A simple derivation of the equation of motion for classical electrodynamics

The equation of motion recently obtained by the author is derived by an elementary method. In addition, this paper contains a careful analysis of three well-known derivations of the (incorrect) Lorentz-Dirac equation, identifying their flaws. The fundamental error in each case is a failure to appreciate that the rate of change of field momentum affects the particle differently according to whether it is an applied field or the self-field. This fundamental physical error can be understood with the aid of a simple analogy.     

A stochastic particle system modeling the Carleman equation

Two species of Brownian particles on the unit circle are considered; both have diffusion coefficient >0 but different velocities (drift), 1 for one species and –1 for the other. During the evolution the particles randomly change their velocity: if two particles have the same velocity and are at distance ( being a positive parameter), they both may simultaneously flip their velocity according to a Poisson process of a given intensity. The analogue of the Boltzmann-Grad limit is studied when goes to zero and the total number of particles increases like ^–1. In such a limit propagation of chaos and convergence to a limiting kinetic equation are proven globally in time, under suitable assumptions on the initial state. If, furthermore, depends on and suitably vanishes when goes to zero, then the limiting kinetic equation (for the density of the two species of particles) is the Carleman equation.Dedicated to the memory of Paola Calderoni.     

Self-turbulence in the motion of a free particle

A deterministic equation of the Hamilton-Jacobi type is proposed for a single particle:S _t+(1/2m)(?S)²+U{S}=0, whereU{S} is a certain operator onS, which has the sense of the potential of the self-generated field of a free particle. Examples are given of potentials that imply instability of uniform rectilinear motion of a free particle and yieldrandom fluctuations of its trajectory. Galilei-invariant turbulence-producing potentials can be constructed using a single universal parameter—Planck's constant. Despite the fact that the classical trajectory concept is retained, the mechanics of the particle then admits quantum-type effects: an uncertainty relation, de Broglie-type waves and their interference, discrete energy levels, and zero-point fluctuations.     

A stochastic derivation of the multivariate master equation describing reaction diffusion systems

We derive the multivariate master equation describing reaction diffusion systems from a discrete form master equation in phase space, assuming that the elastic collisions of the chemically active substances with the inert carrier gas have relaxed. In this state of collisional equilibrium the stochastic operator modelling the displacement of the particles between spatial cells reduces to the random wall operator and the reactive collision term yields the usual birth and death operator. Correlation functions are derived and their validity is discussed.     

On the derivation of the incompressible Mavier-Stokes equation for Hamiltonian particle systems

We consider a Hamiltonian paticle system interacting by means of a pair potetial. We look at the behavior of the system on a space scale of order ^-1, times of order ^-2 and mean velocities of order , with a scale parameter. Assuming that the phase space density of the particles is give by a series in (the analog of the Chapman-Enskog expansion), the behavior of the system under this rescaling is described, to the lowest order in , by the incompressible Navier-Stokes equations. The viscosity is given in terms of microscopic correlations, and its expression agrees with the Green-Kubo formula.     

A special class of solutions of the Schrödinger equation for a free particle

The fundamental solution of the Schrödinger equation for a free particle is modified by the inclusion of an arbitrary scalar and an arbitrary vector, both imaginary. This gives a field free from singularities. By choosing the scalar small and the vector large, one obtains a model of a wavepacket which moves fast and remains concentrated over a long range.     

A simple derivation of the stochastic eigenvalue equation in the transition from quasiperiodicity to chaos

A simple derivation of the stochastic eigenvalue equation, previously obtained by irrational decimation of functional integrals, is given to show the universal scaling behavior of external noise in the transition from quasiperiodicity to chaos in dissipative systems.     

General form of the equation of motion of a charged particle

The most general form of the equation of motion of a charged particle is derived. The equation allows for an arbitrary self-field and is consistent with energy-momentum conservation, angular momentum conservation and the Dirac mass renormalization.     

On the derivation of the Schrödinger equation from stochastic mechanics

It is shown that the existing formulations of stochastic mechanics are not equivalent to the Schrödinger equation, as had previously been believed. It is argued that this is a reflection of fundamental inadequacies in the physical foundations of stochastic mechanics.Some relatively minor difficulties with the demonstration of equivalence are already known for the special case in which the nodal surface separates the manifold of the diffusion into disjoint components.^(1,11) The problems described in this paper are much more general and quite unrelated.     

A derivation of the Broadwell equation

We consider a stochastic system of particles in a two dimensional lattice and prove that, under a suitable limit (i.e.N, 0,N²const, whereN is the number of particles and is the mesh of the lattice) the one-particle distribution function converges to a solution of the two-dimensional Broadwell equation for all times for which the solution (of this equation) exists. Propagation of chaos is also proven.Research partially supported by CNR-PS-MMAIT     

Symmetry of equations of motion for a free particle in riemann space

Spaces are classified in which the equations of motion of a free test particle admit a partial separation of variables aided by a set of first integrals of the second order.     

A derivation of the compressibility equation for liquid metals

The Kirkwood and Buff compressibility equation for multicomponent systems is applied to an assembly of ions and electrons, where the Coulomb potential is replaced by the Yukawa potential. It is then shown that the resulting formula gives the Watabe-Hasegawa-Chihara compressibility equation in the limit of infinitesimal damping constant of the Yukawa potential.     

质点沿可自由移动的光滑凹槽的运动分析

分析了质点在可自由移动的凹槽上运动的运动平面与凹槽的交线是一般曲线的情况,给出了质点在静止参考系中运动的轨迹方程和严格等时振动的条件,求出了质点沿几种常见曲线运动的周期.     

A new derivation of the Silin-Severne-Balescu-Misguich equation

Using the projection-like technique for classical systems, a rederivation of the Silin-Severne-Balescu-Misguich equation is presented.I wish here to express my gratitude to Dr. J. uczka for his interest and valuable aid in the elaboration of this paper. The author is also indebted to the referee for remarks which permitted to improve the presentation.     

Relativistic motion of a free particle in a uniform gravitational field

The problem of the motion of a free particle in a uniform gravitational field is considered. A relativistic solution based on the assumption that the motion is a consequence of the curvature of spacetime is obtained. The results are compared with various results based on the assumption that spacetime is flat in a region in which the gravitational field is uniform. In the curved spacetime approach, if a particle is projected from a point in a uniform gravitational field, the vertical distance covered by the particle in infinite coordinate time is infinite, but the horizontal distance covered and the elapsed proper time of the particle are finite. If spacetime is assumed to be flat and the gravitational motion of a particle a consequence of a relativistic force proportional to the relative mass of the particle, then the results obtained for the motion of a particle in a uniform gravitational field are close to the curved spacetime results. All other assumptions, including the assumption that the motion of a particle in a uniform gravitational field is equivalent to the motion of a particle in a uniformly accelerating frame of reference, lead to results in serious disagreement with the curved spacetime results.     

Entangling the free motion of a particle pair: An experimental scenario

The concept of dissociation-time entanglement provides a means of manifesting non-classical correlations in the motional state of two counter-propagating atoms. In this article, we discuss in detail the requirements for a specific experimental implementation, which is based on the Feshbach dissociation of a molecular Bose-Einstein condensate of fermionic lithium. A sequence of two magnetic field pulses serves to delocalize both of the dissociation products into a superposition of consecutive wave packets, which are separated by a macroscopic distance. This allows to address them separately in a switched Mach-Zehnder configuration, permitting to conduct a Bell experiment with simple position measurements. We analyze the expected form of the two-particle wave function in a concrete experimental setup that uses lasers as atom guides. Assuming viable experimental parameters the setup is shown to be capable of violating a Bell inequality.