From molecular flexibility to shape, curvature and thermotropic transitions in pure surfaces

A material surface of pure constituents with a flexible molecular chain (amphiphilics) is considered; thermodynamic behaviour is studied in the chain length-temperature plane. The Hamiltonian of the system is modelled as the sum of a formation term which refers to the polymer nature of the chain, and of a fluctuation term with a specific elastic form. For closed systems the model exhibits phases with uniform curvature and conformational order/disorder or, alternatively, modulated phases; a critical chain length is found for the existence of modulated phases; the dependence of transition temperature on energy parameters is determined. A critical region is found for open systems, where conformational disorder drives spontaneous generation of curvature; this lies above a characteristic chain length and around the shape transition temperature. Received: 13 November 1996 / Revised: 9 May 1997 / Received in final form: 4 November 1997 / Accepted: 10 November 1997     

Nonlinear waves of spatial polarization for a crystal of the deuterated seignette salt

The dynamic properties of order-disorder systems with an asymmetric double-well single-particle potential and two dipoles per paraelectric unit cell are investigated. The Hamiltonian of the problem has a standard de Gennes form for two sublattices and is used for describing a crystal of the Seignette salt. The conditions of existence of nonlinear waves of spatial polarization in different temperature phases of the system under consideration are analyzed numerically.     

循环量子系统中状态演化的Bloch定理和同步几何相位的统一

对于Hamiltonian随时间作周期变化的量子系统中状态的演化，Bloch定理亦成立，并可据此定义一种新的几何相位———Bloch相位．证明用这种新的几何相位可以把迄今发现的所有同步（即量子态演化一周后获得的）几何相位统一起来，即Bloch相位等于Pancharatnam相位、Aharonov-Anandan相位和Lewis-Riesenfeld相位，并且在绝热条件下化为Bery相位．为此，先对Pancharatnam相位、Aharonov-Anandan相位和Lewis-Riesenfeld相位的定义作等价的改变，使它们变得有物理意义，并把Lewis-Riesenfeld相位和Berry相位推广到简并情形．还讨论了Bloch相位的求解问题 关键词：     

Generating Function Methods for Coefficient-Varying Generalized Hamiltonian Systems

The generating function methods have been applied successfully to generalized Hamiltonian systems with constant or invertible Poisson-structure matrices. In this paper, we extend these results and present the generating function methods preserving the Poisson structures for generalized Hamiltonian systems with general variable Poisson-structure matrices. In particular, some obtained Poisson schemes are applied efficiently to some dynamical systems which can be written into generalized Hamiltonian systems (such as generalized Lotka-Volterra systems, Robbins equations and so on).     

Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree − 3

In this paper, we study the polynomial integrability of natural Hamiltonian systems with two degrees of freedom having a homogeneous potential of degree k given either by a polynomial, or by an inverse of a polynomial. For k=−2,−1,…,3,4, their polynomial integrability has been characterized. Here, we have two main results. First, we characterize the polynomial integrability of those Hamiltonian systems with homogeneous potential of degree −3. Second, we extend a relation between the nontrivial eigenvalues of the Hessian of the potential calculated at a Darboux point to a family of Hamiltonian systems with potentials given by an inverse of a homogeneous polynomial. This relation was known for such Hamiltonian systems with homogeneous polynomial potentials. Finally, we present three open problems related with the polynomial integrability of Hamiltonian systems with a rational potential.     

Off-diagonal geometric phases

We investigate the adiabatic evolution of a set of nondegenerate eigenstates of a parametrized Hamiltonian. Their relative phase change can be related to geometric measurable quantities that extend the familiar concept of Berry phase to the evolution of more than one state. We present several physical systems where these concepts can be applied, including an experiment on microwave cavities for which off-diagonal phases can be determined from published data.     

Finding Short-Range Parity-Time Phase-Transition Points with a Neural Network

The non-Hermitian PT-symmetric system can live in either unbroken or broken PT-symmetric phase. The separation point of the unbroken and broken PT-symmetric phases is called the PT-phase-transition point.Conventionally, given an arbitrary non-Hermitian PT-symmetric Hamiltonian, one has to solve the corresponding Schrodinger equation explicitly in order to determine which phase it is actually in. Here, we propose to use artificial neural network(ANN) to determine the PT-phase-transition points for non-Hermitian PT-symmetric systems with short-range potentials. The numerical results given by ANN agree well with the literature, which shows the reliability of our new method.     

Phase transitions and renormalization group hamiltonian densities in the 80 diperiodic space groups

Phase transitions for systems with diperiodic symmetry are discussed. Direct group-theoretical methods are employed to obtain a list of possible commensurate lower-symmetry phases (subgroups) which are induced by a single order parameter. The lower-symmetry phases for all 80 diperiodic space groups are given, along with specific details of the group-subgroup relationships. Results for the 17 two-dimensional space groups are also contained in our list. The renormalization-group Hamiltonian densities for the diperiodics are calculated. The 12 densities listed constitute the complete set of densities which may arise in the diperiodic space groups. Critical properties for the diperiodics can thus be obtained from analysis of these densities.     

