Selection rules,causality, and unitarity in statistical and quantum physics

The integrodifferential equations satisfied by the statistical frequency functions for physical systems undergoing stochastic transitions are derived by application of a causality principle and selection rules to the Markov chain equations. The result equations can be viewed as generalizations of the diffusion equation, but, unlike the latter, they have a direct bearing onactive transport problems in biophysics andcondensation aggregation problems of astrophysics and phase transition theory. Simple specific examples of the effects of severe selection rules, such as the relaxational Boltzmann transport equation and the diffusion equation, are also given. Finally, partial differential equations for the probability amplitudes of quantum mechanics are derived, usingunitarity instead of causality, and a selection rule is applied directly to obtain ageneralization of the Dirac equation in which infinite transitions between states arenot allowed.     

Dirac-Type Equations in a Gravitational Field,with Vector Wave Function

An analysis of the classical-quantum correspondence shows that it needs to identify a preferred class of coordinate systems, which defines a torsionless connection. One such class is that of the locally-geodesic systems, corresponding to the Levi-Civita connection. Another class, thus another connection, emerges if a preferred reference frame is available. From the classical Hamiltonian that rules geodesic motion, the correspondence yields two distinct Klein-Gordon equations and two distinct Dirac-type equations in a general metric, depending on the connection used. Each of these two equations is generally-covariant, transforms the wave function as a four-vector, and differs from the Fock-Weyl gravitational Dirac equation (DFW equation). One obeys the equivalence principle in an often-accepted sense, whereas the DFW equation obeys that principle only in an extended sense. Part of this work was done while the author was at Dipartimento di Fisica, Università di Bari and INFN Bari, Italy.     

Gaussian sum rules in quantum chromodynamics and local duality

A new type of hadronic sum rules are studied in QCD. They correspond to the Gauss-Weierstrass transform of hadronic spectral functions. There is an interesting analogy between these sum rules and the theory of the heat equation which provides a useful framework for formulating quantitatively the notion of local duality. In particular, finite energy sum rules are shown to follow within this framework from the principle of conservation of total heat. Explicit calculations of these sum rules in QCD - from both perturbative and non-perturbative contributions à la Shifman, Vainshtein and Zakharov [1] - have been made in a rather general way which should cover all the practical cases involving light quark systems. A particular case of the -spectral function and its duality to QCD is discussed in detail as an illustration of this new approach.     

The boson-fermion hybrid representation formulation,I

A boson-fermion hybrid representation is presented. In this framework, a fermion system is described concurrently by the bosonic and the fermionic degrees of freedom. A fermion pair in this representation can be treated as a boson without violating the Pauli principle. Furthermore the “bosonic interactions” are shown to originate from the exchange processes of the fermions and can be calculated from the original fermion interactions. Both the formulation of the BFH representations for the even and odd nuclear systems are given. We find that the basic equation of the nuclear field theory (NFT) is just the usual Schrödinger equation in such a representation with the empirical NFT diagrammatic rules emerging naturally. This theory was numerically checked in the case of four nucleons moving in a single-j shell and the exactness of the theory was established.     

Chemical reactions and fluctuations

Lattice systems with one species diffusion-reaction processes under local complete exclusion rules are studied analytically. We discuss a rigorously derived Fokker-Planck equation for a so-called pseudo-probability. This probability distribution depends on continuous variables in contrast to the original discrete master equation, and their stochastic dynamics may be interpreted as a substitute process which is completely equivalent to the original lattice dynamics. Especially, averages and correlation functions of the continuous variables are connected to corresponding lattice quantities by simple relations. Although the substitute process for diffusion-reaction systems with exclusion rules has some similarities to the well known substitute process for the same system without exclusion rules, their exist a set of remarkable differences. The given approach is not only valid for the discussed single species processes. We give sufficient arguments that arbitrary combinations of uni-molecular and bimolecular lattice reactions under complete local exclusions may be described in terms of our approach.     

非完整转动相对论系统的Lindel?f方程

由转动相对论系统的Hamilton原理分别建立在广义坐标和准坐标下的Lindel?f方程及其改进形式,并从改进的Lindel?f方程导出新Chaplygin方程.最后说明由转动系统的相对论分析力学向普通分析力学过渡的方法. 关键词： 非完整约束 转动系统 相对论 Lindel?f方程 Chaplygin方程     

Birkhoff系统的时间积分定理

研究Birkhoff系统的时间积分定理.建立Birkhoff系统的时间积分等式，由此等式导出系统的类功率方程，Pfaff-Birkhoff积分变分原理以及Pfaff-Birkhoff-d′Alembert微分变分原理. 关键词： Birkhoff系统 时间积分等式 类功率方程 变分原理     

A Fractional Schrödinger Equation and Its Solution

This paper presents a fractional Schrödinger equation and its solution. The fractional Schrödinger equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives to obtain the fractional Euler-Lagrange equations of motion. We present the Lagrangian for the fractional Schrödinger equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Schrödinger equation which is the same as that obtained using the fractional variational principle. As an example, we consider the eigensolutions of a particle in an infinite potential well. The solutions are obtained in terms of the sines of the Mittag-Leffler function.     

