Showing papers in "Journal of Statistical Physics in 2002"
TL;DR: In this article, it was shown that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which is called the Airy process A(y).
Abstract: We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single “time” (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y−2. Roughly the Airy process describes the last line of Dyson's Brownian motion model for random matrices. Our construction uses a multi-layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation.
702 citations
TL;DR: In this article, the authors reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle, and Bowen in the 1970s, called SRB measures, as these objects are called.
Abstract: This is a slightly expanded version of the text of a lecture I gave in a conference at Rutgers University in honor of David Ruelle and Yasha Sinai. In this lecture I reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle, and Bowen in the 1970s. SRB measures, as these objects are called, play an important role in the ergodic theory of dissipative dynamical systems with chaotic behavior. Roughly speaking,
599 citations
TL;DR: In this paper, a dynamical fluctuation theory for stationary non-equilibrium states (SNS) is formulated explicitly in stochastic models of interacting particles, where a crucial role is played by the time reversed dynamics.
Abstract: We formulate a dynamical fluctuation theory for stationary non-equilibrium states (SNS) which is tested explicitly in stochastic models of interacting particles. In our theory a crucial role is played by the time reversed dynamics. Within this theory we derive the following results: the modification of the Onsager–Machlup theory in the SNS; a general Hamilton–Jacobi equation for the macroscopic entropy; a non-equilibrium, nonlinear fluctuation dissipation relation valid for a wide class of systems; an H theorem for the entropy. We discuss in detail two models of stochastic boundary driven lattice gases: the zero range and the simple exclusion processes. In the first model the invariant measure is explicitly known and we verify the predictions of the general theory. For the one dimensional simple exclusion process, as recently shown by Derrida, Lebowitz, and Speer, it is possible to express the macroscopic entropy in terms of the solution of a nonlinear ordinary differential equation; by using the Hamilton–Jacobi equation, we obtain a logically independent derivation of this result.
405 citations
TL;DR: In this article, thermodynamically consistent lattice Boltzmann models for single-component multiphase flows can be derived from a kinetic equation using both Enskog's theory for dense fluids and mean-field theory for long-range molecular interaction.
Abstract: This paper demonstrates that thermodynamically consistent lattice Boltzmann models for single-component multiphase flows can be derived from a kinetic equation using both Enskog's theory for dense fluids and mean-field theory for long-range molecular interaction. The lattice Boltzmann models derived this way satisfy the correct mass, momentum, and energy conservation equations. All the thermodynamic variables in these LBM models are consistent. The strengths and weaknesses of previous lattice Boltzmann multiphase models are analyzed.
353 citations
TL;DR: In this article, the lattice Boltzmann model is used to simulate the velocity slip near the wall in micro-cavity flows and the velocity drop along the channel and mass flow rate variation with the Knudsen number.
Abstract: The lattice Boltzmann model is a simplified kinetic method based on the particle distribution function. We use this method to simulate problems in MEMS, in which the velocity slip near the wall plays an important role. It is demonstrated that the lattice Boltzmann method can capture the fundamental behaviors in micro-channel flow, including velocity slip, nonlinear pressure drop along the channel and mass flow rate variation with Knudsen number. The Knudsen number dependence of the position of the vortex center and the pressure contour in micro-cavity flows is also demonstrated.
353 citations
TL;DR: In this paper, it was shown that the first, second, third, etc. eigenvalues of a Wishart matrix converges to the Tracy-Widom distribution after a proper rescaling.
Abstract: Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X*X(XtX) converges to the Tracy–Widom law as n,p (the dimensions of X) tend to ∞ in some ratio n/p→γ>0. We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy–Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner random matrices in refs. 27, 38, and 39 allows to extend the results by Johansson and Johnstone to the case of X with non-Gaussian entries, provided n−p=O(p1/3). We also prove that λmax≤(n1/2+p1/2)2+O(p1/2 log(p)) (a.e.) for general γ>0.
