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机械运动的英语Mechanical MotionMechanical motion is a fundamental concept in physics that describes the movement of objects and the forces that govern their behavior. This form of motion is essential in various fields, from engineering and transportation to robotics and everyday life. Understanding the principles of mechanical motion is crucial for designing and analyzing a wide range of systems and devices.At its core, mechanical motion involves the displacement of an object from one point to another, often through the application of external forces. These forces can be a result of various factors, such as gravity, friction, or the action of other objects. The study of mechanical motion encompasses the analysis of the motion of rigid bodies, as well as the deformation and behavior of flexible materials.One of the key aspects of mechanical motion is the concept of kinematics, which deals with the description of motion without considering the forces that cause it. Kinematics examines the position, velocity, acceleration, and other related quantities of an object as it moves through space. This information is crucial forunderstanding the behavior of a system and predicting its future states.Another important aspect of mechanical motion is dynamics, which focuses on the forces and torques that act on an object and how they influence its motion. Dynamics provides a deeper understanding of the underlying principles that govern the movement of objects, including the conservation of energy and momentum, as well as the effects of friction and other dissipative forces.The applications of mechanical motion are vast and diverse. In engineering, the study of mechanical motion is essential for the design and analysis of mechanical systems, such as engines, gears, and robotic manipulators. In transportation, the principles of mechanical motion are used to design vehicles, predict their performance, and optimize their efficiency. In everyday life, mechanical motion is encountered in the operation of household appliances, tools, and even the human body.Advances in computational power and numerical simulations have further expanded the understanding and application of mechanical motion. Computational fluid dynamics, for example, can be used to model the complex flow of fluids and the interaction between solids and fluids, enabling the design of more efficient and optimizedsystems.Moreover, the study of mechanical motion has led to the development of innovative technologies, such as energy harvesting devices, which can convert the motion of objects into electrical energy. These advancements have the potential to contribute to the development of more sustainable and efficient energy solutions.In conclusion, the study of mechanical motion is a fundamental and multifaceted field that underpins a wide range of technological and scientific advancements. By understanding the principles of kinematics, dynamics, and the various forces that govern the movement of objects, engineers, scientists, and researchers can continue to push the boundaries of what is possible and create novel solutions to the challenges facing our world.。
FINITE DIMENSIONAL REDUCTION OF NONAUTONOMOUS DISSIPATIVESYSTEMSAlain MiranvilleUniversit´e de Poitiers Collaborators:Long time behavior of equations of the formy′=F(t,y)For autonomous systems:y′=F(y)In many situations,the evolution of the sys-tem is described by a system of ODEs:y=(y1,...,y N)∈R N,F=(F1,...,F N)Assuming that the Cauchy problemy′=F(y),y(0)=y0,is well-posed,we can define the family of solv-ing operators S(t),t≥0,acting on a subset φ⊂R N:S(t):φ→φy0→y(t)This family of operators satisfiesS(0)=Id,S(t+s)=S(t)◦S(s),t,s≥0We say that it forms a semigroup onφQualitative study of such systems:goes back to Poincar´eMuch is known nowadays,at least in low di-mensionsEven relatively simple systems can generate very complicated chaotic behaviorsThese systems are sensitive to perturbations: trajectories with close initial data may diverge exponentially→Temporal evolution unpredictable on ti-me scales larger than some critical value→Show typical stochastic behaviorsExample:Lorenz systemx′=σ(y−x)y′=−xy+rx−yz′=xy−bzObtained by truncature of the Navier-Stokes equationsGives an approximate description of a layer of fluid heated from belowSimilar to what is observed in the atmosphereFor a sufficiently intense heating:sensitive dependence on the initial conditions,repre-sents a very irregular convection→Butterfly effectVery