Branching exclusion process on a strip

We consider a model of stochastically interacting particles on an infinite strip of ²; in this model, known as a branching exclusion process, particles jump to each empty neighboring site with rate /4 and also can create a new particle with rate 1/4 at each one of these sites. The initial configuration is assumed to have a rightmost particle and we study the process as seen from the rightmost vertical line occupied. We prove that this process has exactly one invariant measure with the property thatH, the number of empty sites to the left of the rightmost particle, has an exponential moment. This refines a result presented by Bramson {eaet al.}, who proved that ford=1,H is finite with probability 1.     

Asymptotic and Numerical Analyses for Mechanical Models of Heat Baths

A mechanical model of a particle immersed in a heat bath is studied, in which a distinguished particle interacts via linear springs with a collection of n particles with variable masses and random initial conditions; the jth particle oscillates with frequency j ^p , where p is a parameter. For p>1/2 the sequence of random processes that describe the trajectory of the distinguished particle tends almost surely, as n, to the solution of an integro-differential equation with a random driving term; the mean convergence rate is 1/n ^p–1/2. We further investigate whether the motion of the distinguished particle can be well approximated by an integration scheme—the symplectic Euler scheme—when the product of time step h and highest frequency n ^p is of order 1, that is, when high frequencies are underresolved. For 1/2<p<1 the numerical solution is found to converge to the exact solution at a reduced rate of |log h| h ^2–1/p . These results shed light on existing numerical data.     

Die Anlagerung von radioaktiven Atomen und Ionen an Aerosolteilchen

The attachment of radioactive atoms and ions to spherical aerosol particles has been studied theoretically. For uncharged atoms the deposition is considered to be solely governed by thermal diffusion. With calculations based on the “limiting-sphere”-method ofArendt-Kallman it is found, that the attached activity is proportional toΦ ² for aerosol particle diametersΦ smaller than 0.1 μm, and proportional toΦ forΦ greater than 1 μm. For charged ions the diffusion process is modified by the influence of electrostatic forces between the diffusing ions and the aerosol particles. In the frequently occurring case of a symmetrically bipolar charged aerosol this influence can be expressed by a functionG _p(Φ), which depends on the diameterΦ and the effective numberp of elementary charges on the aerosol particle. For an aerosol particle diameterΦ greater than 0.1 μm the attached activity is proportional toΦ ^1.1, and forΦ smaller than 0.01 μm it is proportional toΦ ^1.55. The effects of neglecting various terms in the calculation are considered. The distribution of natural radioactivity on atmospheric aerosols has been calculated for various particle size distributions according toJunge. The calculation shows that about 90% of the total natural activity should be attached to particles smaller thanΦ=0.5 μm, and about 35% to particles smaller thanΦ=0.1 μm. The time T_1/2, in which the concentration of the radioactive particles decreases to half the initial value, depends on the concentration of the aerosol particles and on their size distribution. For 10⁴ aerosol particles per cm³ and the size distributions mentioned,T _1/2 varies between 15 and 30 seconds for radioactive ions. For radioactive atomsT _1/2 is greater than it is for ions in the range of aerosol particle diameters belowΦ=0.25 μm, and smaller ifΦ greater than 0.25 μm.     

Quantum space-time and gravitational consequences

Relativistic particle dynamics and basic physical quantities for the general theory of gravity are reconstructed from a quantum space-time point of view. An additional force caused by quantum space-time appears in the equation of particle motion, giving rise to a reformulation of the equivalence principle up to values ofO(L ²), whereL is the fundamental length. It turns out that quantum space-time leads to quantization of gravity, i.e., the metric tensorg _v () becomes operator-valued and is not commutative at different pointsx andy in usual space-time on a large scale, and its commutator depending on the vielbein field (gaugelike graviton field) is proportional toL ² multiplied by a translation-invariant wave function propagated between pointsx andy . In the given scheme, there appears to be an antigravitational effect in the motion of a particle in the gravitational force. This effect depends on the value of particle mass; when a particle is heavy its free-fall time is long compared to that for a light-weight particle. The problem of the change of time scale and the anisotropy of inertia are discussed. From experimental data from testing of the latter effect it follows thatL10^–22cm.     

