Simultaneous Brownian motion of N particles in a temperature gradient

A system of N Brownian particles suspended in a nonuniform heat bath is treated as a thermodynamic system with internal degrees of freedom, in this case their velocities and coordinates. Applying the scheme of nonequilibrium thermodynamics, one then easily obtains the Fokker-Planck equation for simultaneous Brownian motion of N particles in a temperature gradient. This equation accounts for couplings in the motion as a result of hydrodynamic interactions between particles.     

Probability and complex quantum trajectories

It is shown that in the complex trajectory representation of quantum mechanics, the Born’s ΨΨ probability density can be obtained from the imaginary part of the velocity field of particles on the real axis. Extending this probability axiom to the complex plane, we first attempt to find a probability density by solving an appropriate conservation equation. The characteristic curves of this conservation equation are found to be the same as the complex paths of particles in the new representation. The boundary condition in this case is that the extended probability density should agree with the quantum probability rule along the real line. For the simple, time-independent, one-dimensional problems worked out here, we find that a conserved probability density can be derived from the velocity field of particles, except in regions where the trajectories were previously suspected to be nonviable. An alternative method to find this probability density in terms of a trajectory integral, which is easier to implement on a computer and useful for single particle solutions, is also presented. Most importantly, we show, by using the complex extension of Schrodinger equation, that the desired conservation equation can be derived from this definition of probability density.     

On quantum electrodynamics of two-particle bound states containing spinless particles

We develop here the general treatment arising from the Bethe-Salpeter equation for a two-particle bound system in which at least one of the particles is spinless. It is shown that a natural two-component formalism can be formulated for describing the propagators of scalar particles. This leads to a formulation of the Bethe-Salpeter equation in a form very reminiscent of the fermion-fermion case. It is also shown, that using this two-component formulation for spinless particles, the perturbation theory can be systematically developed in a manner similar to that of fermions. Quantum electrodynamics for scalar particles is then developed in the two component formalism, and the problem of bound states, in which one of the constituent particles is spinless, is examined by means of the means of the Bethe-Salpeter equation. For this case, the Bethe-Salpeter equation is cast into a form which is convenient to perform a Foldy-Woutyhuysen transformation which we carry out, keeping the lowest-order relativistic corrections to the nonrelativistic equation. The results are compared with the corresponding fermion-fermion case. It is shown, as might have been expected, that the only spin-independent terms that occur for the fermion-fermion system which do not occur for bound scalar particle cases, is the zitterbewegung contribution. The relevance of the above considerations for systems that are essentially bound by electromagnetic interactions, such as kaonic hydrogen, is discussed.     

A discussion of the relativistic equal-time equation

Ruan Tu-nan et al. [1] have proposed an equal-time equation for composite particles which is derived from Bethe-Salpeter (B-S) equation. Its advantage is that the kernel of this equation is a completely definite single rearrangement of the B-S irreducible kernel without any artificial assumptions. In this paper we shall give a further discussion of the properties of this equation. We discuss the behaviour of this equation as the mass of one of the two particles approaches the limitM ₂→∞ in the ladder approximation of single photon exchange. We show that up to orderO(α⁴) this equation is consistent with the Dirac equation. If the crossed two photon exchange diagrams are taken into account the difference between them is of orderO(α⁶).     

Certain general questions of fast-electron transport II. Kinetic equations and conditions governing the accelerated motion of fast electrons in dielectrics in an electric field

A general kinetic equation for the differential density of fast particles moving in a medium in an external field is derived on the basis of the continuity equation in phase space. An equation is written for the differential flux in the case of fixed target particles. This equation is used to derive equations for fast electrons; account is taken of the coupling of energy-loss and scattering events in an electric field for various particular problems analogous to those studied in the theory of electron transport in the absence of a field. The kinetic equations are used to analyze the conditions governing accelerated motion of electrons in a dielectric in an external electric field in the continuous-deceleration approximation. Account is taken of fluctuations in the energy loss and of multiple scattering. There are two energy ranges of particles moving in a dielectric in which accelerated motion can occur; in the case of an electron beam with a continuous energy spectrum, this acceleration would be accompanied by monochromatization of the beam.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 7–12, February, 1972.     

