Singularly Perturbed Vector and Scalar Nonlinear Schroedinger Equations with Persistent Hom
简并 量子力学 英语
简并量子力学英语English:In quantum mechanics, the concept of degeneracy refers to the situation where multiple different quantum states correspond to the same energy level. This can occur in systems such as atoms, where different orbital and spin configurations can result in the same total energy. Degeneracy is a crucial concept in quantum mechanics as it has far-reaching implications for the behavior and properties of quantum systems. For example, it can affect the spectral lines observed in atomic emission spectra and the fine structure of energy levels in atoms. Understanding and managing degeneracy is therefore essential for accurately predicting and describing the behavior of quantum systems.中文翻译:在量子力学中，简并性的概念指的是多个不同的量子状态对应相同的能级的情况。
这种情况可以出现在原子等系统中，不同的轨道和自旋配置可能会导致相同的总能量。
简并性是量子力学中一个至关重要的概念，因为它对量子系统的行为和性质有着广泛的影响。
量子力学英语
量子力学英语
随着量子力学的发展和应用，许多新的概念和术语相继出现。
掌握量子力学英语不仅有利于学习和研究，还可以更好地沟通和交流。
以下是一些常用的量子力学英语词汇：
1. Quantum mechanics 量子力学
2. Wave function 波函数
3. Schrdinger equation 薛定谔方程
4. Uncertainty principle 不确定性原理
5. Superposition principle 叠加原理
6. Entanglement 纠缠
7. Quantum state 量子态
8. Eigenvalue 特征值
9. Eigenfunction 特征函数
10. Hamiltonian 哈密顿量
11. Operator 算符
12. Commutation relation 对易关系
13. Quantum tunneling 量子隧穿
14. Quantum entanglement 量子纠缠
15. Quantum superposition 量子叠加
以上是一些常用的量子力学英语词汇，学习量子力学英语需要不断积累和运用。
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[精彩]量子力学英语词汇
![[精彩]量子力学英语词汇](https://img.taocdn.com/s3/m/2f2f73d60d22590102020740be1e650e52eacf14.png)
1、microscopic world 微观世界2、macroscopic world 宏观世界3、quantum theory 量子[理]论4、quantum mechanics 量子力学5、wave mechanics 波动力学6、matrix mechanics 矩阵力学7、Planck constant 普朗克常数8、wave-particle duality 波粒二象性9、state 态10、state function 态函数11、state vector 态矢量12、superposition principle of state 态叠加原理13、orthogonal states 正交态14、antisymmetrical state 正交定理15、stationary state 对称态16、antisymmetrical state 反对称态17、stationary state 定态18、ground state 基态19、excited state 受激态20、binding state 束缚态21、unbound state 非束缚态22、degenerate state 简并态23、degenerate system 简并系24、non-deenerate state 非简并态25、non-degenerate system 非简并系26、de Broglie wave 德布罗意波27、wave function 波函数28、time-dependent wave function 含时波函数29、wave packet 波包30、probability 几率31、probability amplitude 几率幅32、probability density 几率密度33、quantum ensemble 量子系综34、wave equation 波动方程35、Schrodinger equation 薛定谔方程36、Potential well 势阱37、Potential barrien 势垒38、potential barrier penetration 势垒贯穿39、tunnel effect 隧道效应40、linear harmonic oscillator线性谐振子41、zero proint energy 零点能42、central field 辏力场43、Coulomb field 库仑场44、δ-function δ-函数45、operator 算符46、commuting operators 对易算符47、anticommuting operators 反对易算符48、complex conjugate operator 复共轭算符49、Hermitian conjugate operator 厄米共轭算符50、Hermitian operator 厄米算符51、momentum operator 动量算符52、energy operator 能量算符53、Hamiltonian operator 哈密顿算符54、angular momentum operator 角动量算符55、spin operator 自旋算符56、eigen value 本征值57、secular equation 久期方程58、observable 可观察量59、orthogonality 正交性60、completeness 完全性61、closure property 封闭性62、normalization 归一化63、orthonormalized functions 正交归一化函数64、quantum number 量子数65、principal quantum number 主量子数66、radial quantum number 径向量子数67、angular quantum number 角量子数68、magnetic quantum number 磁量子数69、uncertainty relation 测不准关系70、principle of complementarity 并协原理71、quantum Poisson bracket 量子泊松括号72、representation 表象73、coordinate representation 坐标表象74、momentum representation 动量表象75、energy representation 能量表象76、Schrodinger representation 薛定谔表象77、Heisenberg representation 海森伯表象78、interaction representation 相互作用表象79、occupation number representation 粒子数表象80、Dirac symbol 狄拉克符号81、ket vector 右矢量82、bra vector 左矢量83、basis vector 基矢量84、basis ket 基右矢85、basis bra 基左矢86、orthogonal kets 正交右矢87、orthogonal bras 正交左矢88、symmetrical kets 对称右矢89、antisymmetrical kets 反对称右矢90、Hilbert space 希耳伯空间91、perturbation theory 微扰理论92、stationary perturbation theory 定态微扰论93、time-dependent perturbation theory 含时微扰论94、Wentzel-Kramers-Brillouin method W. K. B.近似法95、elastic scattering 弹性散射96、inelastic scattering 非弹性散射97、scattering cross-section 散射截面98、partial wave method 分波法99、Born approximation 玻恩近似法100、centre-of-mass coordinates 质心坐标系101、laboratory coordinates 实验室坐标系102、transition 跃迁103、dipole transition 偶极子跃迁104、selection rule 选择定则105、spin 自旋106、electron spin 电子自旋107、spin quantum number 自旋量子数108、spin wave function 自旋波函数109、coupling 耦合110、vector-coupling coefficient 矢量耦合系数111、many-partic le system 多子体系112、exchange forece 交换力113、exchange energy 交换能114、Heitler-London approximation 海特勒-伦敦近似法115、Hartree-Fock equation 哈特里-福克方程116、self-consistent field 自洽场117、Thomas-Fermi equation 托马斯-费米方程118、second quantization 二次量子化119、identical particles全同粒子120、Pauli matrices 泡利矩阵121、Pauli equation 泡利方程122、Pauli’s exclusion principle泡利不相容原理123、Relativistic wave equation 相对论性波动方程124、Klein-Gordon equation 克莱因-戈登方程125、Dirac equation 狄拉克方程126、Dirac hole theory 狄拉克空穴理论127、negative energy state 负能态128、negative probability 负几率129、microscopic causality 微观因果性本征矢量eigenvector本征态eigenstate本征值eigenvalue本征值方程eigenvalue equation本征子空间eigensubspace (可以理解为本征矢空间)变分法variatinial method标量scalar算符operator表象representation表象变换transformation of representation表象理论theory of representation波函数wave function波恩近似Born approximation玻色子boson费米子fermion不确定关系uncertainty relation狄拉克方程Dirac equation狄拉克记号Dirac symbol定态stationary state定态微扰法time-independent perturbation定态薛定谔方程time-independent Schro(此处上面有两点)dinger equati on 动量表象momentum representation角动量表象angular mommentum representation占有数表象occupation number representation坐标（位置）表象position representation角动量算符angular mommentum operator角动量耦合coupling of angular mommentum对称性symmetry对易关系commutator厄米算符hermitian operator厄米多项式Hermite polynomial分量component光的发射emission of light光的吸收absorption of light受激发射excited emission自发发射spontaneous emission轨道角动量orbital angular momentum自旋角动量spin angular momentum轨道磁矩orbital magnetic moment归一化normalization哈密顿hamiltonion黑体辐射black body radiation康普顿散射Compton scattering基矢basis vector基态ground state基右矢basis ket ‘右矢’ket基左矢basis bra简并度degenerancy精细结构fine structure径向方程radial equation久期方程secular equation量子化quantization矩阵matrix模module模方square of module内积inner product逆算符inverse operator欧拉角Eular angles泡利矩阵Pauli matrix平均值expectation value （期望值）泡利不相容原理Pauli exclusion principle氢原子hydrogen atom球鞋函数spherical harmonics全同粒子identical partic les塞曼效应Zeeman effect上升下降算符raising and lowering operator 消灭算符destruction operator产生算符creation operator矢量空间vector space守恒定律conservation law守恒量conservation quantity投影projection投影算符projection operator微扰法pertubation method希尔伯特空间Hilbert space线性算符linear operator线性无关linear independence谐振子harmonic oscillator选择定则selection rule幺正变换unitary transformation幺正算符unitary operator宇称parity跃迁transition运动方程equation of motion正交归一性orthonormalization正交性orthogonality转动rotation自旋磁矩spin magnetic monent(以上是量子力学中的主要英语词汇，有些未涉及到的可以自由组合。
