Energy balance in deformation mechanics of solids at low temperatures

The model of a solid in the form of an ensemble of independent anharmonic oscillators arranged in a uniform stress field has been considered to analyze the energy balance during adiabatic mechanical loading of a solid at low temperatures. Oscillator elongation is determined as the average over the ensemble, and a part of its energy is matched to this quantity. This part has the physical meaning of the mechanical energy of sample deformation and becomes a part of the energy balance upon deformation. After averaging, the uniform force field is replaced by the resultant force associated with the average deformation. Another component of the balance at low temperatures is the energy of zero-point vibrations of oscillators. Thus, upon mechanical deformation of a solid, the energy exchange occurs between two scale levels: the atomic vibration energy at a microlevel and the macroscopic deformation energy of the sample as a whole.     

Analysis of energy distribution in a loaded quantum anharmonic oscillator in a wide temperature range

Analysis of the energy distribution in an ensemble of quantum anharmonic oscillators loaded by an external force in a wide temperature range (from T = 0) is carried out using a general approach based on the virial theorem. At T = 0, anharmonic effects are observed: a linear variation of zero-point energy of an oscillator under loading (energy decrease during extension and increase under compression) and a linear variation of the average kinetic and potential energy components. At high temperatures, at which the dynamics of the oscillators becomes classical, the anharmonic effects are manifested in a linear variation in the vibrational energy and a linear variation in the average kinetic and potential energy components upon an increase in force. Mutually compensating variation in the average kinetic and potential energy components of the internal dynamic energy of an oscillator (energy redistribution upon loading) takes place both at low and high temperatures.     

Conditions for the onset of superradiance regime in a relativistic electron bunch

Problems associated with the formation of coherent oscillations of an ensemble of classical oscillators and their superradiance instability are considered. The dispersion properties of an electron bunch and the conditions for the generation of nonequilibrium radiation are determined in the quasi-steady anharmonic oscillator approximation.     

Energy dissipation prediction in a line of colliding oscillators

In this paper the impact of a line of adjacent structures, or oscillators, is studied using an energy formulation. The energy exchange and dissipation from a collision of a pair of oscillators is studied by creating an equivalent oscillator pair, one has the energy of the in-phase motion and the other has the out-of-phase energy. It is found that the energy exchange between colliding oscillators is proportional to the initial kinetic energy difference of the oscillators and that work in the collision is proportional to the out-of-phase energy or difference energy. The kinetic energy at contact is then related to the mean oscillator energy, permitting a power balance equation to be written for each oscillator in line. The power balance equations have three independent variables for each pair of oscillators: the oscillator time averaged energies and the phase difference. This equation is run in a time-stepping procedure, with steps at the mean collision rate. The work in the collisions and internal oscillator dissipation is output as a function of time. A parameter study is conducted to see how the work changes with oscillator: separation, contact stiffness and contact damping.     

Evaluation of the density matrix for an ensemble of anharmonic oscillators by a path integral approach

Considering the Feynman path integral representation for the configuration-space density matrix for an ensemble of anharmonic oscillators, we determine the stationary paths near which the integrand remains stationary. By taking the path integral to be saturated by contributions from the neighborhood of the path which maximizes the integrand we evaluate the density matrix explicitly in analytic form. This seems to be the first such evaluation of a path integral for a system not describable by a quadratic Hamiltonian. We also comment briefly on the question of analyticity with respect to the perturbation parameter.     

Quantum theory with an energy operator defined as a quartic form of the momentum

Quantum theory of the non-harmonic oscillator defined by the energy operator proposed by Yurke and Buks (2006) is presented. Although these authors considered a specific problem related to a model of transmission lines in a Kerr medium, our ambition is not to discuss the physical substantiation of their model. Instead, we consider the problem from an abstract, logically deductive, viewpoint. Using the Yurke–Buks energy operator, we focus attention on the imaginary-time propagator. We derive it as a functional of the Mehler kernel and, alternatively, as an exact series involving Hermite polynomials. For a statistical ensemble of identical oscillators defined by the Yurke–Buks energy operator, we calculate the partition function, average energy, free energy and entropy. Using the diagonal element of the canonical density matrix of this ensemble in the coordinate representation, we define a probability density, which appears to be a deformed Gaussian distribution. A peculiarity of this probability density is that it may reveal, when plotted as a function of the position variable, a shape with two peaks located symmetrically with respect to the central point.     

Sensitivity to anharmonicity of vibrational energy in a diatomic gasdynamic laser

A parameter for evaluating the sensitivity of quantum vibrational energy to anharmonicity in a diatomic gasdynamic laser is defined and calculated by considering the corresponding diatomic molecules as quantum anharmonic oscillators under an interatomic Morse potential. The variation of the above parameter in terms of the vibrational states and in terms of an involved anharmonic coefficient is discussed. In particular, the parameter in question at the classical limit is examined. Both weak and strong anharmonicities are discussed.     

