Monte Carlo simulation of quantum statistical lattice models

In this article we review recent developments in computational methods for quantum statistical lattice problems. We begin by giving the necessary mathematical basis, the generalized Trotter formula, and discuss the computational tools, exact summations and Monte Carlo simulation, that will be used to examine explicit examples. To illustrate the general strategy, the method is applied to an analytically solvable, non-trivial, model: the one-dimensional Ising model in a transverse field. Next it is shown how to generalized Trotter formula most naturally leads to different path-integral representations of the partition function by considering one-dimensional fermion lattice models. We show how to analyze the different representations and discuss Monte Carlo simulation results for one-dimensional fermions. Then Monte Carlo work on one- and two-dimensional spin-$\text{1}\text{2}$ models based upon the Trotter formula approach is reviewed and the more dedicated Handscomb Monte Carlo method is discussed. We consider electron-phonon models and discuss Monte Carlo simulation data on the Molecular Crystal Model in one, two and three dimensions and related one-dimensional polaron models. Exact numerical results are presented for free fermions and free bosons in the canonical ensemble. We address the main problem of Monte Carlo simulations of fermions in more than one dimension: the cancellation of large contributions. Free bosons on a lattice are compared with bosons in a box and the effects of finite size on Bose-Einstein condensation are discussed.     

蚀刻表面面形的分析

采用一个有效的数学模型,分析了在蚀刻工艺中基底自身面形轮廓的曲线形状对基底局部区域的蚀刻速率产生的影响,并通过对数学模型的理论分析和计算机模拟得出受此影响而产生的面形形状,并将结果与实验进行对比.利用这个数学模型对使用离子束蚀刻制作单台阶光栅的台阶与沟槽部分的表面面形随时间的演变过程进行了计算机模拟分析,并通过把理论结果与在实验中得到的蚀刻表面在原子力显微镜(AFM)下拍摄的照片进行比较,结果说明这种模拟分析能够保证对该问题分析所要求的精度,从而也证明了理论模型的合理性和正确性.     

Disorder-Assisted Error Correction in Majorana Chains

It was recently realized that quenched disorder may enhance the reliability of topological qubits by reducing the mobility of anyons at zero temperature. Here we compute storage times with and without disorder for quantum chains with unpaired Majorana fermions ?? the simplest toy model of a quantum memory. Disorder takes the form of a random site-dependent chemical potential. The corresponding one-particle problem is a one-dimensional Anderson model with disorder in the hopping amplitudes. We focus on the zero-temperature storage of a qubit encoded in the ground state of the Majorana chain. Storage and retrieval are modeled by a unitary evolution under the memory Hamiltonian with an unknown weak perturbation followed by an error-correction step. Assuming dynamical localization of the one-particle problem, we show that the storage time grows exponentially with the system size. We give supporting evidence for the required localization property by estimating Lyapunov exponents of the one-particle eigenfunctions. We also simulate the storage process for chains with a few hundred sites. Our numerical results indicate that in the absence of disorder, the storage time grows only as a logarithm of the system size. We provide numerical evidence for the beneficial effect of disorder on storage times and show that suitably chosen pseudorandom potentials can outperform random ones.     

Hysteresis in a superfluid atom circuit

We study a hysteresis phenomenon in a rotating BEC with a weak link in a quasi-one-dimensional torus by proposing a microscopic theoretical model including a dissipation bath. By analyzing the role of dissipation and the decay rates of all the energy levels, we are able to give a microscopic interpretation of hysteresis recently observed in the experiment and confirm that the hysteresis is the result of the presence of metastable state. In particular, we obtain the hysteresis loops in a quench process just as that in the experiment. We also find that the shape and size of the hysteresis loop change drastically with the strength of the link.     

Symbolic dynamics and relaxation oscillations

This paper describes the application of qualitative methods of dynamical systems theory to a specific problem. It examines the forced Van der Pol equation as an example of a relaxation oscillation with aperiodic solutions. The technique of symbolic dynamics, particularly for one-dimensional mappings, is used to give a complete topological characterization of the set of these aperiodic solutions for parameter values for which the equation appears structurally stable. These results are a mathematical interpretation of numerical computations and are not the result of rigorous analysis.     