Multiple Time Series
5.1 Vector Autoregressions
323
5.1.2
Properties of a VAR(1)
The properties of the VAR(1) are fairly simple to study. More importantly, section 5.2 shows that all VAR(P)s can be rewritten as a VAR(1), and so the general case requires no additional effort than the first order VAR.
In many situations, analyzing a time-series in isolation is reasonable; in other cases univariate analysis may be limiting. For example, Campbell (1996) links financially interesting variables, including stock returns and the default premium, in a multivariate system that allows shocks to one variable to propagate to the others. The vector autoregression is the mechanism that is used to link multiple stationary time-series variables together. When variables contain unit roots, a different type of analysis, cointegration, is needed. This chapter covers these two topics building on many results from the analysis of univariate time-series.
On the modelling of multiphase turbulent flows
On the modelling of multiphase turbulent flowsfor environmental and hydrodynamic applicationsDjamel LakehalInstitute of Energy Technology,ETH Zurich,ETH-Zentrum/CLT2,CH-8092Zurich,SwitzerlandReceived 1November 2000;received in revised form 24November 2001AbstractThe paper examines a selection of well-established prediction methods employed for the modelling of multiphase turbulent flows presented in typical environmental and hydrodynamic applications.The main objective is to provide a basic understanding of the subject with a deliberate intention to simplifying the presentation.Turbulence is approached on the basis of the conventional one-point closure context.The experience gathered by the author and by others with various predictive strategies all based on the Eule-rian–Eulerian (field description)and the Eulerian–Lagrangian methods are discussed and summarized;the goals,limitations,and required developments are described.Typical applications of each calculation method are presented,in which the interaction between the transported dispersed-phase and the field turbulence is treated on the basis of both one-way and two-way coupling.The case studies in question include aerosol production and transport over the oceans,pollutant dispersion in the atmospheric surface layer,hydrometeor impact on urban canopies,sedimentation of active sludge in secondary water clarifiers,and mixing and circulation within confined bubble plumes.Analysis of the various models reveals that for most of the reported applications the Reynolds averaged Navier–Stokes approach is inherently ill-posed and should be transcended by the promising large-eddy simulation concept.Ó2002Elsevier Science Ltd.All rights reserved.Keywords:Turbulent-Flow;Droplets;Dilute Suspensions;Bubbles;Particles1.IntroductionThis paper is written in the spirit of an overview of different multiphase modelling methods,with emphasis on the practical motivations for certain selected applications and theexpected International Journal of Multiphase Flow 28(2002)823–/locate/ijmulflowE-mail address:lakehal@iet.mavt.ethz.ch (kehal).0301-9322/02/$-see front matter Ó2002Elsevier Science Ltd.All rights reserved.PII:S0301-9322(01)00086-6824 kehal/International Journal of Multiphase Flow28(2002)823–863returns from computational analyses.The role of simulation strategies in the prediction and design processes is also discussed.The deliberate choice of applications is motivated by the variety of solution methods applied in each case;we aim at discussing them in the sections to follow.The methods are discussed in a comprehensive and simplistic way based on known ideas and prin-ciples.An overview of the state-of-the-art is presented in treating the various subjects using the Eulerian–Eulerian method in both the one-fluid and two-fluid(interpenetrating media)formu-lations,as well as by the Eulerian–Lagrangian variant.Since theflows considered herein are presently out of reach of direct and large-eddy simulation approaches(DNS and LES),we es-sentially focus on the implications of turbulence modelling(by reference to the Reynolds Aver-aged Navier–Stokes Equations,RANS)in the various computational frameworks discussed in this paper,and the way this conventional approach could be improved on by more elaborate ones. In support of this,practical case studies typical of environmental and hydrodynamic applications are presented.Apart from the applications with reference to hydrodynamic applications CS5and CS6,the interaction between the transported phase and thefield turbulence is treated in all other cases on the basis of one-way coupling.Note,too,that the paper does not deal with simplified simulation approaches,for example,with the so-called Gaussian models employed for atmospheric dispersion modelling(cf.Hangan,1999).Eulerian–Eulerian and Eulerian–Lagrangian methods have been extensively used to simulate particle dispersion.Depending on the nature of the case studies in question it is possible to employ a specific form of each of the two solution methods.But prior to that,it is worth highlighting the main differences between these two strategies,i.e.the Eulerian–Eulerian vs.the Eulerian–Lagrangian methods.The choice between these two procedures is in essence problem-dependent. The Eulerian orfield description methodology is commonly adopted for the prediction of in-terpenetrating