often,the trajectories are localized in some subset of the phase space having a very complicated geometric structure(e.g.,locally homeomorphic to the product of R m and a Cantor set)→Strange attractor(Ruelle and Takens)Main feature of a strange attractor:dimen-sionSensitivity to initial conditions:>2(dimen-sion of the phase space≥3,say,3)Contraction of volumes:its volume is equal to0→noninteger,strictly between2and3→Fractal dimensionExample:Lorenz system:dim F A=2.05...Distributed systems:systems of PDEsφis a subset of an infinite dimensional func-tion space(e.g.,L2(Ω)or L∞(Ω))Solution:y:R+→φt→y(t)x→y(t,x)If the problem is well-posed,we can define the semigroup S(t):S(t):φ→φy0→y(t)The analytic structure of a PDE is much more complicated than that of an ODE:the global well-posedness can be a very difficult problemSuch results are known for a large class of PDEs→it is natural to investigate whether the notion of a strange attractor extends to PDEsSuch chaotic behaviors can be observed in dissipative PDEsChaotic behaviors arise from the interaction of•Energy dissipation in the higher part of the Fourier spectrum•External energy income in the lower part•Energyflux from the lower to the higher modesThe trajectories are localized in a”thin”in-variant region of the phase space having a very complicated geometric structure→the global attractor1.The global attractor.S(t)semigroup acting on E:S(t):E→E,t≥0S(0)=Id,S(t+s)=S(t)◦S(s),t,s≥0 Continuity:x→S(t)x is continuous on E,∀t≥0A set A⊂E is the global attractor for S(t)if(i)it is compact(ii)it is invariant:S(t)A=A,t≥0(iii)∀B⊂A,lim t→+∞dist(S(t)B,A)=0dist(A,B)=supa∈A infb∈Ba−b EEquivalently:∀B⊂φbounded,∀ǫ>0,∃t0= t0(B,ǫ)s.t.t≥t0implies S(t)B⊂UǫThe global attractor is uniqueIt is the smallest closed set enjoying(iii)It is the maximal bounded invariant setTheorem:(Babin-Vishik)We assume that S(t)possesses a compact attracting set K, i.e.,∀B⊂E bounded,lim t→+∞dist(S(t)B,K)=0Then S(t)possesses the global attractor A.The global attractor is oftenfinite dimen-sional:the dynamics,restricted to A isfinite dimensionalFractal dimension:Let X be a compact setdim F X=lim supǫ→0+ln Nǫ(X)ǫNǫ(X):minimum number of balls of radius ǫnecessary to cover XIf Nǫ(X)≤c(1Theorem:(H¨o lder-Ma˜n´e theorem)Let X⊂E compact satisfy dim F X=d and N>2d be an integer.Then almost every bounded linear projector P:E→R N is one-to-one on X and has a H¨o lder continuous inverse.This result is not valid for other dimensions (e.g.,the Hausdorffdimension)If A hasfinite fractal dimension,then,fixing a projector P satisfying the assumptions of the theorem,we obtain a reduced dynamical system(S),S= P(A),which isfinite dimensional(in R N)and H¨o lder continuousDrawbacks:(S)cannot be realized as a system of ODEs which is well-posedReasonable assumptions on A which would ensure that the Ma˜n´e projectors are Lipschitz are not knownComplicated geometric structure of A and AThe lower semicontinuitydist(A0,Aǫ)→0asǫ→0is more difficult to prove and may not hold It may be unobservable:∂y∂x2+y3−y=0,x∈[0,1],ν>0y(0,t)=y(1,t)=−1,t≥0A={−1}There are many metastable”almost station-ary”equilibria which live up to t⋆≡eν−12.Inertial manifolds.A Lipschitzfinite dimensional manifold M⊂E is an inertial manifold for S(t)if(i)S(t)M⊂M,∀t≥0(ii)∀u0∈E,∃v0∈M s.t.S(t)u0−S(t)v0 E≤Q( u0 E)e−αt,α>0,Q monotonicM contains A and attracts the trajectories exponentiallyConfirms in a perfect way thefinite dimen-sional reduction principle:The dynamics reduced to M can be realized as a Lipschitz system of ODEs(inertial form)Perfect equivalence between the initial sys-tem and the inertial formDrawback:all the known constructions are based on a restrictive condition,the spectral gap condition→The existence of an inertial manifold is not known for several important equations, nonexistence results for damped Sine-Gordon equations3.Exponential attractors.A compact set M⊂E is an exponential at-tractor for S(t)if(i)It hasfinite fractal dimension(ii)S(t)M⊂M,∀t≥0(iii)∀B⊂E bounded,dist(S(t)B,M)≤Q( B E)e−αt,α>0,Q monotonicM contains AIt is stillfinite dimensional and one has a uni-form exponential control on the rate of at-traction of trajectoriesIt is no longer smoothDrawback:it is not unique→One looks for a simple algorithm S→M(S)Initial construction:non-constructible and valid in Hilbert spaces onlyConstruction in Banach spaces:Efendiev, Miranville,Zelik→Exponential attractors are as general as global attractorsMain tool:Compact smoothing property on the difference of2solutionsLet S:E→E.We consider the discrete dynamical system generated by the iterations of S:S n=S◦...