Scaling limits for interacting diffusions

We consider a large number of particles diffusing on a circle interacting through a drift resulting from the gradient of a pair potential whose support is of the order of the interparticle distance. We derive a nonlinear bulk diffusion equation for the density of the particle distribution on the circle. The diffusion coefficient is determined as a function of density in terms of standard thermodynamical objects.This research was supported by a grant from the National Science Foundation DMS-89-01682     

Influence of anisotropy of the diffusion coefficient on the asymptotic behavior of a spatial atom distribution in structures with complex geometry

Obtained by the method of continual integration in the optimal trajectory approximation is the asymptotic behavior of the Green's function of the Fick equation with anisotropic diffusion coefficient in structures with complex geometry when the domain boundary is impermeable to the diffusing particles. Using the expression found for the Green's function, topological properties of the equal-concentration surfaces, particularly the topological singularities of a diffusion p-n-junction, are determined by means of the given initial distribution of the diffusing atoms.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 9–12, May, 1988.     

Evolution of energy density fluctuations in A + A collisions

Two-particle angular correlation for charged particles emitted in Au + Au collisions at the center-of-mass of 200 MeV measured at RHIC energies revealed novel structures commonly referred to as a nearside ridge. The ridge phenomenon in relativistic A + A collisions is rooted probably in the initial conditions of the thermal evolution of the system. In this study we analyze the evolution of the bumping transverse structure of the energy density distribution caused by fluctuations of the initial density distributions that could lead to the ridge structures. We suppose that at very initial stage of collisions the typical one-event structure of the initial energy density profile can be presented as the set of longitudinal tubes, which are boost invariant in some space-rapidity region and are rather thin. These tubes have very high energy density comparing to smooth background density distribution. The transverse velocity and energy density profiles at different times of the evolution till the chemical freeze-out (at the temperature T = 165 MeV) will be reached by the system are calculated for sundry initial scenarios.     

Spin Echo Attenuation of Restricted Diffusion as a Discord of Spin Phase Structure

By using the particle probability density we analyze the spin echo attenuation of particles, diffusing in a bounded region. It provides a means to expand a nonuniform spin phase distribution into a series of waves that characterize the geometry and boundary conditions of confinement. Random motion disrupts the initial phase structure created by applied gradients and consequently discords its structure waves. By assuming the spin phase fluctuation and/or the randomness of spin phase distribution in the subensemble as a Gaussian stochastic process, we derive a new analytical expression for the echo attenuation related to the particle velocity correlation. For a diffusion in porous structure we get the expression featuring the same “diffusive diffraction” patterns as those being found and explained by P. T. Callaghan and A. Coy (“Principles of Nuclear Magnetic Resonance Microscopy,” Oxford Univ. Press, Oxford (1991);J. Chem. Phys.101, 4599–4609 (1994)) with the use of propagator theory. With the new approach we cast a new light on the phenomena and derive analitically how the diffusive diffractions appear when the sequence of finite or even modulated gradients are applied. The method takes into account the non-Markovian character of restricted diffusion, and therefore the echo dependence on the diffusion lengths and on the strength of applied gradient differs from the results of authors assuming the Markovian diffusion either by dealing with the diffusion propagators or by the computer simulation of Fick's diffusion.     

Bulk Viscosity and Decaying Vacuum Density in Friedmann Universe

We present an isotropic and homogeneous flat cosmological model for bulk viscous fluid distribution. We consider the vacuum density proportional to Hubble expansion parameter and time dependent bulk viscosity related to the velocity and acceleration of universe. The behaviour of resulting solutions are in accordance with recent astronomical observations. The model obtained evolves with a decelerating expansion followed by late time acceleration. Cosmological term Λ being very large at initial epoch relaxes to a genuine cosmological constant asymptotically. Presence of bulk viscosity prevents the matter density to vanish asymptotically and the matter density continues to be of the order of vacuum density after a finite time. Thus, we obtain a universe having the possibility of cosmic coincidence.     

Scaling limit for a mechanical system of interacting particles

A system of a large number of classical particles moving on a onedimensional segment with virtually reflecting boundaries is studied. The particles interact with one another through repulsive pair-potential forces and are subjected to resistance proportional to their velocities. Because of the latter it is only the number of particles that is conserved under the evolution of the system. It is proved that in the hydrodynamic limit of diffusion type scaling the normalized counting measure of particle locations converges and its limiting density is governed by a non-linear diffusion equation which in typical cases is of porous media equation type.     