media situations,including both highly particle loaded systems such asfluidized beds,dilute particle-ladenflows as in the case of dilute suspensions of aerosols,droplets and particles,and gas–liquid mixtures such as bubblyflows.This approach can be employed in two distinctive forms:The one-fluid formulation and the two-fluid approach.In thefirst approach, generally employed in the form of a one-field description of highly-loaded or dilute suspensions formed by concentrations of droplets and particles,the particle concentrations are assumed to have some characteristics of a continuous phase(e.g.the local concentration)and,when ap-propriate,some of a dispersed phase(e.g.the inertial slip).In other words,the method essentially consists in solving an extra conservation law for the concentration of particles or for their mean spatial density.Modifications of the transport equations are also needed to consider buoyancy forces whenever the two phases exhibit differences in density due to the presence of a heavier dispersed phase(e.g.sedimentation problems,snow avalanches,etc.),or when the carrier phase features thermal stratification as is often the case in geophysicalflows(e.g.thermal fronts,at-mospheric surface layer,etc.).In addition,the transport equations must include interfacial ex-change laws to account for mass transfer whenever the dispersed phase evaporates or condenses, e.g.evaporative marine droplets over the ocean.The combination of all these processes leads to a system of equations with a multitude of closure laws.In this respect,the closure relationships for the turbulent concentration or heatflux arising from Reynolds averaging conceptually follow the manner in which the mechanical turbulent stresses are approximated.This important issue is examined herein,too,in particular when the closure law for turbulence is a two-equation based approach,in which the buoyancy-induced contributions are represented in terms of additionalkehal/International Journal of Multiphase Flow28(2002)823–863825 source terms in the turbulence equations with some adjustable coefficients.In the second ap-proach,also known as the six equation model approach,the phases are treated as two inter-penetrating continua evolving within a single system:Each point in the mixture is occupied simultaneously(in variable proportions)by both phases.Each phase is then governed by its own conservation and constitutive equations;these are then coupled through interphase in-teraction properties.More precisely,in contrast to the one-fluid formulation,convective and diffusive processes are explicitly taken into account in each of the two phases.For example, mixtures of two immisciblefluids such as air bubbles in water cannot be considered as mixtures of dilute suspensions evolving within a liquid phase;they have to be simulated via the two-fluid approach.In the Lagrangian reference frame individual particles or clouds of particles are treated in a discrete way.The reference frame moves with the particles,and the instantaneous location of each particle is determined by reference to its origin and the time grangian methods em-ployed for particle tracking are conventionally based on the equation of motion for spherical particles at high-Reynolds numbers,as given by Clift et al.(1978),also known as Basset–Bous-sinesq–Oseen(BBO)equation(cf.Crowe et al.,1996).The dispersed phases are assumed to be heavy and smaller than the Kolmogorov microscales.As a prerequisite computational sequence theflowfield has to be known since tracking individual particles directly relies on its properties, i.e.velocityfield and turbulence statistics.In practical applications theflowfield is modelled by use of RANS,whereas the resort to DNS(Squires and Eaton,1990;Mosyak and Hetsroni,1999; Ahmed and Elghobashi,2000;Sawford and Yeung,2001)or LES(Yeh and Lei,1991;Wang and Squires,1996;Armenio et al.,1999;Boivin et al.,2000;Okong’o and Bellan,2000)is still confined to research studies dealing for example with turbulence–particle interactions.A variety of models accounting for the effects of turbulence on particle motion are available in the literature.A critical review of the variants employed for heavy particles in atmospheric tur-bulence is proposed by Wilson(2000).Another interesting review is that of Shirolkar et al.(1996) focusing on models used for dispersion in combustion problems.On the upper level of classifi-cation the models differ depending on whether they are applied to passive tracers(see,for ex-ample,Thomson,1987)or to inertial particles(IP).The present work places emphasis on the second class of models only.A subcategory of IP dispersion models is an approach based on a Markov chain process,which is afinite discrete form of the Langevin equation supposed to model thefluctuating particle velocities in a purely stochastic way.This equation wasfirst employed for the study of Brownian motion by Wang and Uhlenbeck(1945),and was only later applied to describe dispersion in homogeneous turbulence by Lin and Reid(1962).The other often employed random-flight algorithms treated in this paper are based on the generation of non-miscible(un-correlated)random eddies,in which particle trajectories are purely deterministic.These are known as eddy interaction models(EIM),perhaps initially proposed by Gossman and Ioannides(1981). Here it is assumed that individual particles are subject to a series of interactions with randomly sampled eddies;the particle velocity remains constant during each particle–eddy interaction time, during which the eddy velocity remains unchanged.The difference between the two methods is that Markov chain type models provide a continuousfluctuating velocityfield,whereas in EIMs thefluctuating velocity changes only when individual particles encounter a new eddy.This is the reason why MacInnes and Bracco(1992)refer to thefirst class as continuous random walk models and to the second as discontinuous random walk models.826 kehal/International Journal of Multiphase Flow28(2002)823–863The Eulerian–Lagrangian formalism thus amounts