Semiclassical Theory of Quasiparticles in the Superconducting State

We have developed a semiclassical approach to solving the Bogoliubov-de Gennes equations for superconductors. It is based on the study of classical orbits governed by an effective Hamiltonian corresponding to the quasiparticles in the superconducting state and includes an account of the Bohr-Sommerfeld quantisation rule, the Maslov index, torus quantisation, topological phases arising from lines of phase singularities (vortices), and semiclassical wave functions for multidimensional systems. The method is illustrated by studying the problem of an SNS junction and a single vortex.     

Comparison of several approaches to the relativistic dynamics of directly interacting particles

Several approaches to the relativistic dynamics of directly interacting particles are compared. The equivalence between constrained Hamiltonian relativistic systems and a priori Hamiltonian predictive ones is completely proved. Coordinate transformations are obtained to express these systems in the framework of noncovariant predictive mechanics. The world line condition for constrained Hamiltonian relativistic systems is analyzed and is proved to be also necessary in the predictive Hamiltonian framework.     

由含时边界条件的两种有限深量子势阱构造的哈密顿算符和它们的复BERRY相位

在量子基础理论的框架下分别从一维含时边界条件的半壁无限高量子势阱和三维含时边界条件的有限深球方势阱构造了两种描述带电粒子在电磁场作用下的新哈密顿算符.在绝热近似下，计算了这两种新量子体系的复Berry相位.     

The intrinsic linearization of equilibria of nonholonomic systems

The linearization of equilibria of Hamiltonian systems is Hamiltonian; this has well-known and important implications for the spectrum. The analogous statement for nonholonomic systems is provided. It follows, for example, that the linearization of the ground state of a nonholonomic system is always Hamiltonian.     

On chaotic dynamics in "pseudobilliard" Hamiltonian systems with two degrees of freedom

A new class of Hamiltonian dynamical systems with two degrees of freedom is studied, for which the Hamiltonian function is a linear form with respect to moduli of both momenta. For different potentials such systems can be either completely integrable or behave just as normal nonintegrable Hamiltonian systems with two degrees of freedom: one observes many of the phenomena characteristic of the latter ones, such as a breakdown of invariant tori as soon as the integrability is violated; a formation of stochastic layers around destroyed separatrices; bifurcations of periodic orbits, etc. At the same time, the equations of motion are simply integrated on subsequent adjacent time intervals, as in billiard systems; i.e., all the trajectories can be calculated explicitly: Given an initial data, the state of the system is uniquely determined for any moment. This feature of systems in interest makes them very attractive models for a study of nonlinear phenomena in finite-dimensional Hamiltonian systems. A simple representative model of this class (a model with quadratic potential), whose dynamics is typical, is studied in detail. (c) 1997 American Institute of Physics.     

Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems

This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of nonholonomic Hamiltonian systems are derived from discrete Hamiltonian action. Secondly, the determining equations and structure equation of Lie symmetry of the system are obtained. Thirdly, the Lie theorems and the conservation quantities are given for the discrete nonholonomic Hamiltonian systems. Finally, an example is discussed to illustrate the application of the results.     

Controllability of multiple qubit systems

This paper explores the problem of manipulating multiple-qubit systems when only single-qubit operations or two-qubit-interactive operations are permitted. It is demonstrated that if there exist 2 directional control Hamiltonian for each individual qubit, and one interactive Hamiltonian for each pair of qubits, then multiple qubit systems are open-loop controllable. An important observation of physical interest is emphasized: when only single-qubit operations or two-qubit-interactive operations are permitted, only n(n+3)/2 control Hamilton may guarantee open-loop controllability of n qubit systems, and n(n+3) is, in the restricted sense, also the lower limit on the number of operators needed for controllability. At last, we demonstrate that an n-quantum-dot system is open-loop controllable even when only single-qubit operations or two-qubit-interactive operations are permitted.     

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.     

Theory of Tensor Invariants of Integrable Hamiltonian Systems. II. Theorem on Symmetries and Its Applications

The theorem on symmetries is proved that states that a Liouville-integrable Hamiltonian system is non-degene\-rate in Kolmogorov's sense and has compact invariant submanifolds if and only if the corresponding Lie algebra of symmetries is abelian. The theorem on symmetries has applications to the characterization problem, to the integrable hierarchies problem, to the necessary conditions for the strong dynamical compatibility problem, and to the problem on master symmetries. The invariant necessary conditions for the non-degenerate C-integrability in Kolmogorov's sense of a given dynamical system V are derived. It is proved that the C-integrable Hamiltonian system is non-degenerate in the iso-energetic sense if and only if the corresponding Lie algebra of the iso-energetic conformal symmetries is abelian. An extended concept of integrability of Hamiltonian systems on the symplectic manifolds M ⁿ , n= 2k, is introduced. The concept of integrability describes the Hamiltonian systems that have quasi-periodic dynamics on tori or on toroidal cylinders of an arbitrary dimension . This concept includes, as a particular case, all Hamiltonian systems that are integrable in Liouville's classical sense, for which . The A-B-C-cohomologies are introduced for dynamical systems on smooth manifolds. Received: 16 January 1996 / Accepted: 3 July 1996