The influence of temperature gradients on the kinetic boundary layer problem for a condensing droplet

We extend an earlier method for solving kinetic boundary layer problems to the case of particles moving in aspatially inhomogeneous background. The method is developed for a gas mixture containing a supersaturated vapor and a light carrier gas from which a small droplet condenses. The release of heat of condensation causes a temperature difference between droplet and gas in the quasistationary state; the kinetic equation describing the vapor is the stationary Klein-Kramers equation for Brownian particles diffusing in a temperature gradient. By means of an expansion in Burnett functions, this equation is transformed into a set of coupled algebrodifferential equations. By numerical integration we construct fundamental solutions of this equation that are subsequently combined linearly to fulfill appropriate mesoscopic boundary conditions for particles leaving the droplet surface. In view of the intrinsic numerical instability of the system of equations, a novel procedure is developed to remove the admixture of fast growing solutions to the solutions of interest. The procedure is tested for a few model problems and then applied to a slightly simplified condensation problem with parameters corresponding to the condensation of mercury in a background of neon. The effects of thermal gradients and thermodiffusion on the growth rate of the droplet are small (of the order of 1%), but well outside of the margin of error of the method.     

Geometric field theory and weak Euler–Lagrange equation for classical relativistic particle-field systems

A manifestly covariant, or geometric, field theory of relativistic classical particle-field systems is developed. The connection between the space-time symmetry and energy-momentum conservation laws of the system is established geometrically without splitting the space and time coordinates; i.e., space-time is treated as one entity without choosing a coordinate system. To achieve this goal, we need to overcome two difficulties. The first difficulty arises from the fact that the particles and the field reside on different manifolds. As a result, the geometric Lagrangian density of the system is a function of the 4-potential of the electromagnetic fields and also a functional of the particles’ world lines. The other difficulty associated with the geometric setting results from the mass-shell constraint. The standard Euler–Lagrange (EL) equation for a particle is generalized into the geometric EL equation when the mass-shell constraint is imposed. For the particle-field system, the geometric EL equation is further generalized into a weak geometric EL equation for particles. With the EL equation for the field and the geometric weak EL equation for particles, the symmetries and conservation laws can be established geometrically. A geometric expression for the particle energy-momentum tensor is derived for the first time, which recovers the non-geometric form in the literature for a chosen coordinate system.     

概率波和非概率波

对于把克莱因-戈尔登方程当作是玻色子的方程的看法提出异议,认为它是所有微观粒子均要满足的方程,但它却不能成为任何一类粒子的波动方程.提出了克-戈方程中包含着概率和非概率两类波的概念,认为概率波还要遵从一个对时和空都是一阶导数的方程,这才是粒子的波动方程.不同种类粒子性质的不同则体现在他们概率波类型的不同上.     

Path of Momentum Integral in the Skorniakov-Ter-Martirosian Equation

The Skorniakov-Ter-Martirosian (STM) integral equation is widely used for the quantum three-body problems of low-energy particles (e.g., ultracold atom gases). With this equation these three-body problems can be efficiently solved in the momentum space. In this approach the boundary condition for the case that all the three particles are gathered together is described by the upper limit of the momentum integral, i.e., the momentum cutoff. On the other hand, in realistic systems, the three-body recombination (TBR) process can occur when all these three particles are close to each other. In this process two particles form a deep dimer and the other particle can gain high kinetic energy and then escape from the low-energy system. In the presence of the TBR process, the momentum-cutoff in the STM equation would include a non-zero imaginary part. As a result, the momentum integral in the STM equation should be done in the complex-momentum plane. In this case the result of the integral depends on the choice of the integral path. Obviously, only one integral path can lead to the correct result. In this paper we consider how to correctly choose the integral path for the STM equation. We take the atom-dimer scattering problem in a specific ultracold atom gas as an example, and show the results given by different integral paths. Based on the result for this case we explore the reasonable integral paths for general case.     

