Decay of correlations for piecewise expanding maps

This paper investigates the decay of correlations in a large class of non-Markov one-dimensional expanding maps. The method employed is a special version of a general approach recently proposed by the author. Explicit bounds on the rate of decay of correlations are obtained.     

Decay of correlations in surface models

A convergent low-temperature expansion for a variety of models of twodimensional surfaces is presented. It yields existence of the thermodynamic limit for the pressure and correlation functions as well as analyticity inz =e ^– In addition, the estimates give exponential decay of truncated correlations, which proves the existence of a gap in the spectrum of the transfer matrix below the ground state eigenvalue. Two particular examples included in the general framework are the solid-on-solid and discrete Gaussian models.Supported in part by the National Science Foundation under grant No. PHY 79-16812.     

Decay of correlations. IV. Necessary and sufficient conditions for a rapid decay of correlations

We considerN-particle systems whose probability distributions obey the master equation. For these systems, we derive the necessary and sufficient conditions under which the reducedn-particle (n) probabilities also obey master equations and under which the Ursell functions decay to their equilibrium values faster than the probability distributions. These conditions impose restrictions on the form of the transition rate matrix and thus on the form of its eigenfunctions. We first consider systems in which the eigenfunctions of theN-particle transition rate matrix are completely factorized and demonstrate that for such systems, the reduced probabilities obey master equations and the Ursell functions decay rapidly if certain additional conditions are imposed. As an example of such a system, we discuss a random walk ofN pairwise interacting walkers. We then demonstrate that for systems whoseN-particle transition matrix can be written as a sum of one-particle, two-particle, etc. contributions, and for which the reduced probabilities obey master equations, the reduced master equations become, in the thermodynamic limit, those for independent particles, which have been discussed by us previously. As an example of suchN-particle systems, we discuss the relaxation of a gas of interacting harmonic oscillators.Supported in part (grants to D.B. and K.E.S.) by the Advanced Research Projects Agency of the Department of Defense as monitored by the U.S. Office of Naval Research under Contract N00014-69-A-0200-6018, and in part (grant to I.O.) by the National Science Foundation.     

Decay of correlations in the regular Lorentz gas

The regular Lorentz gas on triangular lattice is studied numerically and analytically. The velocity correlation function is shown to decay exponentially in the number of collisions with a decay rate which vanishes as the scatterers approach close packing. The crossover to power law decay at close packing is described by a scaling function.     

Spin correlations in electron liquids: Analytical results for the local-field correction in two and three dimensions

We study the spin correlations in two- and three-dimensional electron liquids within the sum-rule version of the self-consistent field approach of Singwi, Tosi, Land, and Sj?lander. Analytic expressions for the spin-antisymmetric static structure factor and the corresponding local-field correction are obtained with density dependent coefficients. We calculate the spin-dependent pair-correlation functions, paramagnon dispersion, and static spin-response function within the present model, and discuss the spin-density wave instabilities in double-layer electron systems. Received: 22 September 1997 / Revised: 1 December 1997 / Accepted: 4 December 1997     

Decay of correlations in spin systems

We investigate the decay of initial correlations in a spin system where each spin relaxes independently by an intramolecular mechanism. The equation of motion for the spin density matrix is assumed to be the Redfield equation, which is of the form of a quantum mechanical master equation. Our analysis of this problem is based on the techniques of Shuler, Oppenheim, and coworkers, who have studied the decay of correlations in systems which can be described by classical stochastic master equations. We find that the off-diagonal elements of the reduced spin density matrices approach their equilibrium values faster than the diagonal elements. The Ursell functions, which are a measure of the correlations in the system, decay to their zero equilibrium values faster than the spin density matrix except for the furthest off-diagonal elements. Far off-diagonal matrix elements of the spin density matrix approach equilibrium at the same rate as the Ursell functions, which is the important difference between the quantum mechanical model studied here and the classical models studied earlier.Supported in part by the National Science Foundation.     