to the combination of two separate ap-proaches:The Eulerian part delivers theflowfield with its turbulent statistical properties,and the Lagrangian module employs these data to track individual particles.The parametrization of particle dispersion is therefore intimately tied to the dynamics offield turbulence.This is,of course,the case for dispersed phases smaller than the Kolmogorov micro-scales,whose inter-action with turbulence is commonly termed one-way coupling by reference to the weak effect of particle momentum on turbulence.These two methods are here discussed in their original modelling context and in the LES framework.Still,the Eulerian approach for simulating turbulent dispersion has its own advantages as compared to Lagrangian methods.Forflow laden with a large amount of particles the quanti-tative description of the variation in particle concentration is much simpler by means of the Eulerian method since,for the same purpose,statistical sampling is required with the Lagrangian grangian methods may also face problems whenever the cloud of particles tracked is larger than thefluid parcel over which volume averaging is performed.And apart from that,the Eulerian approach allows both phases to be computed over a single grid,whereas the Lagrangian methods require the interpolation of quantities between thefixed grid nodes and the local position of particles.However,treating particles via the Lagrangian formalism is in essence natural be-cause their motion is tracked as they move through theflowfield,which preserves their actual non-continuum behaviour and accounts for their history effects in a natural way.In addition,if attention is now redirected towards turbulence modelling,the Lagrangian approach holds a fundamental advantage over the Eulerian one in the sense that it does not require closure as-sumptions for turbulence correlations of tracer concentration and velocityfluctuations.More about the relative merits of these approaches is given by Durst et al.(1994)and Mostafa and Mongia(1987).The present paper is structured as follows:Selected applications arefirst introduced to grad-ually highlight the expected results of computational analyses.These selected case studies(see Section2)are referred to as CS1,CS2;...;CS6,respectively.Based on an extended literature survey the solution procedures employed so far in each case are introduced in Section3.Section4 is devoted to computational examples,where the solution methods are examined in the light of calculation results.Finally,key remarks are made in connection with computational strategies and turbulence models together with the presentation of an outlook on future developments. 2.Typical applications in environmental and hydrodynamic research2.1.Pollutant transport in the urban canopyThis type of study enters within the large framework of computational wind engineering(CWE), a discipline that has been progressing since the late1970s,boosted by its potential to overcome the limitations of earlier simplified physical models such as the Gaussian models evoked previously in Section1.Pollutant dispersion within the atmospheric surface layer encompasses a variety of aspects of vital interest that need to be explored:For example,predicting the transport of con-taminants from hazardous releases,analyzing the traffic-induced dispersion(Rafailidis,2000; Kastner-Klein et al.,1997;Meroney et al.,1999),and studying the effects of neighbouringkehal/International Journal of Multiphase Flow28(2002)823–863827 building topography on domestic gas-releases(Cowan et al.,1997;Delaunay et al.,1997;Hangan, 1999;Castro et al.,1999).Without considering the thorny question of predicting the behaviour of hazardous gas releases(Chernobyl type of tragedies)to the atmosphere,we could evoke a similar problem that draws less attention,but may nevertheless have an impact on daily life:The quality of air inside a single or a group of buildings and its relation to external aerodynamic conditions. Theseflow conditions can,for example,connect an external source of pollutants(chimneys re-leasing exhaust gases from centralized heating devices)with fresh-air admission(windows,etc.) which could in turn be contaminated.Although recent contributions to thefield have taken further steps by dealing with dispersion around complex(several buildings)configurations(e.g. Hangan,1999;Castro et al.,1999),the example selected here consists of the three-dimensional prediction of gas dispersion around an isolated,generic building model placed within a simulated urban canopy studied by Delaunay et al.(1997).The aim of this investigation was to provide architects and civil engineers with sufficient indications regarding theflow structure to help them design a group of buildings in which the recirculation of contaminants through fresh air admis-sions can be minimized.2.2.Car-induced pollution in urban areasCar-induced pollution in urban areas is a serious health concern,in particular within cities1 featuring many street canyons.Most often building aggregates placed within the atmospheric boundary layer may act as artificial obstacles to the wind and cause stagnant conditions.Ex-perimental and numerical studies of such problems aim in general at predicting the time evolution of pollutant concentrations and their implications for the comfort of pedestrians as a function of geometry and pollutant doses(Mestayer et al.,1993;Sini et al.,1996;Moussiopoulos et al.,1998; Rafailidis,2000).Previous studies showed the number and arrangement of vortex structures within the street canyon to strongly influence vertical exchange rates.It has also been shown that differential heating of street surfaces can grossly influence the capability of theflow to transport and exchange pollutants(Sini et al.,1996).In particular,differential heating could also shift the in-streetflow structure from a single-vortexflow to aflow with several counter-rotating vortices. We report here on the results of a recent simulation,conducted by Theodoridis and Moussio-poulos(2000),of theflow and contaminant transport within a typical street-canyon configuration studied experimentally by Rafailidis(2000).In contrast to earlier studies,a number of interesting and original issues typical for this type of problem have been dealt with by the authors,focusing for example on the determination of the subsequent production of NO x and ozone.2.3.Dispersion of marine dropletsThe fundamental issues of surface layer meteorology have been reviewed by many specialists, e.g.H€o gstro€m(1996).More specifically with respect to marine climatology Smith et al.