Construction of exact complex dynamical invariant of a two-dimensional classical system

We present the construction of exact complex dynamical invariant of a two-dimensional classical dynamical system on an extended complex space utilizing Lie algebraic approach. These invariants are expected to play a vital role in understanding the complex trajectories of both classical and quantum systems.     

The reliability of a stochastically complex dynamical system

The first-passage problem is an important issue in the theory of dynamical systems; it determines whether the system is safe and reliable or destroyed. Therefore, taking a suitable force to control first-passage failure is very significant for such systems. In this paper, we study the reliability of a stochastically complex dynamical system by using the stochastic averaging method. The reliability results show that the reliability function is monotonically decreasing with respect to time, and that the reliability of the system is relatively worse, with external and parametric white-noise excitations. A bang–bang feedback control is used to improve the reliability for the original system. Numerical results display that even a small control force can greatly enhance the reliability of the original system.     

Jump of the adiabatic invariant at a separatrix crossing: Degenerate cases

An expression for the quasi-random jump of the adiabatic invariant at a separatrix crossing is obtained for a slow-fast Hamiltonian system with two degrees of freedom in the case when the separatrix passes through a degenerate saddle point in the phase plane of the fast variables. The general case with an arbitrary degree of degeneracy was considered, and this degree is assumed to remain fixed in the process of evolution of the slow variables. The typical value of the jump is larger than in the non-degenerate case studied earlier. Though strongly degenerate, such a setting can be relevant for physical problems. The influence of the asymmetry of a phase portrait on the magnitude of adiabatic invariant jumps was considered as well. An example of this kind is studied, namely the motion of ions in current sheets with complex inner structure.     

Majorana bound state of a Bogoliubov-de Gennes-Dirac Hamiltonian in arbitrary dimensions

We study a Majorana zero-energy state bound to a hedgehog-like point defect in a topological superconductor described by a Bogoliubov-de Gennes (BdG)-Dirac type effective Hamiltonian. We first give an explicit wave function of a Majorana state by solving the BdG equation directly, from which an analytical index can be obtained. Next, by calculating the corresponding topological index, we show a precise equivalence between both indices to confirm the index theorem. Finally, we apply this observation to reexamine the role of another topological invariant, i.e., the Chern number associated with the Berry curvature proposed in the study of protected zero modes along the lines of topological classification of insulators and superconductors. We show that the Chern number is equivalent to the topological index, implying that it indeed reflects the number of zero-energy states. Our theoretical model belongs to the BDI class from the viewpoint of symmetry, whereas the spatial dimension d of the system is left arbitrary throughout the paper.     

On a time-dependent system of noncentral anharmonic oscillators in two dimensions

A new system of time-dependent anharmonic and anisotropic oscillators in two space-dimensions which corresponds to unequal but related spring constants, having Hamiltonian structure, admitting quadratic invariants and accounting also for the fractional powers in the coupling terms, is found.     

Symmetry of Hamiltonian and conserved quantity for a system of generalized classical mechanics

This paper focuses on a new symmetry of Hamiltonian and its conserved quantity for a system of generalized classical mechanics.The differential equations of motion of the system are established.The definition and the criterion of the symmetry of Hamiltonian of the system are given.A conserved quantity directly derived from the symmetry of Hamiltonian of the generalized classical mechanical system is given.Since a Hamilton system is a special case of the generalized classical mechanics,the results above are equally applicable to the Hamilton system.The results of the paper are the generalization of a theorem known for the existing nonsingular equivalent Lagrangian.Finally,two examples are given to illustrate the application of the results.     

