been answered, or that essential new ideas are needed. In this survey paper, we discuss the most relevant results, open questions, and heuristics.