Metastability and the Ising model

We discuss a recent theorem which establishes a precise connection between (i) the approximate degeneracy of the zero eigenvalue for the generator of the Glauber dynamics of the Ising model in a small nonzero field and below the critical temperature, (ii) the existence of a partition of the configuration space into a normal region and a metastable region. This enables us to demonstrate that the recent approach to metastability of Davies and Martin may be viewed as a simple (although in some ways fairly crude) approximation to the conventional approach. We also obtain what appear to be the first results concerning the stability of metastable states under small perturbations.     

Metastability and analyticity in a dropletlike model

We consider an urn model closely related to the Fisher-Felderhof droplet model for the purpose of studying the relation between metastability and analytic continuation. For this model both the statics and dynamics can be solved and we confirm the relation between the metastable decay rate and the imaginary part of the analytically continued free energy (actually, pressure, in this model). We also find that eigenvalue degeneracy, an old theme for static aspects of phase transitions, appears in the dynamics as well. When approaching the phase transition from the stable side it is a degeneracy in the eigenvalues of the linear operator appearing in the master equation that causes the system to lock into a particular phase.Supported in part by the US-Israel Binational Science Foundation and the Technion Fund for the Encouragement of Research.     

On the hydrodynamic limit of a one-dimensional Ginzburg-Landau lattice model. Thea priori bounds

The simplest Ginzburg-Landau model with conservation law is investigated. The initial state is specified by an inhomogeneous profile of the chemical potential associated with the conserved quantity, that is, the mean spin. It is shown that the mean spin satisfies a nonlinear diffusion equation in the hydrodynamic limit. The proof is based on the nice, parabolic structure of the model. A standard perturbation technique is used.     

On the asymptotic behaviour of Spitzer's model for evolution of one-dimensional point systems

A nearest-neighbor gradient dynamics of one-dimensional infinite particle systems is considered; the model admits a two-parameter family of stationary configurations. Some domains of attraction of stationary configurations are described, and the continuum (hydrodynamical) limit of the system is investigated. It is shown that the mean density of points satisfies a nonlinear diffusion equation in the hydrodynamical limit.Research supported by I.H.E.S., Bures-sur-Yvette, France.     

Derivation of a hydrodynamic equation for Ginzburg-Landau models in an external field

The lattice approximation to a time-dependent Ginzburg-Landau equation is investigated in the presence of a small external field. The evolution law conserves the spin, but is not reversible. A nonlinear diffusion equation of divergence type is obtained in the hydrodynamic limit. The proof extends to certain stochastically perturbed Hamiltonian systems.     

Accurate Monte Carlo tests of the stochastic Ginzburg-Landau model with multiplicative colored noise

A accurate and fast Monte Carlo algorithm is proposed for solving the Ginzburg-Landau equation with multiplicative colored noise. The stable cases of solution for choosing time steps and trajectory numbers are discussed.     

On the Long Time Behavior of Infinitely Extended Systems of Particles Interacting via Kac Potentials

We analyze the long time behavior of an infinitely extended system of particles in one dimension, evolving according to the Newton laws and interacting via a non-negative superstable Kac potential (x)=(x), (0, 1]. We first prove that the velocity of a particle grows at most linearly in time, with rate of order . We next study the motion of a fast particle interacting with a background of slow particles, and we prove that its velocity remains almost unchanged for a very long time (at least proportional to ^–1 times the velocity itself). Finally we shortly discuss the so called Vlasov limit, when time and space are scaled by a factor .     