280 citations
TL;DR: In this paper, the authors considered an open one dimensional lattice gas on sites i=1,..., N, with particles jumping independently with rate 1 to neighboring interior empty sites, the simple symmetric exclusion process, and the particle fluxes at the left and right boundaries, corresponding to exchanges with reservoirs at different chemical potentials, create a stationary nonequilibrium state (SNS) with a steady flux of particles through the system.
Abstract: We consider an open one dimensional lattice gas on sites i=1,..., N, with particles jumping independently with rate 1 to neighboring interior empty sites, the simple symmetric exclusion process. The particle fluxes at the left and right boundaries, corresponding to exchanges with reservoirs at different chemical potentials, create a stationary nonequilibrium state (SNS) with a steady flux of particles through the system. The mean density profile in this state, which is linear, describes the typical behavior of a macroscopic system, i.e., this profile occurs with probability 1 when N→∞. The probability of microscopic configurations corresponding to some other profile ρ(x), x=i/N, has the asymptotic form exp[−N\(F\)({ρ})]; \(F\) is the large deviation functional. In contrast to equilibrium systems, for which \(F\)eq({ρ}) is just the integral of the appropriately normalized local free energy density, the \(F\) we find here for the nonequilibrium system is a nonlocal function of ρ. This gives rise to the long range correlations in the SNS predicted by fluctuating hydrodynamics and suggests similar non-local behavior of \(F\) in general SNS, where the long range correlations have been observed experimentally.
247 citations
TL;DR: In this article, the authors studied the regularity properties of the Lyapunov exponent of one-frequency quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency.
Abstract: We study regularity properties of the Lyapunov exponent L of one-frequency quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We prove joint continuity of L, in frequency and energy, at every irrational frequency.
216 citations
TL;DR: Various topics concerning the entanglement of composite quantum systems are considered with particular emphasis concerning the strict relations of such a problem with the one of attributing objective properties to the constituents.
Abstract: Various topics concerning the entanglement of composite quantum systems are considered with particular emphasis concerning the strict relations of such a problem with the one of attributing objective properties to the constituents. Most of the paper deals with composite systems in pure states. After a detailed discussion and a precise formal analysis of the case of systems of distinguishable particles, the problems of entanglement and the one of the properties of subsystems of systems of identical particles are thoroughly discussed. This part is the most interesting and new and it focuses in all details various subtle questions which have never been adequately discussed in the literature. Some inappropriate assertions which appeared in recent papers are analyzed. The relations of the main subject of the paper with the nonlocal aspects of quantum mechanics, as well as with the possibility of deriving Bell's inequality are also considered.
205 citations
TL;DR: In this paper, the transition matrix Monte Carlo method is used to compute the density of states, as well as to construct multi-canonical and equal-hit algorithms, respectively.
Abstract: We present a formalism of the transition matrix Monte Carlo method. A stochastic matrix in the space of energy can be estimated from Monte Carlo simulation. This matrix is used to compute the density of states, as well as to construct multi-canonical and equal-hit algorithms. We discuss the performance of the methods. The results are compared with single histogram method, multi-canonical method, and other methods. In many aspects, the present method is an improvement over the previous methods.
155 citations
TL;DR: In this article, the authors discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the six-vertex model and the XXZ chain.
Abstract: We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the six-vertex model and XXZ chain, and for the eight-vertex model. In particular we discuss the “beyond the equator,” infinite momenta and exact complete string problems. We show how they can be overcome and conclude that the coordinate Bethe ansatz does indeed give a complete set of states, as expected.
TL;DR: In this paper, the velocity distribution approaches in the scaling limit to a similarity solution with a power law tail for general classes of initial conditions and derive a transcendental equation from which the exponents in the tails can be calculated.