Frenkel-Kontorova模型中基底势振动的影响

基于一维Frenkel-Kontorova模型, 研究了振动的基底势对系统纳米摩擦现象的影响. 分别在相邻原子间的距离与周期势场的周期比为不公度(incommensurate)、可公度(commensurate)两种情形下, 探讨了基底势振动的振幅和频率对滞回现象(hysteresis)、最大静摩擦力以及超滑现象的作用机理. 两种情形下, 固定频率, 随着振幅的增大, 滞回区域的面积以及最大静摩擦力都将减小, 对于不同的频率, 减小的趋势不同. 系统甚至产生了超滑现象. 但当频率过大时, 振幅的改变不会影响滞回区域的面积以及最大静摩擦力的大小, 此时与基底不加振动时的情形一致; 当振幅固定, 随着频率的增大, 滞回区域的面积将增大, 对于不同振幅, 增大的趋势不同. 特别地, 对于某些固定的振幅, 最大静摩擦力随着振动频率的增大先逐步减小直至出现超滑现象, 再进一步增大频率, 最大静摩擦力又转而逐步增大. 这一现象类似于共振, 表明存在最佳的振动频率促进系统内所有原子的共同运动, 使得整个系统的最大静摩擦力几乎消失. 另外, 两种情形的区别是, 对于某些固定的频率(如ω= 0.5)和不同的小振幅, 不可公度情形往往具有相同的平均终止速度, 而可公度情形则不同, 表明相同前提下后者具有更复杂的动力学行为. 关键词： Frenkel-Kontorova模型 滞回 最大静摩擦力 超润滑     

Modeling of the optical properties of porous silicon photonic crystals in the visible spectral range

Optical devices based on photonic crystals are of great interest because they can be efficiently used in laser physics and biosensing. Photonic crystals allow one to control the propagation of electromagnetic waves and to change the emission characteristics of luminophores embedded into photonic structures. One of the most interesting materials for developing one-dimensional photonic crystals is porous silicon. However, an important problem in application of this material is the control of the refractive index of layers by changing their porosity, as well as the refractive index dispersion. In addition, it is important to have the possibility of modeling the optical properties of structures to choose precisely select the fabrication parameters and produce one-dimensional photonic crystals with prescribed properties. In order to solve these problems, we used a mathematical model based on the transfer matrix method, using the Bruggeman model, and on the dispersion of silicon refractive index. We fabricated microcavities by electrochemical etching of silicon, with parameters determined by the proposed model, and measured their reflection spectra. The calculated results showed good agreement with experimental data. The model proposed allowed us to achieve a microcavity Q-factor of 160 in the visible region.     

Electro–magneto interaction in fractional Green-Naghdi thermoelastic solid with a cylindrical cavity

A unified mathematical model of Green–Naghdi’s thermoelasticty theories (GN), based on fractional time-derivative of heat transfer is constructed. The model is applied to solve a one-dimensional problem of a perfect conducting unbounded body with a cylindrical cavity subjected to sinusoidal pulse heating in the presence of an axial uniform magnetic field. Laplace transform techniques are used to get the general analytical solutions in Laplace domain, and the inverse Laplace transforms based on Fourier expansion techniques are numerically implemented to obtain the numerical solutions in time domain. Comparisons are made with the results predicted by the two theories. The effects of the fractional derivative parameter on thermoelastic fields for different theories are discussed.     

Mathematical modeling of laser ablation in liquids with application to laser ultrasonics

The use of laser ablation as a means of generating ultrasonic waves in liquid metals is studied in this paper. A mathematical model for predicting the onset of ablation is developed, as is a model of the ablation process based on steady state, one-dimensional gas dynamics in which the vapor phase is treated as an ideal gas. The results of this model are then used in a quasi-two-dimensional model of laser ablation that accounts for the spatial distribution of intensity in the laser beam. Model predictions are compared with experiments conducted on liquid mercury and excellent agreement is obtained. Based on these results, a simplified model is developed that shows excellent agreement with both the theory and the experiments.     

Spectral properties of linear chain with one-point dynamical disorder

We have investigated the one-dimensional model of an exciton moving along a (finite or semi-infinite) chain of sites with the local energy at one specific site being randomly modulated. The modulation is described by the Markoff stochastic process and we do not invoke the white-noise assumption. For this model, we give an approximation-free calculation of the density of states, the projected density of states and the optical absorption spectrum. The exact solution is found for a broad family of stochastic processes which bridges (and includes) the dichotomic process and the Gaussian one. The results are discussed in various physical limits such as whitenoise limit, weak-perturbation limit and weak-tunneling limit. Particularly, in the slow-modulation (or static) limit, our model represents a chain with a static distribution of energy at the distinguished impurity site and the spectral characteristics reveal a nontrivial dependence on the width of this energy distribution.     

