Relativistic dynamics of stochastic particles

Particle motion in stochastic space, i.e., space whose coordinates consist of small, regular stochastic parts, is considered. A free particle in this space resembles a Brownian particle the motion of which is characterized by a dispersionD dependent on the universal length l. It is shown that in the first approximation in the parameter l the particle motion in an external force field is described by equations coincident in form with equations of stochastic mechanics due to Nelson, Kershow, and de la Pena-Auerbach. A method is proposed for the relativization of the scheme used to describe the processes in the stochastic space; by using this method, the equations of particle motion can be written in a covariant form.     

Relativistic corrections to the Lagrangian for interacting charged particles

We give the Lagrangian of a system of moving charged particles up to the fourth approximation in 1/c neglecting dipole radiation of the system. In this case the appearance of the electromagnetic waves (quadrupole radiation) by moving charges occurs in the fifth approximation in 1/c.     

Relativistic dynamics for spin-zero and spin-half particles

The classical and quantum mechanics of a system of directly interacting relativistic particles is discussed. We first discuss the spin-zero case, where we basically follow Rohrlich in introducing a set of covariant centre of mass (CM) and relative variables. The relation of these to the classic formulation of Bakamjian and Thomas is also discussed. We also discuss the important case of relativistic potentials which may depend on total four-momentum squared. We then consider the quantum mechanical case of spin-half particles. The negative energy difficulty is solved by introducing a number of first class constraints which fix the spinor structure of physical solutions and ensure the existence of proper CM and relative variables. We derive the form of interactions consistent with Lorentz invariance, space inversion, time reversal and charge conjugation and with the above mentioned first class constraints and find that it is analogous to that for the non-relativistic case. Finally the relationship of the present work with some previous work is briefly discussed.     

Relativistic dynamics of cylindrical shells of counter-rotating particles

Although infinite cylinders are not astrophysical entities, it is possible to learn a great deal about the basic qualitative features of generation of gravitational waves and the behavior of the matter conforming such shells in the limits of very small radius. We study an analytical model of a relativistic cylindrical shell of counter-rotating particles using kinetic theory for the matter and the junction conditions through the shell to obtain its equation of motion. The nature of the static solutions are analyzed, both for a single shell as well as for two coaxial shells. In the latter case, we integrate numerically the time dependent equation of motion of the external shell, when we neglect the wave components of the gravitational field at the shells locations. We obtain solutions that correspond to shells that perform damped oscillations, collapse, or are locally expanding. The collapse ends (numerically) when the external shell hits the interior shell. The numerically work also shows that the radiation becomes important after the bounce of the external shell.     

Relativistic kinetic equations for inelastically interacting particles in a gravitational field

The relativistic canonical formalism is used to construct the kinetic equations for a gas in a gravitational field, whose particles interact with one another via numerous inelastic collisions. Boltzmann's H-theorem is proved for T-invariant interactions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 19–23, August, 1983.     

Memory function approach to the dynamics of interacting Brownian particles

We investigate some of the dynamic properties of diffusing particles described by a many-body Smoluchowski equation. The dynamic structure factor is expressed in terms of a memory function which is evaluated in the cases of i) weak interaction and ii) low particle density, but arbitrary interaction. A one-dimensional system with a hard-core pair potential is treated explicitly. Furthermore, by including a periodic single-particle potential, a model is obtained which has relevance to superionic conductors. For this case we discuss how the frequency-dependent conductivity is affected by the correlated motion of particles.     

Tracer dynamics in Dyson's model of interacting Brownian particles

We prove that the mean square displacement of a tracer particle grows as logt for larget. We point out a connection to the low-temperature floating phase of the ANNNI model.     

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.     

Semiclassical dynamics of Dirac particles interacting with a static gravitational field

The semiclassical limit for Dirac particles interacting with a static gravitational field is investigated. A Foldy–Wouthuysen transformation which diagonalizes at the semiclassical order the Dirac equation for an arbitrary static spacetime metric is realized. In this representation the Hamiltonian provides for a coupling between spin and gravity through the torsion of the gravitational field. In the specific case of a symmetric gravitational field we retrieve the Hamiltonian previously found by other authors. But our formalism provides for another effect, namely, the spin hall effect, which was not predicted before in this context.     