Hamiltonian for a Relativistic Particle with Linear Dissipation

A general formalism for obtaining the Lagrangian and Hamiltonian for a one dimensional dissipative system is developed. The formalism is illustrated by applying it to the case of a relativistic particle with linear dissipation. The hamiltonian for a relativistic particle with linear dissipation is obtained. An example of this approach is given. PACS: 45.20.Jj     

量子系统哈密顿量的必要条件

在此前的量子理论中,哈密顿量被要求是厄米算符,这既保证了其本征值谱为实又保证了动力学演化过程几率守恒。近年来,一些非厄米哈密顿量被发现同样满足这两条要求。然而,这两条要求都是哈密顿量可描述量子系统动力学的充分条件而非必要条件。文章中我们从量子化条件和动力学演化方程出发,考察一般形式的哈密顿量,表述为产生算符和湮灭算符之正规积的形式,欲为恰当的哈密顿量所应满足的必要条件。对于形如$\\hat{H}=\\hat{p}^2+\\mathrm{i} \\hat{x}^3$和$\\hat{H}=\\hat{p}^2-\\hat{x}^4$这样的近年来得到广泛关注的非厄米哈密顿量,容易验证它们满足必要性条件。必要性条件可以用来及时排除那些不恰当的非厄米哈密顿量形式。     

A dynamical system with a strange attractor and invariant tori

This paper describes a simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities and the unusual property that it is exhibits conservative behavior for some initial conditions and dissipative behavior for others. The conservative regime has quasi-periodic orbits whose amplitude depend on the initial conditions, while the dissipative regime is chaotic. Thus a strange attractor coexists with an infinite set of nested invariant tori in the state space.     

Wavelet analysis and Arnold web picture for detecting energy transfer in a Hamiltonian dynamical system

Our motivation is to understand how, in chemical reactions, the reaction coordinate effectively gains dynamical energy from the other degrees of freedom (i.e., bath coordinates) avoiding thermalization of the redistributed energy. In such a system, the phase space structure should be not homogeneous; i.e., the system is never ergodic. In this study, we introduce a way to capture the inhomogeneity of the phase space and to monitor energy transfers among their partial degrees of freedom in nonergodic systems using wavelet analysis and a picture of the Arnold web. First, we examine several simple energy transfer processes, i.e., a motion on a resonance line, between resonance lines, and around a resonance junction in a simple three-degree-of-freedom (DOF) system and show how the elemental processes of the intramolecular vibrational energy redistribution (IVR) are detected by our tools. We especially note that the structure of the higher order resonance of the system can be detected by wavelet analysis and motion in the action space. Next, we analyze a reaction process in a simple Hamiltonian system of 3 DOF with a double-well potential, i.e., a system with a transition state of the center-saddle-center type, and detect energy transfers in the reactive process. The aim of the study is to propose a way to characterize the inhomogeneity of the phase space, e.g., the reactive doorway, which leads to controllability of the chemical reaction by light, i.e., control of the reaction by selectively preparing an initial state in the reactive doorway by optical excitation.     

Estimates for Non-Resonant Normal Forms in Hamiltonian Perturbation Theory

We make a remark about an estimate of the rest for the non-resonant action-angle normal forms and exhibit a simple example suggesting the optimality of this estimate when there are no small divisors. Given a polynomial perturbation of degree P and an integer k, calling the size of the small denominators up to order k, we prove that the kth order remainder is bounded by (2/ ₀) ^k+1 with ₀=const ²/(kP ²). Thus, fixing the degree of the perturbation, if is independent of k (i.e., if there are no small divisors), we obtain a rest bounded by (const k) ^k+1. These estimates are also applied to the case in which the small divisors are absent, and they are conjectured to be optimal in this context. To support this idea we present a simplified model problem with no small denominators, formally related to the above calculations, and we show that it indeed has factorial divergence of its Birkhoff series. We also obtain Nekhoroshev's Theorem for harmonic oscillators. We hope that our simple approach makes more accessible to a general audience this important (although quite technical) topic.     