Tb0.3Dy0.7Fe2合金磁畴偏转的滞回特性研究

研究了不同载荷作用下Tb_0.3Dy_0.7Fe₂合金在压磁和磁弹性效应中磁畴偏转的滞回特性. 基于Stoner-Wolhfarth模型的能量极小原理, 采用绘制自由能-磁畴偏转角度关系曲线的求解方法, 研究了压磁和磁弹性效应中载荷作用下的磁畴角度偏转和磁化过程, 计算分析了不同载荷作用下磁畴偏转的滞回特性. 研究表明, 压磁和磁弹性效应中磁畴偏转均存在明显的滞回、跃迁效应, 其中磁化强度的滞回效应来源于磁畴偏转的角度跃迁; 压磁效应中预加磁场的施加将增大磁化强度的滞回, 同时使滞回曲线向大压应力方向偏移; 磁弹性效应中磁畴偏转的滞回存在两个临界磁场强度, 不同磁场强度下合金具有不同的磁畴偏转路径和磁化滞回曲线, 临界磁场强度的大小取决于预压应力的施加. 理论分析对类磁致伸缩材料磁畴偏转模型的完善和材料器件的设计应用非常有意义.     

Current progress on heat conduction in one-dimensional gas channels

We give a brief review of the past development of model studies on one-dimensional heat conduction. Particularly, we describe recent achievements on the study of heat conduction in one-dimensional gas models including the hard-point gas model and billiard gas channel. For a one-dimensional gas of elastically colliding particles of unequal masses, heat conduction is anomalous due to momentum conservation, and the divergence exponent of heat conductivity is estimated as α≈0.33 in k ∼ L ^α . Moreover, in billiard gas models, it is found that exponent instability is not necessary for normal heat conduction. The connection between heat conductivity and diffusion is investigated. Some new progress is reported. A recently proposed model with a quantized degree of freedom to study the heat transport in quasi-one dimensional systems is illustrated in which three distinct temperature regimes of heat conductivity are manifested. The establishment of local thermal equilibrium (LTE) in homogeneous and heterogeneous systems is also discussed. Finally, we give a summary with an outlook for further study about the problem of heat conduction.     

一维铁磁链中量子孤波的能级和磁矩

使用Hartree近似和一种简化的准离散多标度方法,研究了具有交换作用和经典磁矩相互作用的一维铁磁链中的量子孤波解. 这种一维铁磁链中不仅存在着运动的量子孤波,也存在着静态的量子孤波(即量子内禀局域模).利用所获得的量子孤波解,进一步研究了量子孤波的能级和由量子孤波所携带的磁矩. 研究表明,量子孤波的能量和磁矩都是量子化的,这些结果为正确理解磁性材料中像磁滞回线的量子台阶等宏观量子特性提供了一条可能的途径.     

Universality of One-Dimensional Fermi Systems,I. Response Functions and Critical Exponents

The critical behavior of one-dimensional interacting Fermi systems is expected to display universality features, called Luttinger liquid behavior. Critical exponents and certain thermodynamic quantities are expected to be related among each other by model-independent formulas. We establish such relations, the proof of which has represented a challenging mathematical problem, for a general model of spinning fermions on a one dimensional lattice; interactions are short ranged and satisfy a positivity condition which makes the model critical at zero temperature. Proofs are reported in two papers: in the present one, we demonstrate that the zero temperature response functions in the thermodynamic limit are Borel summable and have anomalous power-law decay with multiplicative logarithmic corrections. Critical exponents are expressed in terms of convergent expansions and depend on all the model details. All results are valid for the special case of the Hubbard model.     