Abstract: This paper deals with solutions of the nonlinear Boltzmann equation for spatially uniform freely cooling inelastic Maxwell models for large times and for large velocities, and the nonuniform convergence to these limits. We demonstrate how the velocity distribution approaches in the scaling limit to a similarity solution with a power law tail for general classes of initial conditions and derive a transcendental equation from which the exponents in the tails can be calculated. Moreover on the basis of the available analytic and numerical results for inelastic hard spheres and inelastic Maxwell models we formulate a conjecture on the approach of the velocity distribution function to a scaling form.
TL;DR: In this paper, the authors review and further develop a mathematical framework for non-equilibrium quantum statistical mechanics and introduce notions of entropy production and heat fluxes, and study their properties in a model of a small finite quantum system coupled to several independent thermal reservoirs.
Abstract: We review and further develop a mathematical framework for non-equilibrium quantum statistical mechanics recently proposed in refs 1–7 In the algebraic formalism of quantum statistical mechanics we introduce notions of non-equilibrium steady states, entropy production and heat fluxes, and study their properties Our basic paradigm is a model of a small (finite) quantum system coupled to several independent thermal reservoirs We exhibit examples of such systems which have strictly positive entropy production
TL;DR: In this paper, the authors considered the Langevin equation from an unusual point of view, that is as an archetype for a dissipative system driven out of equilibrium by an external excitation.
Abstract: In this paper, we consider the Langevin equation from an unusual point of view, that is as an archetype for a dissipative system driven out of equilibrium by an external excitation. Using path integral method, we compute exactly the probability density function of the power (averaged over a time interval of length τ) injected (and dissipated) by the random force into a Brownian particle driven by a Langevin equation. The resulting distribution, as well as the associated large deviation function, display strong asymmetry, whose origin is explained. Connections with the so-called “Fluctuation Theorem” are thereafter discussed. Finally, considering Langevin equations with a pinning potential, we show that the large deviation function associated with the injected power is completely insensitive to the presence of a potential.
TL;DR: In this paper, a comparison of particle simulation with numerical solution of the Navier-Stokes (NS) equations for pressure-driven plane Poiseuille flow is presented and contrasted with that of the acceleration-driven Poiseueille flow.
Abstract: Comparison of particle (DSMC) simulation with the numerical solution of the Navier–Stokes (NS) equations for pressure-driven plane Poiseuille flow is presented and contrasted with that of the acceleration-driven Poiseuille flow. Although for the acceleration-driven case DSMC measurements are qualitatively different from the NS solution at relatively low Knudsen number, the two are in somewhat better agreement for pressure-driven flow.
TL;DR: This work shows how to implement the pivot algorithm so that the time required per accepted pivot is O(Nq) with q<1, and estimates that q is less than 0.57 in two dimensions, and less than0.85 in three dimensions.
Abstract: The pivot algorithm is a Markov Chain Monte Carlo algorithm for simulating the self-avoiding walk. At each iteration a pivot which produces a global change in the walk is proposed. If the resulting walk is self-avoiding, the new walk is accepted; otherwise, it is rejected. Past implementations of the algorithm required a time O(N) per accepted pivot, where N is the number of steps in the walk. We show how to implement the algorithm so that the time required per accepted pivot is O(N
q
) with q<1. We estimate that q is less than 0.57 in two dimensions, and less than 0.85 in three dimensions. Corrections to the O(N
q
) make an accurate estimate of q impossible. They also imply that the asymptotic behavior of O(N
q
) cannot be seen for walk lengths which can be simulated. In simulations the effective q is around 0.7 in two dimensions and 0.9 in three dimensions. Comparisons with simulations that use the standard implementation of the pivot algorithm using a hash table indicate that our implementation is faster by as much as a factor of 80 in two dimensions and as much as a factor of 7 in three dimensions. Our method does not require the use of a hash table and should also be applicable to the pivot algorithm for off-lattice models.
TL;DR: In this article, a two-phase lattice Boltzmann simulation of the hydrodynamic behavior inside a capillary tube is presented, where the diameter of the tube is large enough, typically at least 30 lattice units.