(1996) have made available a complete overview leading to a better understanding of air–sea interaction.1Some Mediterranean cities suffer today because their developers opted in the past for street-canyon type conglomerations seeking for shadow(e.g.Medina and Casbah in North Africa).828 kehal/International Journal of Multiphase Flow28(2002)823–863The authors review the progress achieved in the study of air–sea interaction over the past three decades and its role in the modelling of the coupled system of ocean and atmosphere.Melville (1996)placed emphasis on the role of surface–wave breaking in air–sea interaction and the subsequent impact of aerosol production and transport.More precisely,it is the impact of marine droplets and aerosols on the heatflux balance that represents the key point in this branch,as discussed by Smith et al.(1996)and Fairall et al.(2000).Indeed,the evaporative droplets are known to distort the normal sensible/latent heatflux balance,whereas in their absence the entire surface moistureflux produces a latent heat loss by the ocean leading to an increase in the salinity at the surface.The central issue here is to understand the contribution of sea spray droplets to the transfer of moisture and latent heat from the sea to the atmosphere.The case study reported in the present review refers to the two-dimensional simulation of the turbulent transport and evapo-ration of droplets ejected by bursting bubbles within various simulated air–sea boundary layers (Edson and Fairall,1994;Edson et al.,1996).An integrated Eulerian–Lagrangian strategy was employed to compute theflow,temperature and moisturefields,and the trajectory of each ejected droplet;in particular,the particle trajectories were computed by means of a Markov chain based on the discretization of the Langevin equation for dispersed particles,modified to account for the effects of turbulence,gravity and inertia.This type of Lagrangian technique is presently being employed within the LES framework for other related subjects such as the prediction of pollution dispersion in the atmosphere(e.g.Sorbjan and Uliasz,1999).Studying the generation,transfer mechanisms and aerosol deposition over the ocean has also been migrating gradually from RANS (e.g.Ling et al.,1980;Burk,1984)to LES(e.g.Glendening and Burk,1992),although the Eulerian description is still preferred to the Lagrangian one.2.4.Impacting hydrometeors on buildingsThe deterioration experienced by buildings and monuments is caused in part by the direct impact of hydrometeors and subsequent deposition of moisture on the surface.In contrast to the effects caused by the spectacular impact of heavy hydrometeors such as hail,the more subtle degradations caused by moisture deposited by rain,snow and fog are less well assessed.In these instances,the deposited moisture can cause mechanical disruptions by freezing withinfissures or by actually dissolving the materials.In addition,atmospheric pollutants dissolved or suspended in water droplets can be carried to the surface.Once these pollutants have been deposited on the surface,capillarity can transport the moisture and pollutants into the interior of porous materials. This often results in chemical transformations and deterioration deep within these structures.For this class offlow the literature reports on a very limited number of computational investigations; the earlier ones have adopted simplified formulations relating the intensity of driving rains to the free-falling rain intensity and wind speed(cy,1977;Beguin,1985;Hilaire and Savina, 1989).More elaborate strategies based upon the Eulerian–Lagrangian approach appeared only recently(e.g.Choi,1994;Lakehal et al.,1995;Sankaran and Paterson,1997;Karagiozis et al., 1997).However,the only contribution in thisfield combining in a single model the effects of turbulence,gravity,and inertia is due to Lakehal et al.(1995).The example reported here(from these authors’work)centers around the prediction of wind-driven raindrop trajectories inside a two-dimensional street canyon;thefinal aim was to evaluate the impacting water rate on the facades.The solution procedure was again based on an integrated Eulerian–Lagrangian method,kehal/International Journal of Multiphase Flow28(2002)823–863829 and the particle trajectories were computed by means of a Markov chain modified to account for the effects of turbulence,gravity,and inertia.2.5.Sedimentation in water clarifiersSettling and sedimentation phenomena are complex processes,repeatedly evoked in hydro-dynamic applications;their presence within wastewater treatment plants as the most important unit operations is one example among various others.It is well known that gravity-induced sedimentation and the subsequent thickening process may be subdivided into four different types: Discrete particle settling,flocculent settling,hindered settling,and compression(see,for example, Karl and Wells(1999)for classification).The thickening process in water clarifiers occurs most often as a combination of the last three forms,which poses challenges to the modeler.The recent critical review of Parker et al.(2001)reports on the important design aspects properly applicable to clarifier technology that need to be observed.Investigating this type offlow is dictated by design interests:It is aimed at helping to design secondary clarifiers,whose efficiency is such that the overall performance of the entire wastewater treatment does not require post operations (Krebs et al.,1996).An intensive scientific effort has recently been made in order to understand this type offlow,and various numerical models have been developed for the purpose,most of which are based on two-equation turbulence models describing theflow pattern and sediment-induced density currents(Lyn et al.,1992;Zhou et al.,1992;Zhou and McCorquodale,1992; Szalai et al.,1994;Vitasovic et al.,1997;Armbruster et al.,2001).Apart from Lyn et al.