Quantum entanglement and squeezing in coupled harmonic and anharmonic oscillator systems

We investigate the fundamental connection between quadrature squeezing and continuous variable entanglement within a general class of two-coupled oscillator systems. We determine the quantitative relationship between them through the squeezing parameter and the entanglement entropy of the lowest energy eigenstate of the coupled oscillator systems numerically. Unlike the relation between entanglement and uncertainty product, we found that this relationship is, by no means, the same for the whole class of coupled oscillator systems: to a large extent it depends on the order and strength of the anharmonic potential, which implies that knowledge of the anharmonic potential of the coupled oscillator system is required before one can characterize the degree of entanglement through the squeezing parameter. Our results reveal that a more effective approach to enhance squeezing is to adjust the anharmonicity of the system potential, instead of increasing the quantum correlations between the oscillators. In addition, by probing into a quantum catastrophe model, we uncover transitions in the entanglement entropy and squeezing relation as the potential changes from a single well to a triple well, and then a double-well structure. The transitions appear through distinct entropy–squeezing relation, with a multi-well structure displaying a larger change in the antisqueezing behavior of the position quadrature than the single-well structure, for the same change in the entanglement entropy.     

Study of Certain Aspects of Anharmonic,Time-Dependent and Damped Harmonic Oscillator Systems

The present review is devoted to the study of certain aspects of anharmonic, time-dependent and damped oscillator(s) system using different theoretical techniques. A theoretical understanding of these systems is important for application in many problems in physics, mechanics and other fields. We discuss in detail the difficulties in the theoretical analysis of the problem. In particular we discuss here the regular, well-behaved perturbative solution, the large quantum number behaviour of anharmonic oscillator(s) using the technique of coherent states, exact solution of quantum anharmonic oscillators, the electromagnetic radiation emitted by a charged particle executing damped anharmonic oscillator motion using Krylov-Bogoliubov approximation method, use of invariants to obtain solution and coherent states of time-dependent oscillator(s), the derivation of perturbative frequencies of a damped coupled anharmonic oscillators system using suitable canonical transformation in the framework of Hamilton-Jacobi formalism and the quantisation and construction of coherent states of a damped oscillator using time-dependent operators.     

Ergodicity in computer simulation of quantum dissipative processes

The ergodicity principle in quantum theory is employed for elaboration of a new quantum trajectory technique which is used for numerical simulation of quantum dissipative systems. With this purpose the density matrix of a quantum system is represented as a sum over an ensemble of quantum states in time intervals. The method is employed for computations of a quantum anharmonic oscillator.     

Molecular dynamics study of effects of radius and defect on oscillatory behaviors of C60-nanotube oscillators

Using the micro-canonical ensemble, we investigate the oscillatory behaviors of some selected C60-nanotube oscillators by the classical molecular dynamics (MD) simulations method. The second-generation empirical bond-order potential and the van der Waals potential are used to describe bonding and nonbonding atomic interactions, respectively. In the process of simulation, two factors of the radius and vacancy defect of single-walled carbon nanotubes (SWCNTs) are discussed to investigate their effects on the oscillatory behaviors of C60-nanotube oscillators. The simulation results show that the energy dissipation of the C60-nanotube oscillator is sensitive to the radius and vacancy defect, and that the effect of the vacancy defect on the oscillatory behaviors of oscillator depends obviously on the radius of the outer tube. It is found that a single vacancy defect placed on the outer tube of the C60-(17,0) nanotube oscillator can significantly reduce energy dissipation. For C60-(18,0), C60-(19,0) and C60-(11,11) nanotube oscillators, however, the results show that an oscillator containing a vacancy defect is less stable than the one without defect.     

Ideal bose gas in an oscillator potential (exact solution)

Exact expressions for the statistical sum of the grand canonical ensemble and the one-particle density matrix are derived based on the definition of the density matrix for a system of N identical noninteracting Bose particles in an oscillator potential as a sum with respect to the symmetric exchange of the density matrix coordinates of distinguishable particles. A quasi-classical scenario is analyzed in detail.     

Study of a class of models for self-organization: equilibrium analysis

A new class of nonlinear stochastic models is introduced with a view to explore self-organization. The model consists of an assembly of anharmonic oscillators, interacting via a mean field of system size range, in presence of white, Gaussian noise. Its properties are explored in the overdamped regime (Smoluchowski limit). The single oscillator potential is such that for small oscillator displacements it leads to a highly nonlinear force but becomes asymptotically harmonic. The shape of the potential can be a single-or double-well and is controlled by a set of parameters. Through equilibrium statistical mechanical analysis, we study the collective behavior and the nature of phase transition. Much of the analysis is analytic and exact. The treatment is not restricted to the thermodynamic limit so that we are also able to discuss finite size effects in the model.     