Über die Amputierung der Vertexfunktionen instabiler Teilchen

The normalized form of the amputated vertex functions of an unstable particle is derived from the requirement that the general Ansatz (1) for the scattering amplitude of two stable particles with a resonance in the energy-variable is unitary in the energy interval below the threshold for three particle production. A Klein-Gordon equation for the unstable particle associated with the resonance is derived. The results a re applied to the creation of ⁺ and ⁰ mesons in inelastic scattering processes.     

Scalar particle with polarizability in a uniform external magnetic field

The model of a scalar structured particle is considered, which possesses polarizability in an external electromagnetic field. The expression for the 4-dimensional current density is found. The exact solution of the equations describing a scalar particle with polarizability in a uniform external magnetic field is obtained. Up to the terms of order O(H²), the energy spectrum can be formally obtained by the substitution of the particle mass in the expression for a pointlike scalar particle: mm–H²/2, where is the magnetic polarizability of the particle. It is shown that the rms radius of a trajectory can be obtained by the substitution of the charge in the well-known formula for a structureless scalar particle: ee(1{-H²/m)^1/2 (where is the electric polarizability).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 91–94, January, 1991.I thank A. I. L'vov for discussions.     

On one new effect of the Einsteinian theory of gravitation

It is shown by using the equation of geodesic deviation that a small test particle, placed in the centre of inertia of a terrestrial spherical satellite, will vibrate if a small initial momentum dⁱ/dsbe given to this particle at a certain moment of the satellite's proper time S=0. These vibrations may be in the satellite's orbital plane and the plane perpendicular to it with the frequencies determined by the formulae (14) and (16). The difference between the periods of these vibrations determined by the formula (20) is one new effect of the Einsteinian theory of gravitation (GTR). The shifting of the point of intersection on the lines of these vibrations corresponds to this difference of periods (20). This shifting, taking place in the time of several complete periods, will be of a distance 10^–6–10^–7 cm if plausible assumptions about the quantities of the test particle velocities and amplitudes are made.     

Low-temperature and long-time anomalies of a damped quantum particle

The time evolution of a damped quantum particle is discussed. Dissipation is modeled by the bilinear coupling to a set of harmonic oscillators. Using a functional integral technique that accounts for initial correlations between the particle and the reservoir, one can express the dynamics of the damped particle entirely in terms of equilibrium correlation functions. The long-time behavior of these correlations is determined for memory damping arising from the coupling to a reservoir with spectral densityI() at low frequencies, where > 0. The time evolution of nonequilibrium initial states of the damped particle is discussed. At finite temperatures an initially localized state is found to spread subdiffusively or superdiffusively, depending on . For > 2 the damping becomes ineffective for long times, and the width of a state grows kinematically. At zero temperature and for < 1, an initially localized state remains localized for all times. For 1 the state spreads, but with a slower rate than at finite temperatures. Study of arbitrary initial states indicates that the process is ergodic at finite temperatures only for 2 and at zero temperature for 1 2.     

The reversible measures for symmetric nearest-particle systems

Symmetric nearest-particle systems are certain spin systems on {0, 1}^z in which the flip rate is a function of the distances to the nearest particle of different type to the left and right. The process differs from the ordinary nearest-particle system in that the rates are preserved if zeros and ones are interchanged. The only reversible measure for the symmetric nearest-particle system is a renewaltype measure (the natural analog to the nonsymmetric case). Also as in the nonsymmetric case, reversibility only occurs when the rates are of a specific form. By imposing additional conditions on the rates it can be shown that the reversible measure is the only translation-invariant, invariant measure which concentrates on configurations having infinitely many zeros and ones to either side of the origin. This can be used to prove that for a large class of translation-invariant initial distributions, weak limits are reversible measures. Then we can conclude that the process is convergent for several examples of initial distributions.     

Probability Backflow for a Dirac Particle

The phenomenon of probability backflow, previously quantified for a free nonrelativistic particle, is considered for a free particle obeying Dirac's equation. It is shown that probability backflow can occur in the opposite direction to the momentum; that is to say, there exist positive-energy states in which the particle certainly has a positive momentum in a given direction, but for which the component of the probability flux vector in that direction is negative. It is shown that the maximum possible amount of probability that can flow backwards, over a given time interval of duration T, depends on the dimensionless parameter = (4/mc²T)^1/2, where m is the mass of the particle and c is the speed of light. At = 0, the nonrelativistic value of approximately 0.039 for this maximum is recovered. Numerical studies suggest that the maximum decreases monotonically as increases from 0, and show that it depends on the size of m, , and T, unlike the nonrelativistic case.     