A fluctuation problem in particle transport theory II

A balance equation is formulated for the probability that a particle injected into an infinite, amorphous medium will have suffered N collisions and have given rise to n new particles in a given energy range at time t. The method of regeneration points has been employed and this leads, in the case of two particle production, to a non-linear, integro-differential equation for the probability generating function. This equation is solved for the case of foreign particles slowing down, in which case it becomes linear and results are obtained which include the effects of electronic stopping and absorption, thus generalizing the work in part I. In the cascade problem, a single particle gives rise to two new particles in every collision and it is shown, for a simple hard-sphere model with 1/v scattering and absorption, how the non-linear equation may be solved. The probability for the number of particles and the number of collisions suffered to absorption is obtained in the case of zero absorption, the probability law is shown to obey a Furry distribution. The limitations of the method described in part I for dealing with cascades are highlighted.     

A classical theory of superparamagnetic relaxation

A Fokker-Planck equation for the relaxation of a classical ferromagnetic particle coupled to a classical heat bath is derived from the Nakajima-Zwanzig equation. The equation of motion for the mean magnetization of an ensemble of particles is found to be closed only under special circumstances. In the strong motional narrowing limit the equation of motion reduces to the Bloch equations in the limit MH ? k_BT, i.e. for small particles, and to the Landau-Lifshitz equation in the opposite limit. For the motional narrowing region in toto the particular case of uniaxial anisotropy is analysed, giving an equation of motion which for large particles reduces to a modified Landau-Lifshitz equation with g-shift and a reduced damping constant. This equation cannot be meaningfully identified with the Gilbert equation.Approximate expressions for superparamagnetic relaxation rates by Kramers' method are obtained for the case of (i) triaxial (i.e. orthorhombic) and (ii) cubic (K +ve and ?ve) anisotropy, assuming large energy barriers. The results supplement Brown's expression for uniaxial anisotropy and show a more complicated dependence on the Landau-Lifshitz parameter λ than the linear dependence found by Brown. For small λ the rates tend to constant values compatible with the transition.     

Uncertainty features of microscopic particles described by nonlinear Schrüdinger equation

The uncertainty relationship between position and momentum of the microscopic particles is calculated by nonlinear quantum theory in which the states of the particles are described by a nonlinear Schrüdinger equation. The results show that the uncertainty relation differs from that in the quantum mechanics and has a minimum value in this case. This means that the position and momentum of the particles could be determined simultaneously to a certain degree, which could be caused by the wave–corpuscle duality of the microscopic particles described by the nonlinear Schrüdinger equation.     

Kinetic equation for high-energy low-mass particles interacting with a crystal

From a previously obtained kinetic equation describing the motion of an attenuated gas of fast particles in a crystal, an equation for low-mass particles is obtained, taking into account the recoil of particles of the crystal in the interaction; in particular, an equation is obtained for the case of the potential of absolutely solid spheres of sufficiently small radius.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 7–12, October, 1978.     

A fluctuation problem in particle transport theory I

The fluctuation in the number of collision suffered by particles as they slow down in a moderating medium is studied via a probability balance equation. The equation describes the collision history of foreign particles slowing down in a host medium and also accounts for the recoil particles produced in the collision. The equations are solved by the introduction of a generating function from which the space and time dependent probability distributions are obtained. That is, the probability that a particle will suffer just N collisions to reach energy E at a time t after injection. In the space dependent case it is the probability that a particle suffers just N collisions to travel a given path length before coming to rest.Explicit expressions for the means and variances are obtained by solving a difference equation. From this solution it has been possible to obtain exact expressions for hard spheres and for a variety of models based on the inverse power law approximation. A number of new results are presented and some old ones rederived in a more efficient and general manner. The results are of value in the understanding of radiation damage cascades and in neutron slowing down in moderating materials.     