基于变分方法的混沌系统参数估计

提出一种基于变分原理的估计混沌系统未知参数的方法,对以x= F(x,θ) 为控制方程的所有混沌系统具有普适性.首先将混沌系统方程引入到目标泛函中;接着利用变分原理导出了混沌系统的伴随方程和待辨识参数泛函梯度的通用公式;然后设计了估计混沌系统未知参数的算法;最后对典型的Lorenz混沌系统和超混沌Chen系统的未知参数进行了估计.数值仿真结果表明该方法是一种非常有效的估计混沌系统未知参数的方法. 关键词： 混沌系统 参数估计 变分方法 伴随方程     

Multi-symplectic Birkhoffian structure for PDEs with dissipation terms

A generalization of the multi-symplectic form for Hamiltonian systems to self-adjoint systems with dissipation terms is studied. These systems can be expressed as multi-symplectic Birkhoffian equations, which leads to a natural definition of Birkhoffian multi-symplectic structure. The concept of Birkhoffian multi-symplectic integrators for Birkhoffian PDEs is investigated. The Birkhoffian multi-symplectic structure is constructed by the continuous variational principle, and the Birkhoffian multi-symplectic integrator by the discrete variational principle. As an example, two Birkhoffian multi-symplectic integrators for the equation describing a linear damped string are given.     

The dynamic natures of microscopic particles described by nonlinear Schrödinger equation

There are a lot of difficulties and troubles in quantum mechanics, when the linear Schrödinger equation is used to describe microscopic particles. Thus, we here replace it by a nonlinear Schrödinger equation to investigate the properties and rule of microscopic particles. In such a case we find that the motion of microscopic particle satisfies classical rule and obeys the Hamiltonian principle, Lagrangian and Hamilton equations. We verify further the correctness of these conclusions by the results of nonlinear Schrödinger equation under actions of different externally applied potential. From these studies, we see clearly that rules and features of motion of microscopic particle described by nonlinear Schrödinger equation are greatly different from those in the linear Schrödinger equation, they have many classical properties, which are consistent with concept of corpuscles. Thus, we should use the nonlinear Schrödinger equation to describe microscopic particles.     

Coupled mode theory for irregular acoustic waveguides with loss

The coupled mode theory of Shevchenko for irregular acoustic waveguides is extended to account for loss. An energy dissipation principle for truncated systems of the coupled mode equations is established. Two-point boundary value problems for these equations are solved by means of a differential Riccati equation. Solutions of a related algebraic Riccati equation are used to approximate solutions of the differential Riccati equation and to formulate radiation boundary conditions.     

The bogolyubov variational principle in the theory of an anharmonic crystal

The Bogolyubov variational principle for classical systems is used to obtain the variation equation for a self-consistent field. With the help of this equation, various approximations of the self-consistent field method in the theory of a crystal with a multiparticle interaction are studied. The free energy of the crystal is calculated.     

Toward a Finite-Dimensional Formulation of Quantum Field Theory

Rules of quantization and equations of motion for a finite-dimensional formulation of quantum field theory are proposed which fulfill the following properties: (a) Both the rules of quantization and the equations of motion are covariant; (b) the equations of evolution are second order in derivatives and first order in derivatives of the spacetime coordinates; and (c) these rules of quantization and equations of motion lead to the usual (canonical) rules of quantization and the (Schrödinger) equation of motion of quantum mechanics in the particular case of mechanical systems. We also comment briefly on further steps to fully develop a satisfactory quantum field theory and the difficuties which may be encountered when doing so.     

On the possibility of a new electric effect in ultrathin superfluid films

Two integrable cases of two-dimensional Schrödinger equation with a magnetic field are proposed. Using the polar coordinates and the symmetrical gauge, we will obtain solutions of these equations through biconfluent and confluent Heun functions. The quantization rules will be derived for both systems under consideration.     

MODELLING OF VIBRATING SYSTEMS USING TIME-DOMAIN FINITE ELEMENT METHOD

In this paper, several dynamic systems are modelled using the time-domain finite element method. Galerkin's weak principle is used to model the general second order mechanical system, and is applied to the dynamics of the simple pendulum. Problems that arise during the approximation of the final momentum are also investigated. Furthermore, additional dynamic analysis methods are suggested for hybrid co-ordinate systems that have both slew and flexible modes. The proposed methods are based on both extended Hamilton's principle and Galerkin's weak principle. The matrix wave equation is propagated in a space domain, satisfying the geometric/natural boundary conditions. As a result, the flexible motion can be obtained, and this was compatible with the applied control input. A numerical example is used to demonstrate the effectiveness of the proposed modelling methods for the hybrid co-ordinate systems.     

Non-extensive approach to quark matter

We review the idea of generating non-extensive stationary distributions based on aaabstract composition rules of the subsystem energies, in particular the parton cascade method, using a Boltzmann equation with relativistic kinematics and modified two-body energy composition rules. The thermodynamical behavior of such model systems is investigated. As an application hadronic spectra with power law tails are analyzed in the framework of a quark coalescence model. This paper is part of the Topical Issue Statistical Power Law Tails in High-Energy Phenomena.     

Dynamics of quantum correlations

The density matrix describing the state of a sybsystem of a physical system whose time dependence is assumed to follow a Schrödinger equation does not itself obey a von Neumann equation. The behavior of the eigenvalues and eigenfunctions of this density matrix is studied. An expression for the rate at which initially separating systems are de-separated is derived in perturbation theory. Indications are given that coherent photon states are more stable in the presence of charged particles than photon number eigenstates. The possible dynamical origin of super selection rules is discussed. A simple model is solved analytically.     

相对论性Birkhoff系统的Lie对称性和守恒量

给出相对论性Birkhoff系统的Pfaff-Birkhoff原理和Birkhoff方程.由微分方程在无限小变换下的不变性,定义相对论性Birkhoff系统无限小变换生成元,建立Lie对称的确定方程,得到结构方程和守恒量.并研究该系统的Lie对称性逆问题.给出实例以说明结果的应用. 关键词：