(1992)the above cited works did not consider particle decompositions and were thereby based on the de-termination of an average settling velocity for suspended particles.Jin et al.(2000)have recently taken a step ahead by proposing a one-dimensional model for non-uniform sediment transport capable of handlingflocculation,coagulation,andfiltration.This type offlow raises additional complexities as compared to pollutant dispersion problems.Buoyancy effects may be more im-portant than those induced by turbulent stresses.The transported phase settles at a velocity strongly influenced by its concentration.Finally,the non-Newtonian behaviour of the activated sludge requires appropriate definition of its rheological properties.The results of modelling the sedimentation of a sludge blanket in a circular,center-fed secondary clarifier with inclined bottom and central withdrawal are presented.Axisymmetry is assumed and theflow and settling processes (with variable settling velocities)are computed in a radial section.The non-Newtonian behaviour of the sludge is also taken into account.2.6.Bubble plumesThree-dimensional mixing of multiphaseflows may occur in industrial applications as well as in environment protection processes.Industrial applications include gas stirring by liquid metal ladles in several metallurgical processes,or venting of vapour mixtures to liquid pools in chemical and nuclear reactors.Bubble plumes may also be involved in environment protection problems such as the aeration of lakes,mixing of stagnant water and,generally,de-stratification of water reservoirs.For all these applications the basic need is to determine the currents induced by the gaseous phase evolving in the surrounding liquid and thereby to establish the consequent mixing and partition of energy,or species concentration in the body of the liquid.Here the computationalmethodology to be followed is the two-fluid approach of Ishii (1975)evoked previously.However,more important is the fact that predicting bubbly flows cannot be achieved without suitable models capable of correctly representing interphase momentum transfer mechanisms and tur-bulence modulation induced by the bubbles.For the latter issue,various models have been published in the past,though all of them resort to a single-phase two-equation turbulence model modified to account for these exchange mechanisms (Malin and Spalding,1984).This includes the effect of bubble migration through the liquid (Simonin and Viollet,1988),and more often the interactions between the eddies and the dispersed phase via what is known as turbulent dispersion models (see,for example,Moraga et al.,2001,for a recent review).In practice the idea of tur-bulence dispersion induced by the dispersed phase has most often been reflected in terms of a superposition of the shear-induced and bubble-induced stress tensors in the equations for the liquid phase;the latter being constructed on the basis of scaling arguments.The example reported here consists of the prediction of a confined bubble plume studied experimentally by Anagbo and Brimacombe (1990).The numerical results reported here were obtained by Smith and Milelli (1998),who made a critical assessment of various models that have so far been advanced to support modelling of bubbly flows.3.Outline of the solution methods3.1.The Eulerian–Eulerian one-fluid approach3.1.1.BackgroundTo handle the transport of a dilute continuum acting as a passive scalar within a turbulent flow one generally resorts to the so-called Eulerian–Eulerian one-field formalism.In this approach the particle concentrations are assumed to have some characteristics of a continuous phase and some characteristics of a dispersed phase via the inertial slip,when appropriate (e.g.when the particles settle).An inherent concept in this formalism is the assumption that the transported (passive or active scalar)phase obeys the same Navier–Stokes equation governing the mean flow,since there is no interfacial or interphase exchange processes to account for.However,in general,the pres-ence of heavy particles with non-negligible inertia raises simulation problems not yet totally re-solved,such as the lack of appropriate boundary conditions.3.1.2.The transport equationsThe Eulerian–Eulerian approach is essentially based on the solution of the Reynolds Aver-aged 2Navier–Stokes equations (RANS)governing the motion of an incompressible carrier phase (cf.Hinze,1975),together with a transport equation for the dilute phase:@j U j ¼0;ð1ÞD t U i ¼À1=q w @i p þ@j r ij ÀÀs ij Áþ~f ;ð2Þ2Hereinafter each barred symbol represents an ensemble average,while primed letters denote the fluctuating counterparts.830 kehal /International Journal of Multiphase Flow 28(2002)823–863D t C¼D r2CÀ@j u0j c0:ð3ÞIn the above equations,D t¼@tþuÁr stands for the substantial derivative,r ij the viscous stress, q w the mean density of thefluid at a reference state,p the pressure,D the molecular diffusivity coefficient,and s ij u0i u0j the Reynolds stress tensor that requires a model.The buoyancy force term,~f,is effective only in thermally stratifiedflows and/or when the dispersed phase is appre-ciably heavier than the carrierfluid.For instance,case studies CS1,CS2and CS5were treated on the basis of Eqs.