Mass, energy, and the electron

The two-component solutions of the Dirac equation currently in use are not separately a particle equation or an antiparticle equation. We present a unitary transformation that uncouples the four-component, force-free Dirac equation to yield a two-component spinor equation for the force-free motion of a relativistic particle and a corresponding two-component, time-reversed equation for an antiparticle. The particle-antiparticle nature of the two equations is established by applying to the solutions of these two-component equations criteria analogous to those applied for establishing the four-component particle and antiparticle solutions of the four-component Dirac equation. Wave function solutions of our two-component particle equation describe both a right and a left circularly polarized particle. Interesting characteristics of our solutions include spatial distributions that are confined in extent along directions perpendicular to the motion, without the artifice of wave packets, and an intrinsic chirality (handedness) that replaces the usual definition of chirality for particles without mass. Our solutions demonstrate that both the rest mass and the relativistic increase in mass with velocity of the force-free electron are due to an increase in the rate of Zitterbewegung with velocity. We extend this result to a bound electron, in which case the loss of energy due to binding is shown to decrease the rate of Zitterbewegung.     

Analysis of stresses in two-dimensional isostatic granular systems

We analyze a recently proposed continuous model for stress fields that develop in two-dimensional purely isostatic granular systems. We present a reformulation of the field equations, as a linear first-order hyperbolic system, and show that it is very convenient both for analysis and for numerical computations. Our analysis allows us to predict quantitatively the formation and directions of stress paths and, from these, trajectories and magnitudes of force chains, given the structure in terms of a particular fabric tensor. We further predict quantitatively changes of stresses along the paths, as well as leakage and branching of stress from the main paths into the cones that they make in terms of the fabric tensor. Numerical computations in both Cartesian and cylindrical coordinates verify the analytic results and illustrate the rich behavior discovered. All the phenomena predicted by our solutions have been observed experimentally, suggesting that stresses in isostatic systems can form a base model for a more developed stress theory in granular materials.     

Linear relaxation processes governed by fractional symmetric kinetic equations

The fractional symmetric Fokker-Planck and Einstein-Smoluchowski kinetic equations that describe the evolution of systems influenced by stochastic forces distributed with stable probability laws are derived. These equations generalize the known kinetic equations of the Brownian motion theory and involve symmetric fractional derivatives with respect to velocity and space variables. With the help of these equations, the linear relaxation processes in the force-free case and for the linear oscillator is analytically studied. For a weakly damped oscillator, a kinetic equation for the distribution in slow variables is obtained. Linear relaxation processes are also studied numerically by solving the corresponding Langevin equations with the source given by a discrete-time approximation to white Levy noise. Numerical and analytical results agree quantitatively.     

On granular surface flow equations

Conservation equations are written for surface flows (either fluid or granular). The particularity of granular surface flows is then pointed out, namely that the depth of the flowing layer is not a priori fixed, leading to open equations. It is shown how some hypothesis on the flowing layer allows to close the system of equations. A possible hypothesis, similar to that made for a fluid layer, but inspired from granular flow experiments, is presented. The force acting on the flowing layer is discussed. Averaging over the flowing depth, as in shallow water theory, then allows to transform these conservation laws into equations for the evolution of the profile of a granular pile. Apart from their interest for building models, these conservation laws can be used to measure experimentally the effective forces acting on a flowing layer. Received 25 July 1998 and Received in final form 14 January 1999     

A nonlinear granular medium with particle rotation: A one-dimensional model

Equations of nonlinear acoustics are derived from the micromechanical representation of a granular medium as a system of elastically interacting particles possessing translational and rotational degrees of freedom. The structure of the equations is invariant with respect to the shape and size of the particles. The changes in the latter affect only the coefficients in the equations. The inclusion of microrotations and moment interactions of particles leads to the formation of a new type of waves in the medium—microrotational waves. Their dispersion properties are similar to those of spin waves propagating in a magnetoelastic medium. In the low-frequency approximation, the microrotational waves disappear, and the equation describing the transverse waves acquires a term with quadratic nonlinearity. The latter provides an explanation for the generation of the second shear harmonic that is observed in real solids contrary to the predictions of the nonlinear theory of elasticity, which prohibits such phenomena.     