Ground-state properties of the Falicov-Kimball model in one and two dimensions

A new numerical method is used to study the ground-state properties of the spinless Falicov-Kimball model in one and two dimensions. The resultant solutions are used to examine the phase diagram of the model as well as possibilities for valence and metal-insulator transitions. In one dimension a comprehensive phase diagram of the model is presented. On the base of this phase diagram, the complete picture of valence and metal-insulator transitions is discussed. In two dimensions the structure of ground-state configurations is described for intermediate interactions between f and d electrons. In this region the phase separation and metal-insulator transitions are found at low f-electron concentrations. It is shown that valence transitions exhibit a staircase structure. Received 20 October 2000     

一类具有两个边界的分段光滑系统中边界碰撞分岔现象及混沌

以不连续运行模式下的电流反馈型Buck-Boost变换器为例，导出了一类具有三段形式的分段光滑迭代映射方程，数值仿真得到了输入电压变化时的分岔图.结果表明，发生分岔时映射雅可比矩阵的特征值以不连续的方式跳跃出复平面上的单位圆，而且映射总有某个或某些轨道点位于相平面中不同区域的边界上，即映射随输入电压的变化会发生边界碰撞分岔现象，如由周期态到周期态以及由周期态到混沌态的分岔. 关键词： 分段光滑系统 边界碰撞分岔 混沌     

The depletion potential in one, two and three dimensions

We study the behavior of the depletion potential in binary mixtures of hard particles in one, two, and three dimensions within the framework of a general theory for depletion potential using density functional theory. By doing so we extend earlier studies of the depletion potential in three dimensions to the cases ofd = 1 and 2 about which little is known, despite their importance for experiments. We also verify scaling relations between depletion potentials in sphere-sphere and wall-sphere geometries ind = 3 and in disk-disk and wall-disk geometries ind = 2, which originate from geometrical considerations.     

Topological Classification of Morse Functions and Generalisations of Hilbert’s 16-th Problem

The topological structures of the generic smooth functions on a smooth manifold belong to the small quantity of the most fundamental objects of study both in pure and applied mathematics. The problem of their study has been formulated by A. Cayley in 1868, who required the classification of the possible configurations of the horizontal lines on the topographical maps of mountain regions, and created the first elements of what is called today ‘Morse Theory’ and ‘Catastrophes Theory’. In the paper we describe this problem, and in particular describe the classification of Morse functions on the 2 sphere and on the torus.      

Incoherently coupled photorefractive bright-bright soliton pairs in one and two dimensions

We investigate incoherently coupled bright-bright soliton pairs in photorefractive crystals under steady-state condition in both one and two transverse dimensions. The novel numerical scheme according to the Crank-Nicholson method generalized in three dimensions accompanied by the central difference method was used to solve the coupled wave and potential equations in order to find soliton solution and pairs distribution. Simulation of propagation is performed numerically and show that the presence of both components is required for stable propagation. This simulation approach demonstrates beams evolution after decoupling clearly.     

Decay of correlations and related sum rules in a layered classical plasma

Summary The asymptotic behaviours of particle correlation functions and the related sum rules are discussed for a layered classical plasma withe ²/r interactions in the fluid state, in dependence on the number of layers. These properties derive from consistency conditions imposed by screening on the hierarchical equations, as already treated by A. Alastuey and P. A. Martin (J. Stat. Phys.,39, 405 (1985)) for various Coulomb fluids. The main results concern i) the type of clustering of correlations needed for the validity of multipolar sum rules at various orders, ii) the proof that the pair correlation function in a finite multilayer may carry an electric dipole moment and the calculation of its partioning among the layers, and iii) the dimensionality crossover in an infinitely extended or periodically repeated multiplayer with varying interlayer spacing and wave vector.     