◦S(n times)Theorem:(Efendiev,Miranville,Zelik)We consider2Banach spaces E and E1s.t.E1⊂E is compact.We assume that•S maps theδ-neighborhood Oδ(B)of a bounded subset B of E into B•∀x1,x2∈Oδ(B),≤K x1−x2 ESx1−Sx2 E1Then the discrete dynamical system gener-ated by the iterations of S possesses an ex-ponential attractor M(S)s.t.(i)M(S)⊂B,is compact in E anddim F M(S)≤c1(ii)S M(S)⊂M(S)(iii)dist(S k B,M(S))≤c2e−c3k,k∈N,c3>0 (iv)The map S→M(S)is H¨o lder continu-ous:∀S1,S2,dist sym(M(S1),M(S2))≤c4 S1−S2 c5,c5>0, wheredist sym(A,B)=max(dist(A,B),dist(B,A))S =supSh Eh∈Oδ(B)Furthermore all the constants only depend on B,E,E1,δand K and can be computed explicitly.Remarks:1)We have a mapping S→M(S)and,due to the H¨o lder continuity,we can construct continuous families of exponential attractors2)Exponential attractors for a continuous semigroup S(t):Prove that∃t⋆>0s.t.S⋆=S(t⋆)satisfies the assumptions of the theorem→M⋆for S⋆If(x,t)→S(t)x is Lipschitz(or H¨o lder)on B×[0,t⋆],setS(t)M⋆M=∪t∈[0,t⋆]We again have a mapping S(t)→M(S)which is H¨o lder continuous3)For damped hyperbolic equations:asymp-totically smoothing property4.Finite dimensional reduction of nonau-tonomous systems.Systems of the form∂yDrawback:the uniform attractor has infinite dimension in general.Example:∂yThe family{A(t),t∈R}is a pullback attrac-tor for U(t,τ)if(i)A(t)is compact in E,∀t∈R(ii)U(t,τ)A(τ)=A(t),∀t≥τ(iii)∀B⊂E bounded,dist(U(t,t−s)B,A(t))=0lims→+∞Remarks:1)The pullback attractor is unique2)If the system is autonomous,we recover the global attractor3)In general,A(t)hasfinite fractal dimen-sion,∀t∈RDrawback:The forward convergence does not hold in generalExample:y′=f(t,y),where f(t,y)=−y if y≤0,(−1+2t)y−ty2 if t∈[0,1],and y−y2if t≥1Then A(t)={0},∀t∈R,but every trajectory starting from a neighborhood of0leaves this neighborhood never to enter it againThe forward convergence does not hold be-cause the rate of attraction is not uniform in t→This can be solved by constructing ex-ponential attractorsWe can construct a family{M(t),t∈R}, called nonautonomous exponential attractor, s.t.(i)dim F M(t)≤c1,∀t∈R,c1independent of t(ii)U(t,τ)M(τ)⊂M(t),∀t≥τ,(iii)∀B⊂E bounded,dist(U(t,τ)B,M(t+τ))≤Q( B E)e−αt,t∈R,t≥τ,α>0,Q monotonic(iii)implies the pullback attraction,but also the forward attraction→(i)and(iii)yield a satisfactoryfinite di-mensional reduction principle for nonautono-mous systemsRemarks:1)The time dependence is arbitrary2)The map U(t,τ)→{M(t),t∈R}is also H¨o lder continuous。
拉格朗日力学英文Lagrangian MechanicsLagrangian mechanics is a formulation of classical mechanics that describes the motion of a system in terms of a function called the Lagrangian which is the difference between the kinetic and potential energies of the system. This approach provides a powerful and elegant way to derive the equations of motion for a wide range of physical systems, from simple pendulums to complex multi-body systems.The foundation of Lagrangian mechanics is the principle of stationary action which states that the motion of a system will follow the path that minimizes the action integral over the time interval of interest. This principle can be used to derive the Euler-Lagrange equations which are a set of second-order differential equations that describe the motion of the system. The Lagrangian function is defined as the difference between the kinetic energy and the potential energy of the system, and the Euler-Lagrange equations relate the Lagrangian function to the forces acting on the system.One of the key advantages of Lagrangian mechanics is its ability tohandle systems with constraints. Constraints are restrictions on the motion of the system, such as the motion of a pendulum being restricted to a circular path. In Lagrangian mechanics, these constraints are incorporated into the Lagrangian function, and the Euler-Lagrange equations automatically take them into account. This makes Lagrangian mechanics particularly useful for analyzing complex systems with multiple degrees of freedom and complicated constraints.Another advantage of Lagrangian mechanics is its ability to handle non-conservative forces, such as friction or damping. These forces can be incorporated into the Lagrangian function through the use of generalized coordinates and the principle of virtual work. This allows for the analysis of a wide range of physical systems, including those with dissipative forces.Lagrangian mechanics also provides a powerful tool for analyzing the symmetries of a