(1)–(3),with~f¼0for thefirst two examples.3.1.3.Thermally stratifiedflows with mass transferThe presence of an evaporative medium(e.g.marine droplets)within a thermally stratifiedflow (e.g.a marine sublayer)can be treated on the basis of the RANS equations(1)and(2),using theBoussinesq approximation,in which the buoyancy force term now reads~f¼Àg iðH vÀH rv Þ=H rv.This term is induced by the difference between the instantaneous virtual potential temperature3H v and that of the reference state H rv .According to Stull(1988),the presence of water dropletsrequires the virtual potential temperature to conform to the following relationH v¼H1½þ0:61q VÀq LÀq D ;ð5Þin which q L is the specific humidity of the liquid,and q D the contribution to the total specific humidity from the droplets to be determined by integrating the local droplet volume concen-tration.At high-Reynolds numbers the thermal and moisturefields are represented by the Rey-nolds averaged transport equations for the potential temperature H and the total specific humidity denoted:Q¼q Vþq LD t H¼À@j u0j hþLE S q=C p;ð6ÞD t Q¼À@j u0j qþS q:ð7ÞIn these equations C p stands for the specific heat at constant pressure,LE for the latent heat of vaporization,and S q for the total evaporation rate.The reader can refer to Pruppacher and Klett (1978)for more details on the modelling of the source term S q.Note that the molecular diffusion contributions in the above equations have been dropped,since only high-Re numberflows are of interest in these studies.The turbulentfluxes u0j q and u0j h appearing in Eqs.(6)and(7)need to be modelled,too,as will be discussed later.3.1.4.Density-induced stratification in non-newtonian mixturesIn certain class offlow the dense phase deposited over an impermeable surface forms a structure behaving like a non-Newtonian material.This is the case for biological material settling in water clarifiers.In a similar context,with use of the Boussinesq approximation the momentum equations take the form of Eqs.(1)and(2),but the buoyancy force is now driven by the difference 3The equation of state for‘‘humid air’’,a mixture of dry air and water vapor,is:p¼q a R a Hþq V R V H¼q a R a H v;H v¼H½1þ0:61q V ;ð4Þwhere H stands for the potential temperature,q V for the specific humidity of water vapor,and0.61is the explicit value ofðR VÀR aÞ=R a.kehal/International Journal of Multiphase Flow28(2002)823–863831。
凝聚态物理材料物理专业考博量子物理领域英文高频词汇
凝聚态物理材料物理专业考博量子物理领域英文高频词汇1. Quantum Mechanics - 量子力学2. Wavefunction - 波函数3. Hamiltonian - 哈密顿量4. Schrödinger Equation - 薛定谔方程5. Quantum Field Theory - 量子场论6. Quantum Entanglement - 量子纠缠7. Uncertainty Principle - 不确定性原理8. Quantum Tunneling - 量子隧穿9. Quantum Superposition - 量子叠加10. Quantum Decoherence - 量子退相干11. Spin - 自旋12. Quantum Computing - 量子计算13. Quantum Teleportation - 量子纠缠传输14. Quantum Interference - 量子干涉15. Quantum Information - 量子信息16. Quantum Optics - 量子光学17. Quantum Dots - 量子点18. Quantum Hall Effect - 量子霍尔效应19. Bose-Einstein Condensate - 玻色-爱因斯坦凝聚态20. Fermi-Dirac Statistics - 费米-狄拉克统计中文翻译：1. Quantum Mechanics - 量子力学2. Wavefunction - 波函数3. Hamiltonian - 哈密顿量4. Schrödinger Equation - 薛定谔方程5. Quantum Field Theory - 量子场论6. Quantum Entanglement - 量子纠缠7. Uncertainty Principle - 不确定性原理8. Quantum Tunneling - 量子隧穿9. Quantum Superposition - 量子叠加10. Quantum Decoherence - 量子退相干11. Spin - 自旋12. Quantum Computing - 量子计算13. Quantum Teleportation - 量子纠缠传输14. Quantum Interference - 量子干涉15. Quantum Information - 量子信息16. Quantum Optics - 量子光学17. Quantum Dots - 量子点18. Quantum Hall Effect - 量子霍尔效应19. Bose-Einstein Condensate - 玻色-爱因斯坦凝聚态20. Fermi-Dirac Statistics - 费米-狄拉克统计。
薛定谔方程 英语
薛定谔方程英语English: The Schrödinger equation, also known as the Schrödinger wave equation, is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is named after the Austrian physicist Erwin Schrödinger, who first formulated it in 1926. The equation is a partial differential equation that describes how the wave function of a physical system evolves over time, and it is central to the understanding of quantum mechanics. The Schrödinger eq uation is used to calculate the probability distribution of a particle in a given region of space, and it has been immensely successful in explaining the behavior of particles at the atomic and subatomic levels.中文翻译: 薛定谔方程，也被称为薛定谔波动方程，是量子力学中的基本方程，描述了一个物理系统的量子状态随时间的变化。
它以奥地利物理学家埃尔温·薛定谔的名字命名，他在1926年首次提出了这个方程。
凝聚态物理相关诺贝尔奖(可编辑修改word版)
凝聚态物理相关诺贝尔化学奖1970-1985 年赫伯特·豪普特曼（美）杰罗姆·卡尔勒（美）在测定晶体结构的直接方法上的贡献2000 年艾伦·黑格（美）艾伦·麦克迪尔米德（美/新西兰）白川英树（日）对导电聚合物的研究2011 年丹·谢赫特曼（以）准晶的发现[5]凝聚态物理相关诺贝尔物理学奖1970-1972年约翰·巴丁美国“他们联合创立了超导微观理论，即常说的BCS 理论”"for their jointly developed theory ofsuperconductivity, usually called theBCS-theory"[73]利昂·库珀美国约翰·罗伯特·施里弗国美1973年江崎玲于奈日本“发现半导体和超导体的隧道效应”"for their experimental discoveriesregarding tunneling phenomena insemiconductors and superconductors,respectively"[74]伊瓦尔·贾埃弗挪威国英 “他理论上预测出通过隧道势垒的超电流的性质，特别是那些通常被称为约瑟夫森效应 布赖 的现象” 恩·戴"for his theoretical predictions of the 维·约瑟 properties of a supercurrent through a 夫森 tunnel barrier, in particular those phenomena which are generally known as the Josephson effect"[74]1977 年菲利普·沃伦·安德森国美“对磁性和无序体系电子结构的基础性理论研究”"for their fundamental theoretical investigations of the electronicstructure of magnetic and disorderedsystems"[78]内维尔·莫特 国英约翰·凡扶累克 国美年彼得·列昂尼多维奇·卡皮查苏联“低温物理领域的基本发明和发现” "for his basic inventions and discoveries in the area of low- temperature physics"[79]1982 年肯尼斯·威尔逊美国“对与相转变有关的临界现象理论的贡献”"for his theory for critical phenomena in connection with phase transitions"[83]1985 年克劳斯·冯·克利青德国“发现量子霍尔效应”"for the discovery of thequantized Hall effect"[86]特·鲁斯卡德国“电子光学的基础工作和设计了第一台显微镜"for his fundamental work in electron optics,electron microscope"格尔德·宾宁德国“研制"for their design of the scanningtunneling microscope"海因里希·罗雷尔瑞士约翰内斯·贝德诺尔茨德国“在发现"for their important break-through the discovery of superconductivity ceramic materials"卡尔·米勒瑞士皮埃尔吉勒纳法国“发现研究简单系统中有序现象的方法可以被推广到比较复杂的物质形式，特别是推广到"forfor studying order phenomena in simple systems can be generalized to morecomplexliquid crystals and polymers"1996年戴维·李美国“发现了在氦-3 里的超流动性”"for their discovery ofsuperfluidity inhelium-3"[97]道格拉斯·奥谢罗夫美国罗伯特·理查森美国2000年若雷斯·阿尔费罗夫俄罗斯“发展了用于高速电子学和光电子学的半导体异质结构”"for developing semiconductorheterostructures used in high-speed-and optoelectronics"[101]赫伯特·克勒默德国杰克·基尔比美国“在发明集成电路中所做的贡献”"for his part in the invention of theintegrated circuit"[101]2001年埃里克·康奈尔国美“在碱性原子稀薄气体的玻色-爱因斯坦凝聚态方面取得的成就，以及凝聚态物质属性质的早期基础性研究”"for the achievement of Bose-Einsteincondensation in dilute gases of alkaliatoms, and for early fundamental studiesof the properties of the condensates"[102]卡尔·威曼国美沃尔夫冈·克特勒国德2003年阿列克谢·阿布里科索夫美国俄罗斯“对超导体和超流体理论做出的先驱性贡献”"for pioneering contributions tothe theory of superconductors andsuperfluids"[104]维塔利·金俄罗斯兹堡安东尼·莱格特英国美国2007 年艾尔伯·费尔法国“发现巨磁阻效应”"for the discovery of giant magnetoresistance"[108]彼得·格林贝格德国2009 年高锟英国 美国 [110] “在光学通信领域光在纤维中传输方面的突破性成就” "for groundbreaking achievementsconcerning the transmission of light infibers for optical communication"[111]威拉 德·博伊尔 美国“发明半导体成像器件电荷耦合器件” "for the invention of an imaging semiconductor circuit – the CCD sensor"[111]乔治·史密斯美国2010 年安德烈·海姆荷兰俄罗斯“在二维石墨烯材料的开创性实验”"for groundbreaking experimentsregarding the two-dimensionalmaterial graphene"[112]康斯坦丁·诺沃肖洛夫英国俄罗斯。