Abstract: We report results of extensive two-phase lattice-Boltzmann simulations of capillary rise dynamics. We demonstrate that the method can be used to model the hydrodynamic behaviour inside a capillary tube provided that the diameter of the tube is large enough, typically at least 30 lattice units. We also present results for the dependence of the cosine of the dynamic contact angle on the capillary number Ca. Its deviation from the static advancing contact angle has a power-law form, with the value of the exponent very close to 3/2 for capillary rise at zero gravity, while behaviour is more complex in the presence of gravity.
TL;DR: In this article, it was shown that the Poincare return time of a typical cylinder is at least its length, and that the Lyapunov exponent and dimension can be expressed via return times.
Abstract: We show that the Poincare return time of a typical cylinder is at least its length. For one dimensional maps we express the Lyapunov exponent and dimension via return times.
TL;DR: In this paper, the authors coupled the lattice Boltzmann method with enhanced collisions for hydrodynamics with a model for the anisotropic liquid/solid phase transition, based on a simple reaction model.
Abstract: We coupled the lattice Boltzmann method with enhanced collisions for hydrodynamics with a model for the anisotropic liquid/solid phase transition. The model is based on a simple reaction model. As a test we have performed calculations for dendritic growth of a crystal into an undercooled melt.
TL;DR: A collision integral is derived which enables simple identification of transport coefficients, and which circumvents construction of the equilibrium, which discusses performance of this approach as compared to the standard realizations.
Abstract: In this paper, we present the construction of the Lattice Boltzmann method equipped with the H-theorem Based on entropy functions whose local equilibria are suitable to recover the Navier–Stokes equations in the framework of the Lattice Boltzmann method, we derive a collision integral which enables simple identification of transport coefficients, and which circumvents construction of the equilibrium We discuss performance of this approach as compared to the standard realizations
TL;DR: In this paper, the authors studied a one-dimensional spin (interacting particle) system with product Bernoulli (p) stationary distribution, in which a site can flip only when its left neighbor is in state + 1.
Abstract: We study a one-dimensional spin (interacting particle) system, with product Bernoulli (p) stationary distribution, in which a site can flip only when its left neighbor is in state +1. Such models have been studied in physics as simple exemplars of systems exhibiting slow relaxation. In our “East” model the natural conjecture is that the relaxation time τ(p), that is 1/(spectral gap), satisfies log τ(p)∼\(\tfrac{{\log ^2 (1/p)}}{{\log 2}}\) as p↓0. We prove this up to a factor of 2. The upper bound uses the Poincare comparison argument applied to a “wave” (long-range) comparison process, which we analyze by probabilistic techniques. Such comparison arguments go back to Holley (1984, 1985). The lower bound, which atypically is not easy, involves construction and analysis of a certain “coalescing random jumps” process.
TL;DR: In this paper, the Glauber dynamics of spin systems of Ising type in the limit when the temperature tends to zero in finite volume were considered, and the precise prefactors up to multiplicative errors that tend to 1 as T↓0.
Abstract: We consider Glauber dynamics of classical spin systems of Ising type in the limit when the temperature tends to zero in finite volume. We show that information on the structure of the most profound minima and the connecting saddle points of the Hamiltonian can be translated into sharp estimates on the distribution of the times of metastable transitions between such minima as well as the low lying spectrum of the generator. In contrast with earlier results on such problems, where only the asymptotics of the exponential rates is obtained, we compute the precise pre-factors up to multiplicative errors that tend to 1 as T↓0. As an example we treat the nearest neighbor Ising model on the 2 and 3 dimensional square lattice. Our results improve considerably earlier estimates obtained by Neves–Schonmann,(1) Ben Arous–Cerf,(2) and Alonso–Cerf.(3) Our results employ the methods introduced by Bovier, Eckhoff, Gayrard, and Klein in refs. 4 and 5.