system. Symmetries in the Lagrangian function can lead to conserved quantities, such as momentum or angular momentum, which can simplify the analysis of the system's motion. This is particularly useful in fields such as particle physics, where the underlying symmetries of the system play a crucial role in determining the behavior of the particles.The formulation of Lagrangian mechanics was developed by theFrench mathematician Joseph-Louis Lagrange in the late 18th century. Lagrange's work built upon the earlier work of Euler and Hamilton, and it provided a more general and unified framework for describing the motion of mechanical systems. Lagrangian mechanics has since become a fundamental tool in the study of classical mechanics and has been extended to other areas of physics, such as field theory and quantum mechanics.In Lagrangian mechanics, the motion of a system is described by a set of generalized coordinates which can be any set of independent variables that uniquely specify the configuration of the system. The Lagrangian function is then defined in terms of these generalized coordinates and their time derivatives. The Euler-Lagrange equations are then used to derive the equations of motion for the system, which can be solved to determine the trajectory of the system over time.One of the key applications of Lagrangian mechanics is in the analysis of multi-body systems, such as the motion of planets in the solar system or the motion of robots with multiple joints. In these systems, the Lagrangian function can be used to derive the equations of motion for the entire system, taking into account the interactions between the various components. This makes Lagrangian mechanics a powerful tool for the design and analysis of complex mechanical systems.Another important application of Lagrangian mechanics is in the field of control theory, where it is used to design control systems that can manipulate the motion of a system in a desired way. By using the Lagrangian function to describe the system's dynamics, control engineers can develop control algorithms that can efficiently and effectively control the motion of the system.Overall, Lagrangian mechanics is a powerful and versatile approach to the study of classical mechanics that has had a profound impact on the field of physics and engineering. Its ability to handle complex systems with constraints and non-conservative forces, as well as its connection to symmetries and conserved quantities, make it a essential tool in the study of a wide range of physical phenomena.。
耗散粒子动力学的简单介绍和应用前景mg0424112 徐源一．耗散粒子动力学的发展耗散粒子动力学（dissipative particle dynamics）是一种新欣的计算机模拟和描述流体的方法，是对分子动力学（MD）和LGA 模拟的继承和发展。
分子动力学描述的精度较高，但是计算的代价较高，到目前为止，只能用来成功地处理一些简单的流体。
另外，分子动力学应用条件比较苛刻，只能处理两维问题。
LGA （lattice-gas automata）是1986 年Frisch, Hasslacher, Pomeau,Wolfram 提出的描述流体行为的模拟方法，随后Rothman 和Keller 发展这个方法，使它能描绘不能互融的流体行为。
但是LGA模拟有一个不足：模拟中，LGA引入一个重要的概念：格子(lattice), 格子的存在导致伽利略不变性的消失，因此，在描述压缩流体和多相流体时，误差较大。
耗散粒子动力学，集合了以上两种方法的优点：排除了虚拟格子的概念，从而避免了LGA方法的精度的麻烦，另一方面，保留了分离时间步骤的概念，简化了模型，加快了计算的过程。
更重要的是，与前面两者相比较，耗散粒子动力学更容易和精确的模拟了三维状态下流体的行为，因此，具有更重要的意义。
在耗散粒子动力学中，基本颗粒是“格子”，它表示流体材料的一个小区域，相当于MD模拟中我们所熟悉的原子和分子。
假设所有小于一个格子半径的自由度被调整出去只保留格子间粗粒状的相互作用。
在格子之间存在三种力，使得每个格子对保持格子数和线性动量都守恒：简谐守恒相互作用（保守力），表示运动的格子之间的粘滞阻力（耗散力）和为保持不扩散对系统的能量输入（随机力）。
所有这些力都是短程力并具有一个固定的截止半径。
通过选择适当这些力的大小，可得到一个相应于gibbs-cano系统的稳定态。
对于格子运动方程积分可以产生一条通过系统相空间轨迹线，由它可以计算得到所有的热力学可观测量（如密度场，序参量，相关函数，拉伸张量等）。
22RT-19-1Room sensor TemperatureFor measuring the temperature in the room. The room units can be seamlessly connected to existing third-party controllers. With MP-Bus communication and integrated 0...10 V output. Output signal is selectable via NFC.Type OverviewType CommunicationOutput signal activetemperature 22RT-19-1MP-Bus0...5 V, 0...10 V, 2...10 VTechnical dataElectrical dataNominal voltage AC/DC 24 VNominal voltage range AC 19.2...28.8 V / DC 19.2...28.8 V Power consumption AC 1 VA Power consumption DC 0.5 WElectrical connection Spring loaded terminal 0.25...1.5 mm²Cable entryWire openings at the backside (for In-wall wiring) and top-/bottom side (for On-wall wiring)Functional data Application Air Communication MP-BusVoltage output1x 0...5 V, 0...10 V, 2...10 V, min. load 10 kΩOutput signal active noteOutput 0...5 V, 0...10 V (factory setting), 2...10 V selectable via NFC Measuring data Measured valuesTemperature Measuring range temperature 0...50°C [32...122°F]Accuracy temperature active ±0.5°C @ 25°C [±0.9°F @ 77°F]Long-term stability±0.03°C p.a. @ 25°C [±0.05°F p.a. @ 77°F]Time constant τ (63%) in the room