Stresses in isostatic granular systems and emergence of force chains

Progress is reported on several questions that bedevil understanding of granular systems: (i) Are the stress equations elliptic, parabolic, or hyperbolic? (ii) How can the often-observed force chains be predicted from a first-principles continuous theory? (iii) How do we relate insight from isostatic systems to general packings? Explicit equations are derived for the stress components in two dimensions including the dependence on the local structure. The equations are shown to be hyperbolic and their general solutions, as well as the Green function, are found. It is shown that the solutions give rise to force chains, and the explicit dependence of the force chains trajectories and magnitudes on the local geometry is predicted. Direct experimental tests of the predictions are proposed. Finally, a framework is proposed to relate the analysis to nonisostatic and more realistic granular assemblies.     

Couette颗粒系统中静态应力和侧压力系数的非线性弹性理论分析

经典弹性理论能否适用于静态颗粒物质是颗粒物理的一个基本问题.报道了非线性颗粒弹性方程对环柱几何(或称Couette几何)状样品中应力分布的计算结果.由于这些结果大都可在今后用实验直接测量,因而可将它们用于进一步判断这些弹性方程对颗粒材料的适用性以及澄清这个基本问题.另外,这些结果还可用于分析计算工程中的侧压力系数. 关键词： 颗粒物质 侧压力系数 非线性弹性理论     

垂直振动颗粒混合气体的振荡现象研究

运用事件驱动算法研究颗粒混合物在垂直振动容器中的振荡现象.容器被具有一定高度的隔板分成相等大小的两个小室,并采用半径相差一倍的两种颗粒.研究结果表明,颗粒振荡周期随两种颗粒密度比的减少而急剧增加.通过计算在垂直方向上两种颗粒高度之比随颗粒密度的变化关系,说明决定颗粒振荡与否的主要因素并不是"巴西坚果效应"或"反巴西坚果效应".通过计算颗粒温度,发现颗粒振荡取决于颗粒混合气体的小颗粒温度.当小颗粒温度大于一定值时,颗粒混合气体发生颗粒振荡现象.根据Viridi等提出的流体动力学模型,文中对该模型做出相应的修改,加入密度因子,从而可以解释颗粒振荡周期与颗粒密度比的关系.     

Towards a Landau–Ginzburg-Type Theory for Granular Fluids

In this paper we show how, under certain restrictions, the hydrodynamic equations for the freely evolving granular fluid fit within the framework of the time dependent Landau–Ginzburg (LG) models for critical and unstable fluids. The granular fluid, which is usually modeled as a fluid of inelastic hard spheres (IHS), exhibits two instabilities: the spontaneous formation of vortices and of high density clusters. We suppress the clustering instability by imposing constraints on the system sizes, in order to illustrate how LG-equations can be derived for the order parameter, being the rate of deformation or shear rate tensor, which controls the formation of vortex patterns. From the shape of the energy functional we obtain the stationary patterns in the flow field. Quantitative predictions of this theory for the stationary states agree well with molecular dynamics simulations of a fluid of inelastic hard disks.     

Nonlinear dynamics of density waves in granular flows through narrow vertical channels

We study granular flows through narrow channels driven by gravity in the framework of the kinetic theory for dissipative dense gases. We derive equations of motion for quasi-one-dimensional systems. In a certain range of flow density, the steady homogeneous regime is found to be unstable against the formation of density waves. We show moreover that near the onset of the instability, the governing equation for the flow density is a mixture of the Korteweg-de-Vries equation, which leads to soliton, and the Bürger equation which exhibits spatio-temporal chaos. The competition between chaos and solitons may lead either to regular spatially ordered density waves or to chaotic dynamics. We argue that these two types of dynamics can be encountered experimentally according to the channel width and the dissipative properties of the granular media. Received: 11 March 1998 / Revised and Accepted: 3 July 1998     

Granular statistical mechanics – a personal perspective

The science of granular matter has expanded from an activity for specialised engineering applications to a fundamental field in its own right. This has been accompanied by an explosion of research and literature, which cannot be reviewed in one paper. A key to progress in this field is the formulation of a statistical mechanical formalism that could help develop equations of state and constitutive relations. This paper aims at reviewing some milestones in this direction. An essential basic step toward the development of any static and quasi-static theory of granular matter is a systematic and useful method to quantify the grain-scale structure and we start with a review of such a method. We then review and discuss the ongoing attempt to construct a statistical mechanical theory of granular systems. Along the way, we will clarify a number of misconceptions in the field, as well as highlight several outstanding problems.     