Hysteresis compensation of piezoelectric actuators: The modified Rayleigh model

In this study, we develop a novel modified Rayleigh model for hysteresis compensation in piezoelectric actuators. Piezoelectric actuators suffer from hysteresis, in large drive fields of more than 100 V, which can result in serious displacement errors. The typical phenomenological approach is to use the Rayleigh model; however, this model gives more than 10% difference with experiments at the large electric fields of more than 1 kV/mm. Furthermore, there are no studies that apply the Rayleigh model to the compensation of precision actuators, such as stack actuators; it has only been applied in the study of the physical properties of piezoelectric materials. Therefore, we propose a modified Rayleigh model, in which each coefficient is defined differently according to whether the field is increasing or decreasing to account for asymmetry at the high fields. By applying a computer-based control from an inverse form of this modified Rayleigh model, we show that we can compensate for hysteresis to reduce the position error to less than five percent. This model has the merits of reducing complicated fitting procedures and of saving computation time compared to the Preisach model. Specifically, this model cannot only predict the hysteresis curves in all local fields using only one fitting procedure, but also make it possible to control the displacement of various piezo-based actuators without expensive sensors, based on the charge-based model.     

A new open-loop driving method of piezoelectric actuator for periodic reference inputs

The hysteresis characteristic of piezoelectric actuators is one of the major deficiencies in a wide variety of precise tracking positioning controls. In this paper, a novel mathematical model is proposed to describe hysteresis precisely and an open-loop tracking controller is implemented with the proposed model to compensate the hysteresis for tracking periodic sinusoidal inputs. Finally, some tracking control experiments based on a micro-positioning stage for the desired sinusoidal trajectory are performed according to the proposed method and the experimental results demonstrate that the positioning precision is noticeably improved in the open-loop operation compared to the conventional open-loop control without any compensation. In comparison to the regular PID closed-loop controller, the proposed feedforward control method has better tracking performance.     

Genetic demixing and evolution in linear stepping stone models

Results for mutation, selection, genetic drift, and migration in a one-dimensional continuous population are reviewed and extended. The population is described by a continuous limit of the stepping stone model, which leads to the stochastic Fisher-Kolmogorov-Petrovsky-Piscounov equation with additional terms describing mutations. Although the stepping stone model was first proposed for population genetics, it is closely related to "voter models" of interest in nonequilibrium statistical mechanics. The stepping stone model can also be regarded as an approximation to the dynamics of a thin layer of actively growing pioneers at the frontier of a colony of micro-organisms undergoing a range expansion on a Petri dish. The population tends to segregate into monoallelic domains. This segregation slows down genetic drift and selection because these two evolutionary forces can only act at the boundaries between the domains; the effects of mutation, however, are not significantly affected by the segregation. Although fixation in the neutral well-mixed (or "zero-dimensional") model occurs exponentially in time, it occurs only algebraically fast in the one-dimensional model. An unusual sublinear increase is also found in the variance of the spatially averaged allele frequency with time. If selection is weak, selective sweeps occur exponentially fast in both well-mixed and one-dimensional populations, but the time constants are different. The relatively unexplored problem of evolutionary dynamics at the edge of an expanding circular colony is studied as well. Also reviewed are how the observed patterns of genetic diversity can be used for statistical inference and the differences are highlighted between the well-mixed and one-dimensional models. Although the focus is on two alleles or variants, q-allele Potts-like models of gene segregation are considered as well. Most of the analytical results are checked with simulations and could be tested against recent spatial experiments on range expansions of inoculations of Escherichia coli and Saccharomyces cerevisiae.     

Efficient vector hysteresis modeling using rotationally coupled step functions

Vector hysteresis models are usually used as sub-modules of field computation software tools. When dealing with a massive field computation problem, computational efficiency and practicality of such models become crucial. In this paper, generalization of a recently proposed computationally efficient vector hysteresis model based upon interacting step functions is presented. More specifically, the model is generalized to cover vector hysteresis modeling of both isotropic and anisotropic magnetic media. Model configuration details as well as experimental testing and simulation results are given in the paper.     

雷达天线控制系统频域辨识与仿真方法研究

雷达天线控制系统仿真是雷达系统目标跟踪动态仿真中一个重要环节。为解决雷达天线控制系统仿真中滞后误差大、动态效果差等问题,依据某雷达天线控制系统性能实际测试结果,以典型系统模型作为研究对象,利用频域最小二乘法完成模型参数辨识,建立系统模型,并对辨识模型进行验证。在此基础上,采用基于时域的增广矩阵法对系统进行了数字快速仿真分析。结果表明,辨识模型满足精度,且增广矩阵法能够有效快速地完成系统数字仿真,对天线控制系统建模以及快速动态仿真研究具有一定的参考价值。