Typical 960 s Wall coupling factor52 %Materials HousingPC, white, RAL 9003Safety dataAmbient humidity Max. 95% RH, non-condensing Ambient temperature 0...50°C [30...120°F]Storage temperature -20...60°C [-5...140°F]Protection class IEC/EN III, Protective Extra-Low Voltage (PELV)EU Conformity CE MarkingCertification IEC/EN IEC/EN 60730-1 and IEC/EN 60730-2-9Degree of protection IEC/EN IP30Quality StandardISO 900122RT-19-1General remarks concerning sensorsBuild-up of self-heating by electricaldissipative powerDigital inputSafety notesThis device has been designed for use in stationary heating, ventilation and air-conditioning systems and must not be used outside the specified field of application. Unauthorisedmodifications are prohibited. The product must not be used in relation with any equipment that in case of a failure may threaten humans, animals or assets.Ensure all power is disconnected before installing. Do not connect to live/operating equipment.Only authorised specialists may carry out installation. All applicable legal or institutional installation regulations must be complied during installation.The device contains electrical and electronic components and must not be disposed of as household refuse. All locally valid regulations and requirements must be observed.RemarksThe measuring result is influenced by the thermal characteristics of the wall. A solid concrete wall responds to thermal fluctuations within a room slower than a light-weight structure wall. Room temperature sensors installed in flush-mounted boxes have a longer response time to thermal variations. For example, in extreme cases they will detect the radiant heat of the wall even if the air temperature in the room is lower. The quicker the dynamics of the wall (temperature acceptance of the wall) or the longer the selected inquiry interval of the temperature sensor is, the smaller the deviations are.Temperature sensors with electronic components always have a dissipative power which affects the temperature measurement of the ambient air. The dissipation in active temperature sensors shows a linear increase with rising operating voltage. The dissipative power should be taken into account when measuring temperature. In case of a fixed operating voltage (±0.2 V) this is normally done by adding or reducing a constant offset value. As Belimo transducers work with a variable operating voltage, only one operating voltage can be taken into consideration, for reasons of production engineering. Transducers 0...10 V / 4...20 mA have a standard setting at an operating voltage of DC 24 V. That means, that at this voltage, the expected measuring error of the output signal will be the least. For other operating voltages, the offset error will be increased by a changing power loss of the sensor electronics.If a readjustment directly at the active sensor should be necessary during later operation, this can be done with the following adjustment methods.- For sensors with NFC or dongle by the corresponding Belimo app - For sensors with a trimming potentiometer on the sensor board - For bus sensors via bus interface with a corresponding software variableAuxiliary Digital Input can be used with third-party sensors and switches (window alarm,occupancy detector, etc.). The input values are monitored and transmitted only through the MP-Bus communication protocol.Scope of deliveryScrewsAccessoriesService toolsDescriptionTypeBelimo Assistant App, Smartphone app for easy commissioning, parametrising and maintenance Belimo Assistant AppConverter Bluetooth / NFCZIP-BT-NFC22RT-19-1NFC connectionServiceBelimo equipment marked with the NFC logo can be operated and parameterized with the Belimo Assistant App.Requirement:- NFC- or Bluetooth-capable smartphone- Belimo Assistant App (Google Play & Apple AppStore)Align NFC-capable smartphone on the sensor so that both NFC antennas are superposed.Connect Bluetooth-enabled smartphone via the Bluetooth-to-NFC Converter ZIP-BT-NFC to thesensor. Technical data and operation instructions are shown in the ZIP-BT-NFC data sheet.Wiring diagramGND = 1AC/DC 24 V = 2MP Bus = 5Output Temperature = 8Output Humidity (22RTH / 22RTM) = 9GND = 10Digital Input, e.g. presence detector = 12/13Output CO₂ (22RTM) = 14GND = 1522RT-19-1 DimensionsType Weight22RT-19-10.113 kg。
