Showing papers in "Journal of Statistical Physics in 2000"

Models of the Small World

Abstract: It is believed that almost any pair of people in the world can be connected to one another by a short chain of intermediate acquaintances, of typical length about six This phenomenon, colloquially referred to as the “six degrees of separation,” has been the subject of considerable recent interest within the physics community This paper provides a short review of the topic

Limiting Distributions for a Polynuclear Growth Model with External Sources

Abstract: The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by Prahofer and Spohn. Depending on the strength of the sources, the limiting distribution functions are either the Tracy–Widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero. Moreover, we obtain transition functions between pairs of the above distribution functions in suitably scaled limits. There are also similar results for a discrete totally asymmetric exclusion process.

Hamiltonian Derivation of a Detailed Fluctuation Theorem

Abstract: We analyze the microscopic evolution of a system undergoing a far-from-equilibrium thermodynamic process Explicitly accounting for the degrees of freedom of participating heat reservoirs, we derive a hybrid result, similar in form to both the fluctuation theorem and a statement of detailed balance We relate this result to the steady-state fluctuation theorem and to a free energy relation valid far from equilibrium

On some properties of kinetic and hydrodynamic equations for inelastic interactions

Abstract: We investigate a Boltzmann equation for inelastic scattering in which the relative velocity in the collision frequency is approximated by the thermal speed. The inelasticity is given by a velocity variable restitution coefficient. This equation is the analog to the Boltzmann classical equation for Maxwellian molecules. We study the homogeneous regime using Fourier analysis methods. We analyze the existence and uniqueness questions, the linearized operator around the Dirac delta function, self-similar solutions and moment equations. We clarify the conditions under which self-similar solutions describe the asymptotic behavior of the homogeneous equation. We obtain formally a hydrodynamic description for near elastic particles under the assumption of constant and variable restitution coefficient. We describe the linear long-wave stability/instability for homogeneous cooling states.

Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles

Abstract: We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasigeostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. For each ensemble the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macrostates for the two ensembles. Microcanonical equilibrium macrostates are characterized as the solutions of a certain constrained minimization problem, while canonical equilibrium macrostates are characterized as the solutions of an unconstrained minimization problem in which the constraint in the first problem is replaced by a Lagrange multiplier. The analysis of equivalence and nonequivalence of ensembles reduces to the following question in global optimization. What are the relationships between the set of solutions of the constrained minimization problem that characterizes microcanonical equilibrium macrostates and the set of solutions of the unconstrained minimization problem that characterizes canonical equilibrium macrostates? In general terms, our main result is that a necessary and sufficient condition for equivalence of ensembles to hold at the level of equilibrium macrostates is that it holds at the level of thermodynamic functions, which is the case if and only if the

Natural nonequilibrium states in quantum statistical mechanics

Abstract: A quantum spin system is discussed where a heat flow between infinite reservoirs takes place in a finite region. A time-dependent force may also be acting. Our analysis is based on a simple technical assumption concerning the time evolution of infinite quantum spin systems. This assumption, physically natural but currently proved for few specific systems only, says that quantum information diffuses in space-time in such a way that the time integral of the commutator of local observables converges: ∫0−∞dt ‖[B, α t A]‖<∞. In this setup one can define a natural nonequilibrium state. In the time-independent case, this nonequilibrium state retains some of the analyticity which characterizes KMS equilibrium states. A linear response formula is also obtained which remains true far from equilibrium. The formalism presented here does not cover situations where (for time-independent forces) the time-translation invariance and uniqueness of the natural nonequilibrium state are broken.

Stability of Attractive Bose–Einstein Condensates

Abstract: We propose the critical nonlinear Schrodinger equation with a harmonic potential as a model of attractive Bose–Einstein condensates. By an elaborate mathematical analysis we show that a sharp stability threshold exists with respect to the number of condensate particles. The value of the threshold agrees with the existing experimental data. Moreover with this threshold we prove that a ground state of the condensate exists and is orbital stable. We also evaluate the minimum of the condensate energy.

Replica Symmetry Breaking in Short-Range Spin Glasses: Theoretical Foundations and Numerical Evidences

Abstract: We discuss replica symmetry breaking (RSB) in spin glasses. We update work in this area, from both the analytical and numerical points of view. We give particular attention to the difficulties stressed by Newman and Stein concerning the problem of constructing pure states in spin glass systems. We mainly discuss what happens in finite-dimensional, realistic spin glasses. Together with a detailed review of some of the most important features, facts, data, and phenomena, we present some new theoretical ideas and numerical results. We discuss among others the basic idea of the RSB theory, correlation functions, interfaces, overlaps, pure states, random field, and the dynamical approach. We present new numerical results for the behaviors of coupled replicas and about the numerical verification of sum rules, and we review some of the available numerical results that we consider of larger importance (for example, the determination of the phase transition point, the correlation functions, the window overlaps, and the dynamical behavior of the system).