Self-assembling of gold nanoparticles in one, two, and three dimensions

In the present work, we report the growth of ordered arrays of gold nanoparticles passivated with n-alkylthiol molecules using crystallization in toluene vapor. We kept constant the average particle size and the length of the passivating molecule (1-dodecanethiol). The temperature and time of growth were varied. We show that in the initial stages of the growth, nano-chains of particles are produced. These nano-chains aggregate to form two-dimensional arrays. In the initial stages the nano-chains form a two-dimensional square lattice which then relaxes to a close-packed structure. In latter stages of growth a three-dimensional supercrystal is produced. It is found that the packing of nanoparticles corresponds to an average FCC lattice. However, large variations on the local parameters of the lattice are observed. Near the edges of the supercrystal the anomalous packing reported by Zanchet et al. [20] and Fink et al. [21] was observed. The energy of the observed structures is analyzed using molecular dynamics simulation. It is concluded that using the vapor growth method, it is possible to produce controlled ordered structures from one to three dimensions. Received: 24 February 2000 / Accepted: 28 March 2000 / Published online: 13 July 2000     

Schematic models for fragmentation of brittle solids in one and two dimensions

Stochastic models for the development of cracks in 1D and 2D objects are presented. In one dimension, we focus on particular scenarios for interacting fragments (IF1) and non-interacting fragments (NIF1) during the breakup process. For 2D objects, we consider only NIF1, but analyze isotropic and anisotropic development of fissures. Analytical results are given for many observables. Power-law size distributions are predicted for some of the fragmentation pictures considered.     

Infrared behaviour of metallic systems in one,two and three dimensions

A simple approximate expression for the electron lifetime() in metals is rederived and discussed for different dimensions. In the 3D-case we get the well known Drude behaviour, i.e. a constant. In one dimension() is strongly frequency-dependent in the IR. The 2D-case is intermediate to the preceding ones. These results are essentially due to the different form of the Fermi surface for an electron gas in one, two and three dimensions.     

Infrared behaviour of metallic systems in one, two and three dimensions

A simple approximate expression for the electron lifetime() in metals is rederived and discussed for different dimensions. In the 3D-case we get the well known Drude behaviour, i.e. a constant. In one dimension() is strongly frequency-dependent in the IR. The 2D-case is intermediate to the preceding ones. These results are essentially due to the different form of the Fermi surface for an electron gas in one, two and three dimensions.     

Ground states of the generalized Falicov-Kimball model in one and two dimensions

A combination of small-cluster exact-diagonalization calculations and a well-controlled approximative method is used to study the ground-state phase diagram of the spin-one-half Falicov-Kimball model extended by the spin-dependent on-site interaction between localized (f) and itinerant (d) electrons. Both the magnetic and charge ordering are analysed as functions of the spin-dependent on-site interaction (J) and the total number of itinerant (N_d) and localized (N_f) electrons at selected U (the spin-independent interaction between the f and d electrons). It is shown that the spin-dependent interaction (for N_f=L, where L is the number of lattice sites) stabilizes the fully polarized (FP) and partially polarized (PP) state, while the stability region of the non polarized (NP) phase is gradually reduced. The precisely opposite effect on the stability of FP, PP and NP phases has a reduction of N_f. Moreover, the strong coupling between the f and d-electron subsystems is found for both N_f=L as well as N_f < L.     

Seven basic regimes of steady crystal growth in two dimensions

Using a Markov rate-process model, exact expressions are found for the steady growth rate of an edge of a two-dimensional crystal in terms of the numberM of particles along the edge, the height difference (or number of permanent steps)K along the edge, the nucleation rate , and the speed + of movement of steps. The familiar growth regimes can be identified with asymptotic regimes for the parametersK, (v/)^1/2, andM. From a mathematical viewpoint, there are seven basic regimes, of which the known physical regimes are special cases.     

卤化银微晶中光电子衰减谱的特性

光电子在卤化银材料潜影形成过程中发挥着重要的作用 ,光电子衰减行为在很大程度上决定于卤化银的晶体结构。采用光学与微波双共振技术 ,测量了自由光电子和浅俘获光电子的衰减谱 ,得到了卤化银中电子陷阱的密度和深度分布。以自由光电子寿命为纽带 ,通过分析掺杂卤化银晶体中电子陷阱的分布情况 ,可以确定浅电子陷阱掺杂剂的最佳掺杂浓度。