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a r X i v :c o n d -m a t /0305534v 1 [c o n d -m a t .s t r -e l ] 22 M a y 2003Dissipative dynamics of an extended magnetic nanostructure :Spin necklace in ametallic environmentN.Shah 1and lis 21Center for Materials Theory,Department of Physics &Astronomy,Rutgers University,136Frelinghuysen Road,Piscataway,NJ 08854.2Department of Physics,Columbia University,538W 120th St,NY,NY 10027.We study theoretically the dynamics of an “xxz”spin necklace coupled to a conduction electron sea,a model system for a nanostructure in a dissipative environment.We extract the long-time behavior via a mapping to a multichannel Coulomb gas problem followed by a scaling analysis.The strong quantum fluctuations of the necklace cause a nontrivial dependence of couplings on system size which we extract via an analysis involving the “boundary condition changing operator”,and confirm via a detailed numerical evaluation of one case.The dissipative dynamics of a magnetic nanostructure is important from both technological and fundamental points of view.For example,magnetic recording involves the polarization of small domains,whose stability over long times is crucial.Tunnelling of isolated magnetic particles has been extensively studied [1][2].We shall be interested here in the new physics brought by coupling to a conduction electron bath.There are two effects.Ex-change of spin between the conduction electron system and the nanostructure changes the magnetic state of the nanostructure,leading effectively to a tunneling process.On the other hand,dissipation from the particle-hole ex-citations of the metal suppresses tunnelling [3].A finite cluster of spins coupled to a conduction electron bath is thus a model system for examining general issues of quan-tum coherence and dissipation in a system with many degrees of freedom.Also,recent experimental advances in atom manipulation [4][5]mean that clusters of atoms (eg.quantum corrals)can now be placed in controllable arrays on suitably chosen solid surfaces,so it seems likely that in the near future controlled magnetic nano-arrays may be constructed and studied.In the case of a single spin in a metal,the inter-play between tunnelling and dissipation gives rise to the logarithmic renormalizations characteristic of the one-impurity Kondo effect [6].While the two [7][8]and three [9]impurity Kondo problems have been studied,there is less known about the behavior of systems con-taining larger numbers of spins.The subject has been recently studied in the context of the “Griffiths phase”scenario for the non-Fermi-liquid physics apparently ob-served in certain ‘heavy fermion’metals [10].To study this physics,Castro-Neto and Jones [8]considered a clus-ter of N magnetic impurities,coupled strongly to each other and weakly to a conduction electron bath.They presented a series of qualitative arguments and approxi-mate mappings from which they concluded that the sys-tem could undergo a more or less conventional Kondo screening (which they termed a “cluster Kondo effect”),at an N −dependent temperature which they estimated.The qualitative nature of the treatment suggests how-ever that further analysis would be useful.One of us,with Morr and Schmalian [11],treated a similar problem of N coupled moments from a functional integral point of view.In the Ising case,orthogonality effects were found to suppress tunnelling in all but the smallest nanostruc-tures,while no definitive results were obtained in the continuous symmetry case.In this paper we attack the problem from a different point of view.We identify a specific nanostructure,the finite-size spin necklace depicted in Fig.1.There are three motivations for this choice of cluster.First,it is (as can well be seen)amenable to a detailed analysis.Second,it exhibits large (but tractable)quantum fluctuations whereas in other works [8][11]the quantum fluctuations within the cluster are not taken into account.Third,we can treat the continuous symmetry case as well.Fig.1Sketch of ‘spin necklace’in conduction electron bath for N =5.Heavy dots represent local moments,which are coupled via an exchange coupling K.Shaded area represents conduction electron (c)sea,Kondo-coupled to local moments via exchange J.The Hamiltonian describing the physics of interest isH =H necklace +H cond +H z coupling +H ⊥coupling (1)H necklace =N −1j =0(K ⊥(S x j S x j +1+S y jS y j +1)+K z S z j S z j +1)(2)H cond =kǫk c †kσc kσ(3)H zcoupling=J z N −1 j =0S z j (c †R j ↑c R j ↑−c †R j ↓c R j ↓)(4)H ⊥coupling=J ⊥N −1 j =0(S +j c †R j ↓c R j ↑+S −j c †R j ↑c R j ↓)(5)with R j the position of spin j.Here each spin value is S j=1/2and S N=S0.We restrict to the case of K z>−|K⊥|,so the ground state of H necklace is