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(1.1) (1.2)
where p(t, x) and q (t, x) are subject to periodic boundary condition of period 2π , and are even in x, i.e. p(t, x + 2π ) = p(t, x) , p(t, −x) = p(t, x) , q (t, x + 2π ) = q (t, x) , q (t, −x) = q (t, x) , ω ∈ (1, 2), α > 0 and β are real constants, and ε > 0 is the perturbation parameter. By the transformation, 1 2 1 2 p=p ˜ei 2 ω t , q = q ˜ei 2 ω t ,
4 Persistent Homoclinic Orbits 17 4.1 The Invariant Diagonal Subspace . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3 An Alternative Melnikov Function . . . . . . . . . . . . . . . . . . . . . . . 18 5 Conclusion and Discussion 19
The rest of this article will be dealing with this form of perturbed vector nonlinear Schr¨ odinger equations (PVNLS). The corresponding work for regularly perturbed scalar nonlinear Schr¨ odinger equation 2 has been accomplished in [11], where the singular perturbation ε∂x is replaced by a regular ˆ2 which is a bounded Fourier multiplier obtained from a truncation of perturbation ε∂ x 2 2 ε∂x . The main difficulty introduced by the singular perturbation ε∂x is that it breaks the spectral gap condition of the unperturbed system. Thus the standard invariant manifold results do not hold. Nevertheless, certain invariant manifold results can be established, which are sufficient for establishing the existence of homoclinic orbits. The corresponding work for such singularly perturbed scalar nonlinear Schr¨ odinger equation (PNLS) has been accomplished in [10]. The crucial new features of this article are listed below: 1. Homoclinic orbits asymptotic to the spatially uniform and temporally periodic solutions up to phase shifts, are constructed for the integrable vector nonlinear Schr¨ odinger equations ((1.1, 1.2) at ε = 0). In particular, such homoclinic orbits are built with spatially non-periodic and non-anti-periodic Bloch eigenfunctions. This study provides an elegant formula which is missing in [16] [6] [5]. 2. For the scalar nonlinear Schr¨ odinger equation, a complete sequence of Melnikov vectors can be constructed out of the Floquet discriminant for the Lax pair [12]. Unlike scalar nonlinear Schr¨ odinger equation, the Lax pair for the integrable vector nonlinear Schr¨ odinger equations, does not have a complete isospectral theory built upon a Floquet discriminant. By restricting ω in the interval (1, 2), we are able to deal with one unstable mode problem for which only one Melnikov function is needed. The Melnikov function will be built out of the Hamiltonian and the invariant L2 norms. 3. In the 4D invariant subspace of functions independent of x, the dynamics is quite different from that of perturbed scalar nonlinear Schr¨ odinger equation [11]. The √ eigenvalues of the saddle are: a positive one of order ε, a negative one of order √ ε, and a complex conjugate pair for which the real part is negative, and both real √ and imaginary parts are of order ε. The two eigenvalues of order ε, correspond to a fish-shape dynamics, and the other two eigenvalues correspond to a stable spiraling dynamics. 4. The dynamics in the invariant diagonal subspace given by p = q , is governed by the PNLS. The unstable manifold of the saddle resides in the invariant subspace. This leads to that the persistent homoclinic orbit for the PNLS is the only one for the PVNLS, asymptotic to the saddle.
3
1
Introduction
In recent years, novel results have been obtained on the solutions of the vector nonlinear Schr¨ odinger equations [3] [4] [17]. Abundant ordinary integrable results have been carried through [16] [6], including linear stability calculations [5]. Specifically, the vector nonlinear Schr¨ odinger equations can be written as 1 ipt + pxx + (|p|2 + χ|q |2 )p = 0, 2 1 iqt + qxx + (χ|p|2 + |q |2)q = 0, 2 where p and q are complex valued functions of the two real variables t and x, and χ is a positive constant. These equations describe the evolution of two orthogonal pulse envelopes in birefringent optical fibers [14] [15], with industrial applications in fiber communication systems [7] and all-optical switching devices [8]. For linearly birefringent fibers [14], χ = 2/3. For elliptically birefringent fibers, χ can take other positive values [15]. When χ = 1, these equations are first shown to be integrable by S. Manakov [13], and thus called Manakov equations. When χ is not 1 or 0, these equations are non-integrable. Propelled by the industrial applications, extensive mathematical studies on the vector nonlinear Schr¨ odinger equations have been carried. This article is another piece of mathematical works. Like the scalar nonlinear Schr¨ odinger equation, the vector nonlinear Schr¨ odinger equations also possess figure eight structures in their phase space. Figure eight structures are also called separatrices. For two-dimensional Hamiltonian systems, figure eight structures are given by the singular level sets of the Hamiltonian. Our goal is to understand the consequence of such figure eight structures when these equations are under perturbations, in particular, whether or not the consequence is chaotic dynamics. Specifically, we will investigate the type of chaos created through homoclinic orbits. (cf: the section “Conclusion and Discussion”.) This article will deal with the problem on the existence of homoclinic orbits for the singularly perturbed vector nonlinear Schr¨ odinger equations,