Optimization of the Effective Hamiltonian Parameters in a Semiempirical Method of Calculation of Complex Atomic Spectra

In a semiempirical investigation of complex atomic spectra a new approach is suggested, according to which the number of experimental points for optimization of parameters is to be increased. This is topical when the number of optimization parameters approaches the number of available experimental energy levels. The possible optimization of the effective Hamiltonian parameters has been investigated for the case where not experimental energy levels but rather the wavelengths (wavenumbers) of transitions between the levels of the studied configurations are used. For the spectra of Mo XVII and Fe VI agreement between optimized parameters in the conventional and new approaches has been established.     

Collective Hamiltonian for Multi-O(4) Model

The collective Hamiltonian up to the fourth order for multi-O(4) model is derived based on the self-consistent collective-coordinate (SCC) method,which is formulated in the framework of the time-dependent Hartree-Bogoliubov (TDHB) theory.The validity of the collective Hamiltonian is checked in the two special cases of the multi-O(4) modelthe case where the number of the shells is equal to one (a single j-shell case),and the case where the Hartree-Bogoliubov equilibrium point is spherical (the spherical case).The collective Hamiltonian constitutes a good starting point to study nuclear shape coexistence.     

Treatment of Dirac Fields in Light—Front Coordinates by Dirac‘s Method for Constrained Hamiltonian Systems

Dirac‘s method which itself is for constrained Boson fields and particle systems is followed and developed to treat Dirac fields in light-front coordinates.     

构造哈密尔顿系统jet辛差分格式的生成函数法

考虑哈密尔顿系统的保结构算法,在经典哈密尔顿系统的jet辛算法的基础上,给出了一般哈密尔顿系统的jet辛差分格式的定义.并利用带有变系数辛矩阵的一般哈密尔顿系统中的构造辛差分格式的生成函数法的思想,来建立由一般的反对称矩阵所确定的微分二形式与生成函数的关系,再利用哈密尔顿-雅可比方程来构造jet辛的差分格式.     

Hamiltonian replica exchange in GROMACS: a flexible implementation

A simple and general implementation of Hamiltonian replica exchange for the popular molecular dynamics software GROMACS is presented. In this implementation, arbitrarily different Hamiltonians can be used for the different replicas without incurring in any significant performance penalty. The implementation was validated on a simple toy model – alanine dipeptide in water – and applied to study the rearrangement of an RNA tetraloop, where it was used to compare recently proposed force-field corrections.     

非对易空间的三模谐振子能谱

在非对易空间中,用不变本征算符方法(IEO),对非耦合、坐标耦合、动量耦合三种三模谐振子系统能谱进行求解,并将求解结果与一般对易空间的能谱进行比较分析.通过比较发现,当非对易参数为零时,所求能级差还原到了与普通空间相对应的一般量子系统哈密顿量能级差,验证了推导结果的正确性;同时讨论了耦合系数对非对易空间能谱的影响. 关键词： 不变本征算符 非对易空间 三模谐振子能谱 能级差     

奇异系统Hamilton正则方程的Mei对称性、Noether对称性和Lie对称性

研究奇异系统Hamilton正则方程的形式不变性即Mei对称性，给出其定义、确定方程、限制方程和附加限制方程.研究奇异系统Hamilton正则方程的Mei对称性与Noether对称性、Lie对称性之间的关系，寻求系统的守恒量.给出一个例子说明结果的应用. 关键词： 奇异系统 Hamilton正则方程 约束 对称性 守恒量     

Some applications of stochastic averaging method for quasi Hamiltonian systems in physics

Many physical systems can be modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems can be applied to yield reasonable approximate response statistics. In the present paper, the basic idea and procedure of the stochastic averaging method for quasi Hamiltonian systems are briefly introduced. The applications of the stochastic averaging method in studying the dynamics of active Brownian particles, the reaction rate theory, the dynamics of breathing and denaturation of DNA, and the Fermi resonance and its effect on the mean transition time are reviewed. Supported by the National Natural Science Foundation of China (Grant Nos. 10772159 and 10802074), the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20060335125), and the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y7080070)