not ferromagnetic.We further specialize to N odd so the ground state is characterized by a spin quantum number S z=±1/2.For K⊥<0the total momentum of each of the S z=±1/2states is zero and the ground state is two-fold degenerate.For K⊥>0the ground state also hasnon-zero total momentum P=κ|P|,κ=±1implyingfour-fold degeneracy.In this paper we focus on the limit in which the Kondo coupling,J,of the cluster to the con-duction electrons is weak enough and the temperatures of interest low enough,that it is sufficient to consideronly those spinflip processes that couple the degenerateground states of the cluster.We expect that stronger couplings,implying mixing of higher energy states of thecluster,give rise to qualitatively similar physics.Thiscase will be considered in a future paper[12].If H⊥coupling=0,then the z−component of the clusterspin is a good quantum number.The cluster remains ina definite spin state,the conduction electrons just feel a potential due to the longitudinal coupling J z,and theHamiltonian H0=H necklace+H cond+H z coupling is eas-ily diagonalized.For non-zero J⊥the problem is nolonger trivially solvable.To study it,we use a pertur-bative expansion in J⊥to establish a mapping onto a Coulomb gas,following the approach introduced by An-derson and Yuval[6].Let|Ψ⇑ denote the ground stateof H0when the cluster is in the S z=+1/2state,|⇑ and consider the ground-state-to-ground-state amplitude(al-ternatively the partition function in the imaginary time formulation)given in the interaction representation by Z(t)= Ψ⇑ e iH0t T exp[i t0H′(t′)dt′] Ψ⇑ (6) where H′(t)=e−iH0t H⊥coupling e iH0t and T is the“timeordering operator”.Restricting to the ground state sub-space of the cluster,and expanding to O(H′2),Z(t)=J2⊥ t0dt1 t10dt2 R i R j ⇑|S+i|⇓ ⇓|S−j|⇑ ×F↑(R ij,t12)F↓(R ij,t12)(7) where R ij≡R i−R j,t12≡t1−t2andF↑(R ij,t12)= Ψ⇑ T c↑R i(t1)c†↑R j(t2)S(t12) Ψ⇑ (8) S(t12)=T exp[−i t1t2dτV(τ)](9)V(τ)=−2J z R j ⇑|S z j|⇑ c†R j(τ)c R j(τ).(10)The expressions above are similar to the ones introduced in the context of the X-ray[13][14]and the Kondo prob-lem[6],but with new features arising from the spatially extended nature of the magnetic object.The most im-portant of these new features are the space dependence of F,arising from processes in which an electron is cre-ated at one site and destroyed at another,and a nontriv-ial dependence of matrix elements such as ⇑|S+i|⇓ on necklace size and symmetry.We discuss these in turn. Following Ref.[13]we express F as the product of open line part g=F/ S and a closed loop part S ≡e C which may be computed by an expansion in J z. The crucial object in this expansion is the Green’s func-tions G(r,t)of H cond.We require the long time limit (t≫R/v F with R the radius of the necklace)in which G(r,t)→ρF G(r)/it:for a spherical Fermi surface G(r)=2sin(k F r)/v F r.In order to obtain a closed form for g we exploit the specific geometry of the necklace. We parametrize the positions of the spins by the az-imuthal angleφj=2πj/N,j∈[0,N−1].Using R kl= 2R sin(φkl/2),whereφkl≡φk−φl,G(R kl)→G(φkl). Wefinally define the one-dimensional Fourier transformG m=N−1k−l=0G(φkl)e imφkl(11) G(φkl)=12,...0,...N−1iN(t12)e−imφij τct12V2ρ2F m G2m(14)whereτc is the short-time cut-offseparating two spin-flips and we have neglected terms of O(J3z)in the exponent. We now consider the matrix elements.Wefirst note that if K⊥>0then the necklace ground states|⇑ ,|⇑ have a nonvanishing momentum±P⇑,⇓and the matrix el-ement depends on the momentum transfer: ⇑|S+j|⇓ = ⇑|S+0|⇓ exp[iQj]where Q=(P⇓−P⇑).When K⊥>0, the absolute value of the matrix element decays faster with N when P⇑=P⇓.For K⊥>0when P⇑=−P⇓and for K⊥<0wefind,definingα≤1,⇑|S+j|⇓ = ⇑|S+0|⇓ ∼1N.(16)For|K z|≤|K⊥|,in terms of the Jordan-Wigner fermion representation[17],the states|⇑ and|⇓ have different boundary conditions,one periodic and the other antiperiodic.Considering the low energy form of the Hamiltonian given by the Luttinger(massless)modeland adopting the“boundary-condition changing opera-tor”formalism[15][16]wefindα=1πcos−1K z2and1τc t10dt2t′−t (20)where using2π Q/N=Q=(P⇓−P⇑),εm= J z[G m+G m+ Q]− J z2 m′G2m′(21) u m= J⊥G m G m+ Q(22)with J⊥=J⊥ρF ⇑|S+0|⇓ ∼J⊥ρF/Nαand J z= 2ρF J z ⇑|S z0|⇑ ∼J zρF/N.Note that since m G2m∼N,the two terms inεm are of the same order in N and its leading behavior is given by the term linear in J zρF(as-sumed≪1).In the corresponding expression obtained in Ref.[11]it was assumed that only one channel was important and in the expression forεm,the term lin-ear in J z was overlooked.The above expression for Z, Eq.