Consequences of coarse-grained Vlasov equations

The Vlasov equation is analyzed for coarse-grained distributions resembling a finite width of test particles as used in numerical implementations. It is shown that this coarse-grained distribution obeys a kinetic equation similar to the Vlasov equation, but with additional terms. These terms give rise to entropy production indicating dissipative features due to a nonlinear mode coupling. The interchange of coarse graining and dynamical evolution is discussed with the help of an exactly solvable model for the self-consistent Vlasov equation and practical consequences are worked out. By calculating analytically the stationary solution of a general Vlasov equation we can show that a sum of modified Boltzmann-like distributions is approached dependent on the initial distribution. This behavior is independent of degeneracy and only controlled by the width of test particles. The condition for approaching a stationary solution is derived and it is found that the coarse graining energy given by the momentum width of test particles should be smaller than a quarter of the kinetic energy. Observable consequences of this coarse graining are: (i) spatial correlations in observables, (ii) too large radii of clusters or nuclei in self-consistent Thomas–Fermi treatments, (iii) a structure term in the response function resembling vertex correction correlations or internal structure effects and (iv) a modified centroid energy and higher damping width of collective modes.     

Relativistic remnants in the reduction of the Bethe-Salpeter equation to the Schrödinger equation

Following Salpeter, the Bethe-Salpeter equation for the bound system of two oppositely charged particles is reduced to a Schrödinger equation for each of the following cases: (a) both particles are spin 1/2 particles, (b) one particle is a spinor while the other is spinless, and (c) both particles are spinless. It is shown that ife is the magnitude of charge carried by each of the particles whose masses are set equal to the electron and proton masses then, strictly speaking, only in case (a) do we obtain the familiar Schrödinger equation for the hydrogen atom. The latter equation is recovered in the other two cases only if relativistic remnants—terms of the order of 10^?5 and smaller—are neglected in comparison with unity. Attention is drawn to a situation where such remnants may not be negligibly small, viz. the problem of confinement of quarks.     

Aharonov–Bohm effect for bound states on spin-0 massive charged particles in a Gödel-type space–time with Coulomb potential

In this paper, we investigate the relativistic quantum dynamics of spin-0 massive charged particles in a Gödel-type space–time with electromagnetic interactions. We derive the radial wave equation of the Klein–Gordon equation with an internal magnetic flux field and Coulomb-type potential in the Som–Raychaudhuri space–time with cosmic string. We solve this equation and analyze the analog effect in relation to the Aharonov–Bohm effect for bound states.     

Exact quantum mechanical description of the motion of spin-1/2 particles and of spin motion in a uniform magnetic field

The Dirac-Pauli equation is used to obtain the exact equation of spin motion for spin-1/2 particles with an anomalous magnetic moment in a constant and uniform magnetic field. Exact formulas are established for the angular velocity of the revolution of such particles along circular orbits and the rotation of the particle spin with respect to momentum. Finally, a quantum mechanical equation for the motion of the particles in a strong magnetic field is derived. Zh. éksp. Teor. Fiz. 114, 448–457 (August 1998)     

From the Liouville Equation to the Generalized Boltzmann Equation for Magnetotransport in the 2D Lorentz Model

We consider a system of non-interacting charged particles moving in two dimensions among fixed hard scatterers, and acted upon by a perpendicular magnetic field. Recollisions between charged particles and scatterers are unavoidable in this case. We derive from the Liouville equation for this system a generalized Boltzmann equation with infinitely long memory, but which still is analytically solvable. This kinetic equation has been earlier written down from intuitive arguments.     

The moment method for boundary layer problems in Brownian motion theory

We apply Grad's moment method, with Hermite moments and Marshak-type boundary conditions, to several boundary layer problems for the Klein-Kramers equation, the kinetic equation for noninteracting Brownian particles, and study its convergence properties as the number of moments is increased. The errors in various quantities of physical interest decrease asymptotically as inverse powers of this number; the exponent is roughly three times as large as in an earlier variational method, based on an expansion in the exact boundary layer eigenfunctions. For the case of a fully absorbing wall (the Milne problem) we obtain full agreement with the recent exact solution of Marshall and Watson; the relevant slip coefficient, the Milne length, is reproduced with an accuracy better than 10^–6. We also consider partially absorbing walls, with specular or diffuse reflection of nonabsorbed particles. In the latter case we allow for a temperature difference between the wall and the medium in which the particles move. There is noa priori reason why our method should work only for Brownian dynamics; one may hope to extend it to a broad class of linear transport equations. As a first test, we looked at the Milne problem for the BGK equation. In spite of the completely different analytic structure of the boundary layer eigenfunctions, the agreement with the exact solution is almost as good as for the Klein-Kramers equation.