Nonlinear dynamics of charged particles interacting with combined ac-dc electromagnetic fields

A nonlinear Lorenz model describing interactions between charged particles and combined ac-dc electromagnetic fields is studied for various combinations of frequencies, field strengths and relative angle (θ) between the ac and dc magnetic fields. Strong directional effects on the magnitude and location of resonant particle motion are observed when θ is varied and the regular resonance windows in the aligned field (θ = 0) and linear version of the model studied previously by Durney etaal., break up to form irregular and less well pronounced regions of large and small particle displacements when nonlinearities are taken into account. The length of time takne to achieve resonant behaviour also becomes larger and more variable when nonlinearities are present. The possible relevance of these effects to interactions between electromagnetic fields and biological media is briefly discussed.     

Transient dynamics of charged particles interacting with localized waves of continuous spectra

A modified canonical perturbation method is employed for analyzing the charged particle dynamics as they interact with localized waves with continuous spectrum. In contrast with periodic Hamiltonian models, where the method has already been applied in a multitude of respective systems, the system in hand is inherently aperiodic. The localized waves have the form of amplitude modulated electrostatic fields, ranging from ordinary wave packets to ultrashort pulses. The analytically obtained approximate invariants of the motion contain rich information for the structure of the phase space and the respective distribution functions.     

A positive energy dynamics and scattering theory for directly interacting relativistic particles

Starting from the tensor product of N irreducible positive energy representations of the Poincaré group describing N free relativistic particles with arbitrary spins and positive masses, we construct an interacting positive energy representation by modifying the total 4-momentum operator. We first make a transformation to a Hilbert space on which the free total 4-momentum operator equals the product of a dimensionless center-of-mass 4-vector ($\text{(|k|}{}^2\text{+ 1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, k) and a free “reduced Hamiltonian” H_r⁰, which is a positive operator acting only on internal variables, and then replace H_r⁰ by an interacting reduced Hamiltonian H_r = H_r⁰ + V, where V commutes with the Lorentz group and is such that H_r is a positive operator. The resulting product form is shown to imply that the wave operators interwine the free and interacting representations so that the S-operator is Lorentz invariant. From a physical point of view the scheme is related to the framework first introduced by Bakamjian and Thomas, in which the Hamiltonian and boost generators are modified, but the above procedure makes a mathematically rigorous discussion much simpler. In the spin-zero case we introduce a natural generalization of the pair potentials of nonrelativistic N-particle Schrödinger theory to the present relativistic setting, study its scattering theory, and point out some problems that do not have analogs at the nonrelativistic level. In the $\text{spin}\text{-}\text{1}\text{2}$ case we propose, inspired by the Dirac equation, explicit reduced Hamiltonians to describe atomic energy levels and present arguments making plausible that their eigenvalues are in closer agreement with the experimental data than their nonrelativistic counterparts. We also consider extensions to arbitrary spin and, in the $\text{spin}\text{-}\text{1}\text{2}$ case, coupling of a quantized radiation field. In view of eventual applications to “completely integrable” one-dimensional field theories the case of one space dimension is studied as well, both in quantum mechanics and in classical mechanics.     

Relativistic Navier-Stokes equations in the Meixner-Prigogine scheme

Viscous effects are included in the relativistic Meixner-Prigogine scheme (see: A. Sandoval-Villalbazo, L.S. García-Colín, Physica A 234 (1996) 358). A relativistic generalization of the Navier-Stokes equations is obtained within this framework. The system obtained is analyzed and compared with related work.     

弱相互作用费米气体的相对论顺磁性

通过考虑单粒子的相对论能量并且应用量子统计方法,研究弱磁场中弱相互作用费米气体的相对论顺磁性,求解相对论性的最可几磁化率及平均磁化率,讨论相对论效应对最可几磁化率的影响,给出相对论性的粒子数的临界值. 结果显示, 与非相对论情况比较,当 时, 相对论情况下的最可几磁化率和粒子数的临界值都没有变化. 当 时,相对论效应使系统呈现顺磁性容易,而且相对论效应增加磁化率,同时也放大了相互作用对磁化率的影响.     

弱相互作用费米气体的相对论顺磁性

通过考虑单粒子的相对论能量并且应用量子统计方法，研究弱磁场中弱相互作用费米气体的相对论顺磁性，求解相对论性的最可几磁化率及平均磁化率，讨论相对论效应对最可几磁化率的影响，给出相对论性的粒子数的临界值. 结果显示, 与非相对论情况比较，当 时, 相对论情况下的最可几磁化率和粒子数的临界值都没有变化. 当 时，相对论效应使系统呈现顺磁性容易，而且相对论效应增加磁化率，同时也放大了相互作用对磁化率的影响.