(19),is the second-order term in the expansion of a multicomponent Coulomb gas model in which each spin flip event(Coulomb gas charge)is labelled by the channel (m−value)of the electron-hole pair created.Higher order terms in the expansion in J⊥may be treated similarly, but compact expressions cannot be obtained because(as noted in a related context[18])in the multi-channel case, the higher order terms do not combine into the Cauchy determinant form[6].However we may define a scaling procedure as usual[19]by eliminating close-pair spinflips of the same channel.To the leading order the different channels are not coupled,and standard arguments show that near the weak couplingfixed point,the partition function retains the form Eq.(19)but with parameters εm(τc)and u m(τc)which evolve according todεm/d(lnτc)=2u2m(23)du m/d(lnτc)=u mεm/2(24)with initial conditions given by Eq.(21)and(22).The flow implied by Eq.(23)and(24)is shown inFig.3.m Fig.3Plot showing theflow for each channel,near the weak couplingfixed point.The heavy solid line is the separatrixεm=−u m/2.The shaded oval region shows the generic initial conditions for the xx symmetry.The dotted circle shows the generic initial conditions for the Ising limit.The striped region near u m=0is the range of initial conditions for‘imperfect’necklaces.For each channel theflow is to the no-flipfixed line (u m=0)ifεm<−u m/2and towards the stronglycoupled (rapidly flipping)regime (with εm =u m /2)if εm >−u m /2.Scaling amplifies the differences between channels.For example,in the K ⊥<0case of thexx −model,the initial condition is u 0m =J ⊥ρF G m /N1/4and ε0m =2J z ρF G m /N .For large N ,ε0m is thus para-metrically smaller than u 0m ;all initial conditions are near the vertical axis and the dominant channels are those with the largest G m .Representative results are shown in Fig.4.Note that for the ideal ring studied here,G −m =G m .We see that all channels with m <Nk F /2πcouple with roughly equal strength,but generically there is one largest pair,so after a transient many channel regime,the asymptotic behavior is of the two-channel Kondo problem.Exceptions can occur,and in particu-lar for k F 2π/N ,the m =0channel dominates.The same generic behavior occurs in other cases we have stud-ied,but we note that in the Ising regime the J z and J ⊥couplings are both ∼1/N ,so the initial conditions are shifted as indicated in Fig.3.m0123G mFig.4Plot showing G m in the K ⊥<0case of the xx −model for two values of k F .For k F =π,the pair m =−10and 10and for k F =π/2,the pair m =−4and 4is the highest,constituting the dominant pair which governs the low energy behavior.To summarize:we have shown that the generic long-time dynamics of a spin necklace coupled to a conduction electron bath is of the two-channel Kondo class.We em-phasize that generic deviations from perfect translation invariance break the two-channel symmetry.A key fea-ture of the model is the large quantum fluctuations char-acteristic of the isolated necklace:these ensure that the dominant spin flip amplitude between the ground states of the spin necklace decays only as a power of the sys-tem size (N −αwith 0<α≤1)and that the static field exerted by the necklace on the conduction bath is small,of order 1/N,so “orthogonality”effects do not dominate.To explore the changes occurring as quantum fluctuations are reduced,we consider a necklace in the extreme Ising limit (K z ≫|K ⊥|)and with one weak link (K 01z →K z (1−δ))which acts to localize the frustrated bond.A detailed discussion and extension to other cases will be published elsewhere [12].Here we note that al-though the residual reflection symmetry allows two chan-nel behavior,the spin flip amplitude changes from ∼1/N to ∼exp[−N ln(K ⊥/δK z )](moving the initial condition much closer to the u m =0axis of Fig.3).More im-portantly,because the frustrated bond is localized,the quadratic,“orthogonality”term in εm (Eq.(21))becomes∼J 2zN ,moving the initial condition substantially to the left (independent of the sign of J z );and,for large enough N ,into the stable (no asymptotic flip)region.Extensions of the present work include a treatment of systems with open boundary conditions,and of other nanostructures.We expect generically that internal quantum fluctuations of a nanoscale object (neglected in many treatments,Ref.[8]and [11]for example)will have a dramatic effect on the long time dynamics.Acknowledgments:We thank L.B.Ioffe for helpful discussions and NSF-DMR-00081075for support.。