Classical Skew Orthogonal Polynomials and Random Matrices

Abstract: Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the case that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed-form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre, and Jacobi cases.

Gaussian Fluctuation for the Number of Particles in Airy, Bessel, Sine, and Other Determinantal Random Point Fields

Abstract: We prove the Central Limit Theorem (CLT) for the number of eigenvalues near the spectrum edge for certain Hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions. As another corollary of the Costin–Lebowitz Theorem we prove the CLT for the empirical distribution function of the eigenvalues of random matrices from classical compact groups.

Universal Amplitude Ratios in the Critical Two-Dimensional Ising Model on a Torus

Abstract: Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x★=limL→∞ξ(L)/L and the first four magnetization moment ratios V2n=〈\(M\)2n〉/〈\(M\)2〉 n . As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G*2n. We confirm these predictions by a high-precision Monte Carlo simulation.

On the level spacing distribution in quantum graphs

Abstract: We derive a formula for the level spacing probability distribution in quantum graphs. We apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases. Moreover, we derive an exact and explicit formula for the level spacing distribution of integrable quantum graphs.

On the Trend to Equilibrium for Some Dissipative Systems with Slowly Increasing a Priori Bounds

Abstract: We prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails: in particular, Boltzmann-type equations with (smoothed) soft potentials. We compensate the lack of uniform-in-time estimates by the use of precise logarithmic Sobolev-type inequalities, and the assumption that the initial datum decays rapidly at large velocities. Our method not only gives explicit results on the times of convergence, but is also able to cover situations in which compactness arguments apparently do not apply (even mere convergence to equilibrium was an open problem for soft potentials).

On the Distribution of Long-Term Time Averages on Symbolic Space

Abstract: The pressure was studied in a rather abstract theory as an important notion of the thermodynamic formalism. The present paper gives a more concrete account in the case of symbolic spaces, including subshifts of finite type. We relate the pressure of an interaction function Φ to its long-term time averages through the Hausdorff and packing dimensions of the subsets on which Φ has prescribed long-term time-average values. Functions Φ with values in ℝd are considered. For those Φ depending only on finitely many symbols, we get complete results, unifying and completing many partial results.

Monte Carlo Transition Dynamics and Variance Reduction

Abstract: For Metropolis Monte Carlo simulations in statistical physics, efficient, easy- to-implement, and unbiased statistical estimators of thermodynamic properties are based on the transition dynamics. Using an Ising model example, we demonstrate (problem-specific) variance reductions compared to conventional histogram estimators. A proof of variance reduction in a microstate limit is presented.

A Modified Boltzmann Equation for Bose–Einstein Particles: Isotropic Solutions and Long-Time Behavior

Abstract: Under some strong cutoff conditions on collision kernels, global existence, local stability, entropy identity, conservation of energy, and moment production estimates are proven for isotropic solutions of a modified (quantum effect) Boltzmann equation for spatially homogeneous gases of Bose–Einstein particles (BBE). Then applying these results with the biting-weak convergence, some results on the long-time behavior of the conservative isotropic solutions of the BBE equation are obtained, including the velocity concentration at very low temperatures and the tendency toward equilibrium states at very high temperatures.

Heat Conduction in Two-Dimensional Nonlinear Lattices

Abstract: The divergence of the heat conductivity in the thermodynamic limit is investigated in 2d-lattice models of anharmonic solids with nearest-neighbour interaction from single-well potentials. Two different numerical approaches based on nonequilibrium and equilibrium simulations provide consistent indications in favour of a logarithmic divergence in “ergodic”, i.e., highly chaotic, dynamical regimes. Analytical estimates obtained in the framework of linear-response theory confirm this finding, while tracing back the physical origin of this anomalous transport to the slow diffusion of the energy of hydrodynamic modes. Finally, numerical evidence of superanomalous transport is given in the weakly chaotic regime, typically observed below a threshold value of the energy density.

