也谈概率密度和概率流密度算符

针对<大学物理>2008年第8期上关于粒子概率密度算符和概率流密度算符的讨论一文进行了补充,对一般空间(如动量空间)中的概率和概率流进行了较系统的阐述,并纠正了该文中的有关错误论断.     

谐振子的概率密度、概率流密度与时间的关系

以一维线性谐振子为例,对非定态情况,通过数值计算给出了不同时间的概率密度和概率流密度分布,并且讨论了概率密度和概率流密度随时间的变化的基本特征.     

希尔伯特空间中的概率流和概率守恒

实空间中的概率流概念和概率守恒定理被推广到希尔伯特(Hilbert)状态空间,从而得到了概率流算符的通式.应用一般形式的概率流算符公式,还导出了紧束缚模型中单电子和互作用多电子系统的概率流算符,并以异常简单的方式证明了反射-透射流归一化条件 关键词： 希尔伯特空间 概率流算符 紧束缚模型     

概率假设密度滤波的物理空间意义

为了深入理解概率假设密度滤波,本文在Ozgur Erdinc对随机集的物理空间假设的基础上,采用Bayes公式和全概率公式对概率假设密度滤波的迭代过程进行了推导.为有效改善概率假设密度滤波的目标漏检问题提供了理论基础.     

带势估计的概率假设密度滤波的物理空间意义

为了便于人们深入理解带势估计的概率假设密度滤波,本文在Ozgur Erdin对随机集做的物理空间假设的基础上,采用Bayes公式和全概率公式对带势估计的概率假设密度滤波的迭代过程进行了推导.推导过程详细明了,推导结果与文献一致.这为带势估计的概率假设密度滤波在目标跟踪中的应用及性能改进提供了理论基础.     

二能级系统态函数概率流新形式

首先讨论了一般量子系统的概率流的表达形式,鉴于对于二能级系统态函数的概率流文献中讨论的较少,本文通过定义一些新的变量,给出了二能级系统态函数概率流的新形式,这一新形式中不含有两个态矢的干涉项.     

几类粒子密度算符和粒子流密度算符的讨论

从粒子密度和粒子流密度出发,分析了几类相关的算符,讨论了它们的性质与相互关系;从具体的物理意义出发,明确了合适的粒子密度算符和粒子流密度算符的表达形式.     

求解任意方向辐射强度的广义多流法

本文提出一种基于有限体积法的精确求解任意方向辐射强度的广义多流法(Multi-Flux Method,MFM).以圆柱形吸收、散射、发射性介质为例,采用MFM模拟其沿任意方向的出射辐射强度,并与反向蒙特卡洛法(Backward MonteCarlo,BMC)的计算结果进行了比较分析.结果表明,MFM法与BMC法的计算结果吻合较好,但MFM法的计算效率明显优于BMC法.因此,MFM模型是一种适用于计算任意方向辐射强度问题的高效数值模型.     

从流密度的Fourier分量出发民出多极矩表达式

从运动电荷产生的流密度入手，通过Ｆｏｕｒｉｅｒ变换引入各级多极矩的表达式。     

从流密度的Fourier分量出发导出多极矩表达式

从运动电荷产生的流密度入手,通过Fourier变换引入各级多极矩的表达式.     

Effect of boundary conditions on the invariant density of noisy maps at fully-developed chaos

The invariant density of one-dimensional maps in the regime of fully-developed chaos with uncorrelated additive noise is considered. Boundary conditions are shown to play a significant role in determining the precise form of the invariant density, via the manner in which they handle the spill-over, caused by the noise, of orbits beyond the interval. The known case of periodic boundary conditions is briefly recapitulated. Analytic solutions for the invariant density that are possible under certain conditions are presented with applications to specific well-known maps. The case of ‘sticky’ boundaries is generalized to ‘re-injection at the nearest boundary’, and the exact functional equations determining the invariant density are derived. Interesting boundary layer effects are shown to occur, that lead to significant modifications of the invariant density corresponding to the unperturbed (noise-free) case, even when the latter is a constant — as illustrated by an application of the formalism to the noisy tent map. All our results are non-perturbative, and hold good for any noise amplitude in the interval.     