Pulses in a complex Ginzburg-Landau system: Persistence under coupling with slow diffusion

The Ginzburg-Landau (GL) equation is essential for understanding the dynamics of patterns in a wide variety of physical contexts. It governs the evolutions of small amplitude instabilities near criticality. If the instabilities are, however, driven by two coupled instability mechanisms, of which one corresponds with a neutrally stable mode, their evolution is described by a GL equation coupled to a diffusion equation.In this paper, we study the influence of an additional diffusion equation on the existence of pulse solutions in the complex GL equation. In light of recently developed insights into the effect of slow diffusion on the stability of pulses, we consider the case of slow diffusion, i.e., in which the additional diffusion equation acts on a long spatial scale.In previous work [A. Doelman, G. Hek, N. Valkhoff, Stabilization by slow diffusion in a real Ginzburg-Landau system, J. Nonlinear Sci. 14 (2004) 237-278; A. Doelman, G. Hek, N.J.M. Valkhoff, Algebraically decaying pulses in a Ginzburg-Landau system with a neutrally stable mode, Nonlinearity 20 (2007) 357-389], we restricted ourselves to a model with both real coefficients and, more importantly, a real amplitude A rather than the complex-valued A that is needed to completely describe the pattern formation near criticality. In this simpler setting, we proved that pulse solutions of the GL equation can both persist and be stabilized under coupling with a slow diffusion equation. In the current work, we no longer make these restrictions, so that the problem is higher-dimensional and intrinsically harder. By a combination of a geometrical approach and explicit perturbation analysis, we consider the persistence of the solitary pulse solution of the GL equation under coupling with the additional diffusion equation. In the two limiting situations of the nearly real GL equation and the near nonlinear Schrödinger equation, we show that the pulse solutions can indeed persist under this coupling.     

Time-dependent Ginzburg-Landau equation in a car-following model considering the driver’s physical delay

In this paper, an extended car-following model considering the delay of the driver’s response in sensing headway is proposed to describe the traffic jam. It is shown that the stability region decreases when the driver’s physical delay in sensing headway increases. The phase transition among the freely moving phase, the coexisting phase, and the uniformly congested phase occurs below the critical point. By applying the reductive perturbation method, we get the time-dependent Ginzburg-Landau (TDGL) equation from the car-following model to describe the transition and critical phenomenon in traffic flow. We show the connection between the TDGL equation and the mKdV equation describing the traffic jam.     

Three-dimensional Ginzburg-Landau simulation of a vortex line displaced by a zigzag of pinning spheres

A vortex line is shaped by a zigzag of pinning centers and we study here how far the stretched vortex line is able to follow this path. The pinning center is described by an insulating sphere of coherence length size such that in its surface the de Gennes boundary condition applies. We calculate the free energy density of this system in the framework of the Ginzburg-Landau theory and study the critical displacement beyond which the vortex line is detached from the pinning center.     

Extension of the de Broglie-Bohm theory to the Ginzburg-Landau equation

The de Broglie-Bohm approach permits to assign well defined trajectories to particles that obey the Schroedinger equation. We extend this approach to electron pairs in a superconductor. In the stationary regime this extension is completely natural; in the general case additional postulates are required. This approach gives enlightening views for the absence of Hall effect in the stationary regime and for the formation of permanent currents.     

Metastability for a Stochastic Dynamics with a Parallel Heat Bath Updating Rule

We consider the problem of metastability for a stochastic dynamics with a parallel updating rule with single spin rates equal to those of the heat bath for the Ising nearest neighbors interaction. We study the exit from the metastable phase, we describe the typical exit path and evaluate the exit time. We prove that the phenomenology of metastability is different from the one observed in the case of the serial implementation of the heat bath dynamics. In particular we prove that an intermediate chessboard phase appears during the excursion from the minus metastable phase toward the plus stable phase.     

具有在位势的一维双原子链晶格振动的色散关系

在简谐近似下,求解具有在位势的一维双原子链晶格振动运动方程,得到了具有在位势的晶格振动的色散关系.在位势使色散关系声频支在布里渊区中心的振动频率不再为零,并且随在位势的增大而增大.对于原子之间相互作用势不随在位势大小变化的情况下,晶格振动的色散关系的频隙随在位势的增大而变宽.讨论了原子链由只有在位势的不连续极限(AC极限),通过在位势逐渐减弱而原子间相互作用势逐渐增强,最后演变到只有原子间相互作用势的原子链的情况.随着在位势减弱和相互作用势增强,色散关系的频隙由AC极限的孤立轻、重原子简谐振动频率之差逐渐变化到通常的无在位势的色散关系频隙.     

Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits

We present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter (r, t) of a binary alloy undergoing phase segregation. Our model is ad-dimensional lattice gas evolving via Kawasaki exchange with respect to the Gibbs measure for a Hamiltonian which includes both short-range (local) and long-range (nonlocal) interactions. The nonlocal part is given by a pair potential ^dJ(|x–y|), >0 x and y in ^d, in the limit 0. The macroscopic evolution is observed on the spatial scale ^–1 and time scale ^–2, i.e., the density (r, t) is the empirical average of the occupation numbers over a small macroscopic volume element centered atr=x. A rigorous derivation is presented in the case in which there is no local interaction. In a subsequent paper (Part II) we discuss the phase segregation phenomena in the model. In particular we argue that the phase boundary evolutions, arising as sharp interface limits of the family of equations derived in this paper, are the same as the ones obtained from the corresponding limits for the Cahn-Hilliard equation.     

耦合激光列阵的连续模型及其时空特性的研究

继对整体耦合激光列阵的讨论,本文介绍作者最近对局部耦合激光列阵的研究结果.文中采用连续的近似方法,推导出描述耦合激光列阵的连续模型.正如数字计算结果所表明,耦合激光系统是另外一个很典型的“光学流体”系统.本文还对如何设计耦合激光系统,使之具有高功率、单模输出并保持良好的光束特性,进行了定性的讨论.对波导式耦合的本征不稳定性,运用孤子理论进行了初步的分析,进而讨论了耦合激光列阵中的孤子波现象及其应用.     

Metastability and nucleation for the Blume-Capel model. Different mechanisms of transition

We study metastability and nucleation for the Blume-Capel model: a ferromagnetic nearest neighbor two-dimensional lattice system with spin variables taking values in {–1,0, +1}. We consider large but finite volume, small fixed magnetic fieldh, and chemical potential in the limit of zero temperature; we analyze the first excursion from the metastable –1 configuration to the stable +1 configuration. We compute the asymptotic behavior of the transition time and describe the typical tube of trajectories during the transition. We show that, unexpectedly, the mechanism of transition changes abruptly when the lineh=2 is crossed.     

Van der waals limit and phase separation in a particle model with kawasaki dynamics

A one-dimensional interacting particle system with a stochastic dynamics is studied in the local mean field limit, extending the results of Lebowitz, Orlandi, and Presutti to processes which satisfy detailed balance (with respect to Gibbs measures). The behavior of the system below the critical temperature and inside the unstable (spinodal) region is then investigated by means of computer simulations. The experiments clearly indicate the presence of phase separation and confirm the validity of some conjectures on the dynamics of the spinodal decomposition.     

The Ginzburg-Landau expansion in the simple model of a superconductor with a pseudogap

We propose a simple model of the electron spectrum of a two-dimensional system with hot sections on the Fermi surface that significantly transforms the spectral density (pseudogap) in these sections. Using this model, we set up a Ginzburg-Landau expansion for s and d type Cooper pairing and analyze the effect of the pseudogap in the electron spectrum on the main properties of a superconductor. Zh. éksp. Teor. Fiz. 115, 632–648 (February 1999)     

A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model

We give a closer look at the Central Limit Theorem (CLT) behavior in quasi-stationary states of the Hamiltonian Mean Field model, a paradigmatic one for long-range-interacting classical many-body systems. We present new calculations which show that, following their time evolution, we can observe and classify three kinds of long-standing quasi-stationary states (QSS) with different correlations. The frequency of occurrence of each class depends on the size of the system. The different microscopic nature of the QSS leads to different dynamical correlations and therefore to different results for the observed CLT behavior.