Thermodynamic Properties of the Two-Dimensional Two-Component Plasma

Abstract: The model under consideration is a two-dimensional two-component plasma, stable against collapse for the dimensionless coupling constant β<2. The combination of a technique of renormalized Mayer expansion with the mapping onto the sine-Gordon theory provides the full thermodynamics of the plasma in the whole stability range of β. The explicit forms of the density–fugacity relationship and of the specific heat (at constant volume) per particle are presented.

Nonequilibrium Phase Transition in a Model of Diffusion, Aggregation, and Fragmentation

Abstract: We study the nonequilibrium phase transition in a model of aggregation of masses allowing for diffusion, aggregation on contact, and fragmentation. The model undergoes a dynamical phase transition in all dimensions. The steady-state mass distribution decays exponentially for large mass in one phase. In the other phase, the mass distribution decays as a power law accompanied, in addition, by the formation of an infinite aggregate. The model is solved exactly within a mean-field approximation which keeps track of the distribution of masses. In one dimension, by mapping to an equivalent lattice gas model, exact steady states are obtained in two extreme limits of the parameter space. Critical exponents and the phase diagram are obtained numerically in one dimension. We also study the time-dependent fluctuations in an equivalent interface model in (1+1) dimension and compute the roughness exponent χ and the dynamical exponent z analytically in some limits and numerically otherwise. Two new fixed points of interface fluctuations in (1+1) dimension are identified. We also generalize our model to include arbitrary fragmentation kernels and solve the steady states exactly for some special choices of these kernels via mappings to other solvable models of statistical mechanics.

Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems

Abstract: We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett derived the Fokker–Planck equation with friction for the Wigner distribution of the particle in the large-temperature limit; however, their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large-time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the radial process in phase space and we prove the Einstein relation in this case.

Dynamical Localization for the Random Dimer Schrödinger Operator

Abstract: We study the one-dimensional random dimer model, with Hamiltonian Hω=Δ+Vω, where for all x∈\(\mathbb{Z}\), Vω(2x)=Vω(2x+1) and where the Vω(2x) are i.i.d. Bernoulli random variables taking the values ±V, V>0. We show that, for all values of Vand with probability one in ω, the spectrum of His pure point. If V≤1 and V≠1/\(\sqrt 2\), the Lyapunov exponent vanishes only at the two critical energies given by E=±V. For the particular value V=1/\(\sqrt 2\), respectively, V=\(\sqrt 2\), we show the existence of new additional critical energies at E=±3/\(\sqrt 2\), respectively, E=0. On any compact interval Inot containing the critical energies, the eigenfunctions are then shown to be semi-uniformly exponentially localized, and this implies dynamical localization: for all q>0 and for all ψ∈\(\ell\)2(\(\mathbb{Z}\)) with sufficiently rapid decrease $${\mathop {\sup }\limits_t} r_{\psi ,I}^{\left( q \right)} {\kern 1pt} \left( t \right): = {\mathop {\sup }\limits_t} \left\langle {P_I \left( {H\omega } \right)\psi _t ,\left| X \right|^q P_I \left( {H\omega } \right)\psi _t } \right\rangle < \infty $$ Here \(\psi _t = e^{- iH_{\omega ^t}} \psi\), and PI(Hω) is the spectral projector of Hωonto the interval I. In particular, if V>1 and V≠\(\sqrt 2\), these results hold on the entire spectrum [so that one can take I=σ(Hω)].

Stieltjes Integrals of Hölder Continuous Functions with Applications to Fractional Brownian Motion

Abstract: We give a new estimate on Stieltjes integrals of Holder continuous functions and use it to prove an existence-uniqueness theorem for solutions of ordinary differential equations with Holder continuous forcing. We construct stochastic integrals with respect to fractional Brownian motion, and establish sufficient conditions for its existence. We prove that stochastic differential equations with fractional Brownian motion have a unique solution with probability 1 in certain classes of Holder-continuous functions. We give tail estimates of the maximum of stochastic integrals from tail estimates of the Holder coefficient of fractional Brownian motion. In addition we apply the techniques used for ordinary Brownian motion to construct stochastic integrals of deterministic functions with respect to fractional Brownian motion and give tail estimates of its maximum.