Chaotic diffusion for particles moving in a time dependent potential well

The chaotic diffusion for particles moving in a time dependent potential well is described by using two different procedures: (i) via direct evolution of the mapping describing the dynamics and; (ii) by the solution of the diffusion equation. The dynamic of the diffusing particles is made by the use of a two dimensional, nonlinear area preserving map for the variables energy and time. The phase space of the system is mixed containing both chaos, periodic regions and invariant spanning curves limiting the diffusion of the chaotic particles. The chaotic evolution for an ensemble of particles is treated as random particles motion and hence described by the diffusion equation. The boundary conditions impose that the particles can not cross the invariant spanning curves, serving as upper boundary for the diffusion, nor the lowest energy domain that is the energy the particles escape from the time moving potential well. The diffusion coefficient is determined via the equation of the mapping while the analytical solution of the diffusion equation gives the probability to find a given particle with a certain energy at a specific time. The momenta of the probability describe qualitatively the behavior of the average energy obtained by numerical simulation, which is investigated either as a function of the time as well as some of the control parameters of the problem.     

Analysis of laser-generated ultrasonic force source at specimen surface and display of bulk wave in transversely isotropic plate by numerical method

Taking into account the effects of thermal diffusion and optical penetration, as well as the finite width and duration of the laser source, the laser-generated ultrasonic force source at surface vicinity is presented. The full acoustic fields of laser-generated ultrasonic bulk wave are obtained and displayed in transversely isotropic plate. The features of laser-generated ultrasound bulk waves are analyzed. The features of laser-generated ultrasonic bulk wave are in good agreement with the theoretical results (the phase velocity surfaces), demonstrating the validity of this simulation. The numerical results indicate that the features of laser-generated ultrasound waveforms in anisotropic specimen, different from the case in isotropic materials, have a close relation with the propagating plane and propagation direction. This method can provide insight to the generation and propagation of laser-generated ultrasonic bulk wave in transversely isotropic material.     

Ballistic annihilation with continuous isotropic initial velocity distribution

Ballistic annihilation with continuous initial velocity distributions is investigated in the framework of the Boltzmann equation. The particle density and the rms velocity decay as c approximately t(-alpha) and velocity approximately t(-beta), with the exponents depending on the initial velocity distribution and the spatial dimension d. For instance, in one dimension for the uniform initial velocity distribution beta = 0.230 472ellipsis. In the opposite extreme d-->infinity, the dynamics is universal and beta-->(1-2(-1/2))d(-1). We also solve the Boltzmann equation for Maxwell particles and very hard particles in arbitrary spatial dimension. These solvable cases provide bounds for the decay exponents of the hard sphere gas.     

Simulation of random mixed packing of different density particles

This paper presents the effects of density difference on the three-dimensional (3D) distribution of random mixed packing. The random mixed packing dynamics of particles of two different densities are simulated. The initial state is homogeneous, but the final packing state is inhomogeneous. The segregation phenomenon (inhomogeneous distribution) is also observed. In the final state, the top layers are composed of mostly light particles. The several layers beneath the top contain more heavy particles than light particles. At the bottom, they also contain more heavy particles than light particles. Furthermore, at both the top and the bottom, particle clustering is observed. The current study also analyses the cause of this inhomogeneity in detail. The main cause of this phenomenon is the velocity difference after collision of these two types of particles induced by the density difference. The present study reveals that even if particles were perfectly mixed, the packing process would lead to the final inhomogeneous mixture. It suggests that special treatment may be required to get the true homogeneous packing.     

First study of manganese diffusion in Cr2O3 polycrystals and thin films by SIMS

Chromia layers are formed on many industrial alloys and act as a protective barrier against the corrosion of the materials by limiting the diffusion of oxygen and cations. Most of these alloys contain manganese as an impurity, and manganese oxides are often found at the outer surface of the oxide films. In order to clarify the oxidation mechanism and to check if chromia acts as a barrier, manganese diffusion in chromia was studied in both polycrystals and oxide films formed by oxidation of Ni–30Cr alloy in the temperature range 700–1100°C at an oxygen pressure of 10^?4?atm. After deposition of Mn on the chromia surface and a diffusing treatment, the manganese penetration profiles were established by secondary ion mass spectrometry. In all cases, the diffusion profiles showed two domains. For the first domain, using the solution of Fick's law for diffusion from a thick film into a semi-infinite medium, bulk diffusion coefficients were determined. For the second domain, the Le Claire model allowed the grain boundary diffusion parameter (αD _gbδ) to be obtained. Manganese diffusion does not vary significantly according to the nature and microstructure of chromia. The activation energy of grain boundary diffusion is not far from that obtained for bulk diffusion, probably on account of segregation phenomena. Manganese diffusion was compared to cationic self-diffusion and iron diffusion, and related to the protective character of chromia.     