TL;DR: In this paper, a lattice Boltzmann model is proposed for simulation of combustion, which includes reaction, diffusion, and convection, assuming the chemical reaction does not affect the flow field.
Abstract: Turbulent combustion is ubiquitously used in practical combustion devices. However, even chemically non-reacting turbulent flows are complex phenomena, and chemical reactions make the problem even more complicated. Due to the limitation of the computational costs, conventional numerical methods are impractical in carrying out direct 3D numerical simulations at high Reynolds numbers with detailed chemistry. Recently, the lattice Boltzmann method has emerged as an efficient alternative for numerical simulation of complex flows. Compared with conventional methods, the lattice Boltzmann scheme is simple and easy for parallel computing. In this study, we present a lattice Boltzmann model for simulation of combustion, which includes reaction, diffusion, and convection. We assume the chemical reaction does not affect the flow field. Flow, temperature, and concentration fields are decoupled and solved separately. As a preliminary simulation, we study the so-called “counter-flow” laminar flame. The particular flow geometry has two opposed uniform combustible jets which form a stagnation flow. The results are compared with those obtained from solving Navier–Stokes equations.
TL;DR: The Lees-Edwards boundary conditions (LEbc) as discussed by the authors are an extension of the well known periodic boundary conditions and allow the simulation of bulk systems in a simple shear flow.
Abstract: Lees–Edwards boundary conditions (LEbc) for Molecular Dynamics simulations(1) are an extension of the well known periodic boundary conditions and allow the simulation of bulk systems in a simple shear flow. We show how the idea of LEbc can be implemented in isothermal lattice Boltzmann simulations and how LEbc can be used to overcome the problem of a maximum shear rate that is limited to less then 1/L
y
(with L
y
the transverse system size) in traditional lattice Boltzmann implementations of shear flow. The only previous Lattice Boltzmann implementation of LEbc(2) requires a specific fourth order equilibrium distribution. In this paper we show how LEbc can be implemented with the usual quadratic equilibrium distributions.
TL;DR: In this paper, the authors present some results and open problems about stable ergodicity of partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents.
Abstract: We present some results and open problems about stable ergodicity of partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents. The main tool is local ergodicity theory for non-uniformly hyperbolic systems.
TL;DR: In this paper, the total asymmetric exclusion process is investigated by analyzing the dynamics of the shock and the fluctuations of the number of particles and density profiles not only in the stationary state but also in the transient regime.
Abstract: We investigate the total asymmetric exclusion process by analyzing the dynamics of the shock. Within this approach we are able to calculate the fluctuations of the number of particles and density profiles not only in the stationary state but also in the transient regime. We find that the analytical predictions and the simulation results are in excellent agreement.
TL;DR: In this paper, it was shown that the Q matrix introduced in Baxter's 1972 solution of the eight vertex model has some eigenvectors which are not eigenvctors of the spin reflection operator and conjecture a new functional equation for Q(v) which both contains the Bethe equation that gives the eigenvalues of the transfer matrix and computes the degeneracies of these eigen values.
Abstract: We demonstrate that the Q matrix introduced in Baxter's 1972 solution of the eight vertex model has some eigenvectors which are not eigenvectors of the spin reflection operator and conjecture a new functional equation for Q(v) which both contains the Bethe equation that gives the eigenvalues of the transfer matrix and computes the degeneracies of these eigenvalues.
TL;DR: In this article, the moment equations for a granular material under the simplifying assumption of pseudo-Maxwellian particles approximating dissipative hard spheres were derived. But the assumption was not extended to the case of hard spheres.
Abstract: We compute the moment equations for a granular material under the simplifying assumption of pseudo-Maxwellian particles approximating dissipative hard spheres. We obtain the general moment equations of second and third order and the isotropic moment equations of any order. Our equations describe, in the space homogeneous case, the granular system described by a Boltzmann-like collision term and subject to a Brownian motion due to the interaction with a bath, described by a Fokker–Planck term. The trend to equilibrium is studied in detail.