Entrainment Versus Chaos in a Model for a Circadian Oscillator Driven by Light-Dark Cycles

Abstract: Circadian rhythms occur in nearly all living organisms with a period close to 24 h These rhythms constitute an important class of biological oscillators which present the characteristic of being naturally subjected to forcing by light-dark (LD) cycles In order to investigate the conditions in which such a forcing might lead to chaos, we consider a model for a circadian limit cycle oscillator and assess its dynamic behavior when a light-sensitive parameter is periodically forced by LD cycles We determine as a function of the forcing period and of the amplitude of the light-induced changes in the light-sensitive parameter the occurrence of various modes of dynamic behavior such as quasi-periodicity, entrainment, period-doubling and chaos The type of oscillatory behavior markedly depends on the forcing waveform; thus the domain of entrainment grows at the expense of the domain of chaos as the forcing function progressively goes from a square wave to a sine wave Also studied is the dependence of the phase of periodic or aperiodic oscillations on the amplitude of the light-induced changes in the control parameter The results are discussed with respect to the main physiological role of circadian rhythms which is to allow organisms to adapt to their periodically varying environment by entrainment to the natural LD cycle

Diffraction of Random Tilings: Some Rigorous Results

Abstract: The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar random tilings based on solvable dimer models, augmented by a brief outline of the diffraction from the classical 2D Ising lattice gas. We also give a summary of the measure theoretic approach to mathematical diffraction theory which underlies the unique decomposition of the diffraction spectrum into its pure point, singular continuous, and absolutely continuous parts.

Limit Sets of Cellular Automata Associated to Probability Measures

Abstract: We introduce the concept of limit set associated to a cellular automaton (CA) and a shift invariant probability measure. This is a subshift whose forbidden blocks are exactly those, whose probabilities tend to zero as time tends to infinity. We compare this probabilistic concept of limit set with the concepts of attractors, both in topological and measure-theoretic sense. We also compare this notion with that of topological limit set in different dynamical situations.

Applications of the Stell–Hemmer Potential to Understanding Second Critical Points in Real Systems

Abstract: We consider the novel properties of the StellHemmer core-softened potentials. First we explore how the theoretically predicted second critical point for these potentials is related to the occurrence of the experimentally observed solidsolid isostructural critical point. We then discuss how this class of potentials can generate anomalies analogous to those found experimentally in liquid water.

A Simple Inductive Approach to the Problem of Convergence of Cluster Expansions of Polymer Models

Abstract: We explain a simple inductive method for the analysis of the convergence of cluster expansions (Taylor expansions, Mayer expansions) for the partition functions of polymer models. We give a very simple proof of the Dobrushin–Kotecký–Preiss criterion and formulate a generalization usable for situations where a successive expansion of the partition function has to be used.

Similarity of percolation thresholds on the hcp and fcc lattices

Abstract: Extensive Monte Carlo simulations were performed in order to determine the precise values of the critical thresholds for site (phcpc, S=0.199 255 5±0.000 001 0) and bond (phcpc, B=0.120 164 0±0.000 001 0) percolation on the hcp lattice to compare with previous precise measurements on the fcc lattice. Also, exact enumeration of the hcp and fcc lattices was performed and yielded generating functions and series for the zeroth, first, and second moments of both lattices. When these series and the values of pc are compared to those for the fcc lattice, it is apparent that the site percolation thresholds are different; however, the bond percolation thresholds are equal within error bars, and the series only differ slightly in the higher order terms, suggesting the actual values are very close to each other, if not identical.

Derivation of Maximum Entropy Principles in Two-Dimensional Turbulence via Large Deviations

Abstract: The continuum limit of lattice models arising in two-dimensional turbulence is analyzed by means of the theory of large deviations In particular, the Miller–Robert continuum model of equilibrium states in an ideal fluid and a modification of that model due to Turkington are examined in a unified framework, and the maximum entropy principles that govern these models are rigorously derived by a new method In this method, a doubly indexed, measure-valued random process is introduced to represent the coarse-grained vorticity field The natural large deviation principle for this process is established and is then used to derive the equilibrium conditions satisfied by the most probable macrostates in the continuum models The physical implications of these results are discussed, and some modeling issues of importance to the theory of long-lived, large-scale coherent vortices in turbulent flows are clarified

Spectral Localization by Gaussian Random Potentials in Multi-Dimensional Continuous Space

Abstract: A detailed mathematical proof is given that the energy spectrum of a non-relativistic quantum particle in multi-dimensional Euclidean space under the influence of suitable random potentials has almost surely a pure-point component. The result applies in particular to a certain class of zero-mean Gaussian random potentials, which are homogeneous with respect to Euclidean translations. More precisely, for these Gaussian random potentials the spectrum is almost surely only pure point at sufficiently negative energies or, at negative energies, for sufficiently weak disorder. The proof is based on a fixed-energy multi-scale analysis which allows for different random potentials on different length scales.