Möbius transformation and conformal relativity

The Möbius transformation (MT) was analyzed as a coordinate transformation in the Minkowski form. The transformation function contains three separate light cones. The Weyl spheres were interpreted as basic constituents of local light cones. These cones are related to the denominators of the MT and its inverse, and their apexes define an axis with the top of the global light cone as the centerpoint. That axis represents the local part of the world-line of a moving frame of reference. On the world-line, the scale factor of the MT is proportional to the ratio of the radii of the initial Weyl sphere and the equivalent transformed one. The projection centers, defining the transformation of the world-line, were determined graphically. There are two types of such MT's. Inner transformations have their projetion centers on the axes of the frame at rest, outer transformations on the axes of the observed moving frame. The signature of x ₀ ² –r² is conserved during inner transformations. All possible directions of the world-line of an inertial frame form a timelike mass cone around the time axis of the frame at rest. The mass cone and the related spacelike phase cone may be seen as projections of a lightlike central motion on the surface of a Weyl sphere. Conformal transformations leave both the mass cone and the phase cone invariant: the MT locally, and the Lorentz transformation globally. The moving frame of any lightlike particle rotates by /2 radians, thus exchanging the time axis with one of the space axes. The measurable mass of such a particle is considered to be zero because only the central plane of the particle is space-oriented. The real mass is invariant. To test this hypothesis, the two -quanta produced by the electron pair annihilation should be led to a recollision, for recreating the initial electron-position pair.     

Classical motion in two-dimensional crystals

The classical motion of an electron of high enough energy in a two-dimensional crystal is diffusive for many potentials with Coulomb singularities. A simple model of the dynamics is developed which predicts the dependence of the diffusion constantD on the particle energyE in the high-energy limit:D(E)const·E ^3/2. This diffusion law is checked for a concrete crystal by numerically integrating the Hamilton equations for an ensemble of initial conditions. Finally this method is compared with other models of the classical dynamics in a crystal, especially the Sinai billiard.     

Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum

We consider a system ofN hard spheres in the Boltzmann-Grad limit (i.e.d0,N,Nd ^2–1>0, whered is the diameter of the spheres). If is sufficiently large, and if the joint distribution densities factorize at time zero, with the one particle distribution decaying sufficiently rapidly in space and velocities, we prove that the time evolved one-particle distribution converges for all times to the solution of the Boltzmann equation with the same initial datum. This result improves and is based on a previous paper [1], valid only in two dimensions.Partially supported by MPI and GNFM (CNR)     

Quasiclassical trajectory-coherent states of an anharmonic oscillator

Based on the complex WKB—Maslov method, asymptotic solutions accurate within O(h²) are constructed for a class of wave packets — quasiclassical trajectory-coherent states for a quantum anharmonic oscillator with a potential V(x)=x²+x⁴. We obtain the behavior of the packet width and the deviations of the approximate solution as a function of time. The actual parameter of the quasiclassical expansion is shown to be h^1/2. We find the regions of variation of the potential parameters and of the initial data for which is small, while the breakdown time of the given accuracy and the packet spreading time are comparable to the time of a single particle revolution.Moscow Institute of Electric Machine Construction. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 21–27, September, 1993.     

Geometroelectrodynamics

We present an elementary particle model that can be thought of as a unification of certain topological ideas abstracted from the string model and the standard Yang-Mills theory. The basic dynamical entity of the model is a spacelike 3-surfaceX ³ in some metric spaceH and is interpreted as a particle. The dynamics of the model is based on two ideas. First the model is formally a Yang-Mills theory on the surfaceX ⁴ representing the orbit(s) of the particle(s) inH. Secondly the Yang-Mills structure onX ⁴ is constructed using only the natural geometric structures of the space H by a process which we call induction. It is found that some rather general requirements highly fix the choice of the space H. In fact the minimal model, for which the space H is the product of Minkowski space and the 2-sphere, is obtained by requiring that the symmetry group of the theory is the product of the Poincaré group and the color groupSO(3). The unique feature of the minimal model is that it affords a purely topological mechanism for quark confinement.     