Ballistic aggregation for one-sided Brownian initial velocity

We study the one-dimensional ballistic aggregation process in the continuum limit for one-sided Brownian initial velocity (i.e. particles merge when they collide and move freely between collisions, and in the continuum limit the initial velocity on the right side is a Brownian motion that starts from the origin x=0). We consider the cases where the left side is either at rest or empty at t=0. We derive explicit expressions for the velocity distribution and the mean density and current profiles built by this out-of-equilibrium system. We find that on the right side the mean density remains constant whereas the mean current is uniform and grows linearly with time. All quantities show an exponential decay on the far left. We also obtain the properties of the leftmost cluster that travels towards the left. We find that in both cases relevant lengths and masses scale as t² and the evolution is self-similar.     

The role of dispersal in competition success and in the emerging diversity

The dynamics of dispersal-structured populations, consisting of competing individuals that are characterized by different diffusion coefficients but are otherwise identical, is investigated. Competition is taken into account through demographic processes. The problem addressed models natural selection. It is observed that the mean value and the relative width of the initial distribution of the diffusion coefficients characterizing the individuals together with the temporal fluctuations determine the final distribution of the diffusivities (diffusion coefficients leading to the competition success) as well as the final diversity of the system at finite time (the number of different diffusion coefficients present in the system). Large initial mean diffusivity of the system leads to a rather fast disappearance of the diversity. Instead, small initial mean diffusivity of the system leads to a diversity equal to the number of niches forming in the system due to the competitive interactions. The cluster formation is also associated to the competition success of the slower diffusing individuals. The diversity is diminished by the increase of the temporal fluctuations that give the competition advantage to the faster diffusing individuals. Somewhat counterintuitively, under certain conditions the competition success is given by intermediate values of the diffusion coefficients.     

Random walks on the Comb model and its generalizations

Microscopic models with anomalous diffusion, which include the Comb model and its generalization for the finite width of the backbone, have been considered in this paper. The physical mechanisms of the subdiffusion random walks have been established. The first comes from the permanent return of the diffusing particle to the initial point of the diffusion due to "effective reducing" of the dimensionality of the considered system to the quasi-one-dimensional system. This physical mechanism has been obtained in the Comb model and in the model with a strip. The second mechanism of the subdiffusion is connected with random capture on the traps of diffusing particles and their ensuing random release from the traps. It has been shown that these different mechanisms of subdiffusion have been described by the different generalized diffusion equations of fractional order. The solutions of these different equations have been obtained, and the physical sense of the fractional order generalized equations has been discussed.     

Plasma diffusion across a magnetic field

Using a simple model of a slowly diffusing plasma across a strong magnetic field, it is demonstrated that plasma mass and energy evolves from an initially given density and temperature distribution into isothermal state with a self-similar diffusion profile that depends only on its initial mass and energy.     

Incorporation of surface effects in three-dimensional modelling of complex photo-detector arrays by spot-scan simulation

The diffusion limited leakage current densities of individual photodiodes in three-dimensional focal plane arrays are calculated using a random walk simulation of a spot-scan experiment. We have derived a relationship between the surface recombination velocity and the capture probability at a surface, enabling complex geometries with surfaces of arbitrary surface recombination velocity to be modelled. Due to a unique relationship between the leakage current density and the collection efficiency of carriers generated close to a junction, even in non-uniform material, there is no necessity to sample the effects of carrier generation at all points throughout the array, giving great savings in computation time. An example of the method applied to an array on a hexagonal net is given.     

Heating of a sample with a laser pulse

The transient temperature associated with the bulk absorption of a rectangular laser pulse in a solidstate sample of finite size is calculated analytically and analyzed. Radiation is incident on the frontal surface with an arbitrary surface thermal conductivity. The opposite surface is thermostatically controlled and maintained at a constant equilibrium temperature. The general solution is obtained for pulses of arbitrary duration. The pulse duration is determined with respect to the relaxation time of the nonstationary thermal diffusion (it is the characteristic time of the problem). The limiting cases of the adiabatic insulation and isothermal contact at the frontal surface are considered, and the criteria for surface and bulk light absorption are derived for both cases. The temperature distributions are numerically simulated and examined for long and short pulses, as well as for different values of the light absorption coefficient and the surface thermal conductivity of the frontal surface.     

A Mechanical Model of Brownian Motion for One Massive Particle Including Slow Light Particles

We provide a connection between Brownian motion and a classical mechanical system. Precisely, we consider a system of one massive particle interacting with an ideal gas, evolved according to non-random mechanical principles, via interaction potentials, without any assumption requiring that the initial velocities of the environmental particles should be restricted to be “fast enough”. We prove the convergence of the (position, velocity)-process of the massive particle under a certain scaling limit, such that the mass of the environmental particles converges to 0 while the density and the velocities of them go to infinity, and give the precise expression of the limiting process, a diffusion process.