Phonon absorbing boundary conditions for molecular dynamics

With the goal of minimizing the domain size for molecular dynamics (MD) simulations, we develop a new class of absorbing boundary conditions (ABCs) that mimic the phonon absorption properties of an unbounded exterior. The proposed MD-ABCs are extensions of perfectly matched discrete layers (PMDLs), originally developed as an absorbing boundary condition for continuous wave propagation problems. Called MD-PMDL, this extension carefully targets the absorption of phonons, the high frequency waves, whose propagation properties are completely different from continuous waves. This paper presents the derivation of MD-PMDL for general lattice systems, followed by explicit application to one-dimensional and two-dimensional square lattice systems. The accuracy of MD-PMDL for phonon absorption is proven by analyzing reflection coefficients, and demonstrated through numerical experiments. Unlike existing MD-ABCs, MD-PMDL is local in both space and time and thus more efficient. Based on their favorable properties, it is concluded that MD-PMDL could provide a more effective alternative to existing MD-ABCs.     

First-principle proof of the modified collision boundary conditions for the hard-sphere system

A fundamental issue lying at the foundation of classical statistical mechanics is the determination of the collision boundary conditions that characterize the dynamical evolution of multi-particle probability density functions (PDF) and are applicable to systems of hard-spheres undergoing multiple elastic collisions. In this paper it is proved that, when the deterministic N-body PDF is included in the class of admissible solutions of the Liouville equation, the customary form of collision boundary conditions adopted in previous literature becomes physically inconsistent and must actually be replaced by suitably modified collision boundary conditions.     

Bosonic string theories with new boundary conditions

We show that the classical Nambu-Goto string inD dimensions admits Poincaré invariance ind dimensions (d⩽D) if (i)d − 2 of the transverse co-ordinatesx ⁱ are periodic and the rest quasi-periodic involving a real orthogonal matrix with (D − d) (D − d − 1)/2 free parameters, or if (ii)d − 2 ofx ⁱ obey Neumann and the rest obey a boundary condition involvingN free parameters, whereN=(D − d)²/2 ifD − d is even, andN=[(D − d)² − 1]/2 ifD − d is odd.     

Advanced boundary conditions for eigenmode expansion models

In order to realise the full potential of eigenmode expansion models, advanced boundary conditions are required that can absorb the radiation impinging on the walls of the discretisation volume. In this paper, we will discuss and compare a number of these boundary conditions, like perfectly matched layers (PMLs), open (leaky mode) boundary conditions and transparent boundary conditions (TBCs). We will also introduce the case of PMLs with infinite absorption and discuss its relation to leaky mode expansion, leading to a deeper insight into the physics of PML.     

Advanced boundary conditions for eigenmode expansion models

In order to realise the full potential of eigenmode expansion models, advanced boundary conditions are required that can absorb the radiation impinging on the walls of the discretisation volume. In this paper, we will discuss and compare a number of these boundary conditions, like perfectly matched layers (PMLs), open (leaky mode) boundary conditions and transparent boundary conditions (TBCs). We will also introduce the case of PMLs with infinite absorption and discuss its relation to leaky mode expansion, leading to a deeper insight into the physics of PML.     

Open and traction boundary conditions for the incompressible Navier–Stokes equations

We present numerical schemes for the incompressible Navier–Stokes equations (NSE) with open and traction boundary conditions. We use pressure Poisson equation (PPE) formulation and propose new boundary conditions for the pressure on the open or traction boundaries. After replacing the divergence free constraint by this pressure Poisson equation, we obtain an unconstrained NSE. For Stokes equation with open boundary condition on a simple domain, we prove unconditional stability of a first order semi-implicit scheme where the pressure is treated explicitly and hence is decoupled from the computation of velocity. Using either boundary condition, the schemes for the full NSE that treat both convection and pressure terms explicitly work well with various spatial discretizations including spectral collocation and C⁰ finite elements. Moreover, when Reynolds number is of O(1) and when the first order semi-implicit time stepping is used, time step size of O(1) is allowed in benchmark computations for the full NSE. Besides standard stability and accuracy check, various numerical results including flow over a backward facing step, flow past a cylinder and flow in a bifurcated tube are reported. Numerically we have observed that using PPE formulation enables us to use the velocity/pressure pairs that do not satisfy the standard inf–sup compatibility condition. Our results extend that of Johnston and Liu [H. Johnston, J.-G. Liu, Accurate, stable and efficient Navier–Stokes solvers based on explicit treatment of the pressure term. J. Comp. Phys. 199 (1) (2004) 221–259] which deals with no-slip boundary conditions only.