Statistical Physics on the Space (x, v) for Dissipative Systems and Study of an Ensemble of Harmonic Oscillators in a Weak Linear Dissipative Medium

We use the phase space position-velocity (x, v) to deal with the statistical properties of velocity dependent dynamical systems, like dissipative ones. Within this approach, we study the statistical properties of an ensemble of harmonic oscillators in a linear weak dissipative media. Using the Debye model of a crystal, we calculate at first order in the dissipative parameter the entropy, free energy, internal energy, equation of state and specific heat using the classical and quantum approaches. For the classical approach we found that the entropy, the equation of state, and the free energy depend on the dissipative parameter, but the internal energy and specific heat do not depend of it. For the quantum case, we found that all the thermodynamical quantities depend on this parameter. PACS: 05.20.Gg, 05.30.Ch, 05.20.-y, 05.30.-d     

Classical and quantum oscillators of quartic anharmonicities: second-order solution

An analytical solution up to the second order in the coupling constant λ is obtained for a classical quartic anharmonic oscillator by using Taylor series method. Our solution yields, as a special instance, the corresponding results obtained by using Laplace transform. With the help of correspondence principle, the classical solution is used to obtain the solution corresponding to a quantum quartic anharmonic oscillator. In the weak coupling regime (i.e., anharmonic constant λ⪡1), the so-called secular terms in classical and quantum solutions are tucked in (summed up) to avoid the nonconvergence. Both the classical and quantum solutions are used to obtain the frequency shifts of the quartic oscillators. It is found that these frequency shifts coincide exactly with those of the earlier results obtained by other methods. From the quantum field theoretic point of view, our solution exhibits the so-called Lamb shift. As an application of the solution for the quantum oscillator, we examine the possibility of getting squeezed states out of the input coherent light interacting with a nonlinear medium of inversion symmetry.     

非简谐振子自由能的微扰展开

推导出正则系统的特性函数－自由能分别在经典情况及量子情况下的生扰表达式，并对非简谐振子自由能的微扰展开进行了具体计算。     

Analysis of the causes of formation of negative loss factors

I-IatroductionStatisticalEnergyAnalysis(SEA)isakindofeffective,simpleanddirectapproachforan-alyzingvibrationandsound,andithasbeenfoundwidelyapplicationsinanalysisofmechanicalnoiseandvibrationcolltrolsince198osl1-4].However,forgeneralindustrialmachineswhichalwayconsistofcomPlexandheaVystructures,thedeterminationmethodsofSEAparametersintheclassicalSEAtheoryareinapplicable[5]because:(1)SEAparametersofthesekindsofstructuresaredifficulttoobtainfromthetheory(2)theconditionofconservativeandweak…     

谐振子系统的量子-经典轨道、Berry相及Hannay角

从量子-经典轨道和几何相两方面, 研究了二维旋转平移谐振子系统的量子-经典对应. 通过广义规范变换得到了Lissajous经典周期轨道和Hannay角. 另外, 使用含时规范变换解析推导了旋转平移谐振子系统Schrödinger方程的本征波函数和Berry相, 得出结论: 原规范中的非绝热Berry相是经典Hannay角的-n倍. 最后, 使用SU(2)自旋相干态叠加, 构造一稳态波函数, 其波函数的概率云很好地局域于经典轨道上, 满足几何相位和经典轨道同时对应.     

Molecular motion under the influence of a coherent infrared-laser field

The quantum dynamics of molecular vibrations under the influence of coherent infrared-laser multiphoton excitation is irregular, in most cases, on the time scale of experimental interest. This irregularity arises from the anharmonic nature of the molecular force fields and is already present in one dimensional anharmonic oscillators. We analyse the onset of the irregularity in the model of a Morse oscillator and discuss the role of statistical or “chaotic” behaviour of the time dependent probability density. We further discuss the role of direct multiphoton excitation in these multilevel systems and compare the results with exact solutions for the harmonic oscillator.     

Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities

A model dynamical system with a great many degrees of freedom is proposed for which the critical condition for the onset of collective oscillations, the evolution of a suitably defined order parameter, and its fluctuations around steady states can be studied analytically. This is a rotator model appropriate for a large population of limit cycle oscillators. It is assumed that the natural frequencies of the oscillators are distributed and that each oscillator interacts with all the others uniformly. An exact self-consistent equation for the stationary amplitude of the collective oscillation is derived and is extended to a dynamical form. This dynamical extension is carried out near the transition point where the characteristic time scales of the order parameter and of the individual oscillators become well separated from each other. The macroscopic evolution equation thus obtained generally involves a fluctuating term whose irregular temporal variation comes from a deterministic torus motion of a subpopulation. The analysis of this equation reveals order parameter behavior qualitatively different from that in thermodynamic phase transitions, especially in that the critical fluctuations in the present system are extremely small.Dedicated to Ilya Prigogine on the occasion of his 70th birthday.