利用Grad-Shafranov重构方法研究磁尾通量绳的特征参量和内部结构

2009年2月21日THEMIS-C卫星在磁尾X=-15.7R_E(R_E为地球半径,1R_E=6371 km)观测到典型的磁通量绳事件.采用Grad-Shafranov 重构技术研究该磁通量绳的特性、内部磁场和电流结构.研究表明,磁通量绳不变轴位于GSM(geocentric solar magnetospheric coordinates)坐标为(-0.3975,0.8905,0.221 关键词： 磁尾 磁通量绳 Grad-Shafranov重构 多X线重联     

Low-energy excitations in the three-dimensional random-field Ising model

General conservation equations are derived for 2D dense granular flows from the Euler equation within the Boussinesq approximation. In steady flows, the 2D fields of granular temperature, vorticity and stream function are shown to be encoded in two scalar functions only. We checked such prediction on steady surface flows in a rotating drum simulated through the Non-Smooth Contact Dynamics method even though granular flows are dissipative and therefore not necessarily compatible with Euler equation. Finally, we briefly discuss some possible ways to predict theoretically these two functions using statistical mechanics.     

Boltzmann-Type Equation in Anelastic Case: A Convergence Result

In this Letter, a convergence result for the BBGKY hierarchy to a Boltzmann-like equation, in the case of an Anelastic collision, is shown. Boltzmann-like equations are often used to model dissipative dynamical systems such as granular media. This convergence result aims to make a contribution towards a mathematical foundation to these applications.     

Magneto-vibratory separation of glass and bronze granular mixtures immersed in a paramagnetic liquid

A fluid-immersed granular mixture may spontaneously separate when subjected to vertical vibration, separation occurring when the ratio of particle inertia to fluid drag is sufficiently different between the component species of the mixture. Here, we describe how fluid-driven separation is influenced by magneto-Archimedes buoyancy, the additional buoyancy force experienced by a body immersed in a paramagnetic fluid when a strong inhomogeneous magnetic field is applied. In our experiments glass and bronze mixtures immersed in paramagnetic aqueous solutions of MnCl₂ have been subjected to sinusoidal vertical vibration. In the absence of a magnetic field the separation is similar to that observed when the interstitial fluid is water. However, at modest applied magnetic fields, magneto-Archimedes buoyancy may balance the inertia/fluid-drag separation mechanism, or it may dominate the separation process. We identify the vibratory and magnetic conditions for four granular configurations, each having distinctive granular convection. Abrupt transitions between these states occur at well-defined values of the magnetic and vibrational parameters. In order to gain insight into the dynamics of the separation process we use computer simulations based on solutions of the Navier-Stokes' equations. The simulations reproduce the experimental results revealing the important role of convection and gap formation in the stability of the different states.     

颗粒介质中的粘滞系数

将颗粒介质看成是等效均匀介质, 其中的声衰减系数和声速等于该颗粒介质中的相应的量值(它们可由作者的理论给出), 等效静态密度可以用二元混合规则求得. 此外, 根据浓颗粒介质中相互作用的声传播理论, 当入射波为平面波时, 相互作用的次级波仍然是平面波. 在这样的情况下, 可以将三维非线性方程组简化为一维情况, 从而算得浓颗粒介质中的粘滞系数, 结果表明, 颗粒介质中的粘滞系数不仅依赖于颗粒的体积分数而且还与频率有关. 根据推导过程可知, 对比于爱因斯坦理论所能应用的限制, 本文的结果可以更广泛地应用于实际介质.     

Effect of a force-free end on the mechanical property of a biopolymer——A path integral approach

We study the effect of a force-free end on the mechanical property of a stretched biopolymer.The system can be divided into two parts.The first part consists of the segment counted from the fixed point(i.e.,the origin) to the forced point in the biopolymer,with arclength L_f.The second part consists of the segment counted from the forced point to the force-free end with arclength △L.We apply the path integral technique to find the relationship between these two parts.At finite temperature and without any constraint at the end,we show exactly that if we focus on the quantities related to the first part,then we can ignore the second part completely.Monte Carlo simulation confirms this conclusion.In contrast,the effect for the quantities related to the second part is dependent on what we want to observe.A force-free end has little effect on the relative extension,but it affects seriously the value of the end-to-end distance if △L is comparable to L_f.     

Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems

The general structure of irreducible invariant algebraic curves for a polynomial dynamical system in ${\mathbb{C}}^2$ is found. Necessary conditions for existence of exponential factors related to an invariant algebraic curve are derived. As a consequence, all the cases when the classical force-free Duffing and Duffing–van der Pol oscillators possess Liouvillian first integrals are obtained. New exact solutions for the force-free Duffing–van der Pol system are constructed.