The Crossover Regime for the Weakly Asymmetric Simple Exclusion Process

We consider the asymmetric simple exclusion process in one dimension with weak asymmetry (WASEP) and 0–1 step initial condition. Our interest are the fluctuations of the time-integrated particle current at some prescribed spatial location. One expects a crossover from Gaussian to Tracy-Widom distributed fluctuations. The appropriate crossover scale is an asymmetry of order \(\sqrt{\varepsilon}\), times of order ε ^?2, and a spatial location of order ε ^?3/2. For this parameter window we obtain the limiting distribution function of the integrated current in terms of an integral over the difference of two Fredholm determinants. For large times, on the scale ε ^?2, this distribution function converges to the one of Tracy-Widom.     

A semiclassical interpretation of wave mechanics

The single-particle wave function =Re^iS/h has been interpreted classically: At a given point the particle momentum is S, and the relative particle density in an ensemble is R ² . It is first proposed to modify this interpretation by assuming that physical variables undergo rapid fluctuations, so that S is the average of the momentum over a short time interval. However, it appears that this is not enough. It seems necessary to assume that the density also fluctuates. The fluctuations are taken to be random and to satisfy conditions required for agreement with quantum mechanics. In some cases the fluctuating density may take on instantaneous negative values. One gets agreement with quantum mechanics for the spin correlations of two particles in a singlet state. This comes about because of the correlations between the fluctuations of the variables of the two particles, the effect of which is equivalent to an action at a distance. The relation to Bell's inequality is discussed.     

Physical facets of ultrasonic cavitational synthesis of zinc ferrite particles

This paper addresses the physical features of the ultrasonic cavitational synthesis of zinc ferrite particles and tries to establish the relationship between cavitation physics and sonochemistry of the zinc ferrite synthesis. A dual approach of coupling experimental results with simulations of radial motion of cavitation bubbles has been adopted. The precursors for the zinc ferrite, viz. ZnO and Fe₃O₄ are produced in situ by the hydrolysis of Zn and Fe(II) acetates stimulated by OH radicals produced from the transient collapse of the cavitation bubbles. Experiments performed under different conditions create significant variation in the production of OH radicals, and hence, the rate of acetate hydrolysis. Correlation of the results of experiments and simulations sheds light on the important facets of the physical mechanism of ultrasonic cavitational zinc ferrite synthesis. It is revealed that too much or too little rate of acetate hydrolysis results in smaller particle size of zinc ferrite. The first effect of a higher rate of hydrolysis leads to excessively large growth of particles, due to which they become susceptible to the disruptive action of cavitation bubbles. Whereas, the second effect of too small rate of hydrolysis of Zn and Fe(II) acetates restricts the growth of particles. It has been observed that the initial reactant concentration does not influence the mean particle size or the size distribution of zinc ferrite particles. The present investigation clearly confirms that the rate-controlling step of zinc ferrite synthesis through ultrasonic cavitational route is the rate of formation of OH radicals from cavitation bubbles.     

A Very Smooth Ride in a Rough Sea

It has been known for some time that a 3D incompressible Euler flow that has initially a barely smooth velocity field nonetheless has Lagrangian fluid particle trajectories that are analytic in time for at least a finite time Serfati (C. R. Acad. Sci. Paris Série I 320:175–180, 1995), Shnirelman (Glob. Stoch. Anal., http://arxiv.org/abs/1205.5837v1, 2012). Here an elementary derivation is given, based on Cauchy’s form of the Euler equations in Lagrangian coordinates. This form implies simple recurrence relations among the time-Taylor coefficients of the Lagrangian map, used here to derive bounds for the C ^1,γ Hölder norms of the coefficients and infer temporal analyticity of Lagrangian trajectories when the initial velocity is C ^1,γ.     

Exact cosmological solution with particle creation in JBD theory

Exact solutions are sought by taking the generated particles of spin 1/2 (according to the creation rate of Schäfer and Dehnen [1]) as matter sources of the Cosmological equations of JBD theory. There exists one exact solution for which the gravitational constant decreases linearly with time and the mass of the universe increases proportionally to the square of its age (Dirac's hypotheses). The radius of curvature increases linearly with time while the density decreases inversely with it. It is found that for an age of the universe 10^–22 sec only two particles have populated the universe. This is assumed to be the initial state of the model. The calculated present particle number and their density are in agreement with the observed data. This model implies that all present matter (excluding the two initial particles) has been created by the expansion of the universe.Supported in part by CONACYT grant No. 11358.