Coherence of two-level systems and relaxation

The relaxation of a two-level system interacting with radiation is studied using the method of coherent states. A solution of the equation for the quasiprobability density function is presented in two extreme cases of high and low photon field temperatures. In the first case, the equation has the form of a purely diffusion equation, and from the point of view of a spin interpretation, defines Brownian motion of a magnetic moment on a spherical surface. The second case leads to an equation of the general Fokker-Planck type, in which both diffusion and a systematic term are present. The results of an investigation of spontaneous emission of a system containing a large number of particles, using the quasiclassical approximation, are presented. In this approximation, a Gaussian distribution with a variable variance, whose center moves along a classical trajectory, is obtained for the quasiprobability distribution.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 77–80, October, 1981.     

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.     

Diffusion in inhomogeneous media

The problem is to establish the correct diffusion equation in a medium that is inhomogeneous and whose temperature also varies in space. As a special model we study particles whose phase space distribution obeys Kramers' equation with a generalized collision operator. In the usual limit of strong collisions a diffusion equation is obtained. This equation contains additional drift terms, which depend on the form of the collision operator. They cannot be expressed as a mobility and a diffusion coefficient, unless the decay law of the velocity happens to be linear. Conclusion: no universal form of the diffusion equation exists, but each system has to be studied individually.Dedicated to Professor Harry Thomas on the occasion of his 60th birthday     

The impact approximation in the theory of bimolecular quasi-resonant processes

The kinetic equation for a density matrix, which describes the relaxation of the internal states of encountering particles dissolved in an inert medium, has been derived under the following assumptions: a) the random motion of reacting particles in a liquid is considered to be a classical Markoffian process; b) the concentration of reacting particles is small enough. The equation obtained is shown to be the generalization of that of the familiar impact theory of pressure broadening for the case of any type of encountering particle motion. Our general formulae are concretized in accordance with the physical situations of rectilinear, diffusion, and stochastic jump motion of the encountering particles.     

Scaling limit for interacting Ornstein-Uhlenbeck processes

The problem of describing the bulk behavior of an interacting system consisting of a large number of particles comes up in different contexts. See for example [1] for a recent exposition. In [4] one of the authors considered the case of interacting diffusions on a circle and proved that the density of particles evolves according to a nonlinear diffusion equation. The interacting particles evolved according to a generator that was symmetric in equilibrium. In this article we consider interacting Ornstein-Uhlenbeck processes. Here the diffusion generator is not symmetric relative to the equilibrium and the earlier methods have to be modified considerably. We use some ideas that were employed in [3] to extend the central limit theorem from the symmetric to nonsymmetric cases.This research is supported in part by the National Science Foundation, grant nos. DMS 89-01682 and DMS-88-06727     

Brownian diffusion in a nonuniform gas

An analysis is made of the effects on the diffusion of Brownian particles whose Knudsen number is large compared to unity, of nonuniformities in the host gas. As examples, in one type of nonuniformity of the host gas, the Chapman-Enskog velocity distribution function for the gas molecules is used; in the other, the host gas is a free-molecule Couette flow. In both cases, a new force on the Brownian particles appears. Two techniques are used (extending Kramers' method and utilizing the Chapman-Enskog method) to transform the new Fokker-Planck equation into generalized Smoluchowski and convective diffusion equations. In these equations, the diffusion coefficient appears as a second-order tensor. Thus, it is demonstrated that Brownian diffusion in a nonuniform gas is anisotropic.The work of Slinn was financially supported in part by Battelle Memorial Institute and in part by U.S. Atomic Energy Commission Contract AT(45-1)-1830. The work of Shen was supported in part by U.S. Air Force Office of Scientific Research Contract 49(638)-1346.     

An Accurate Multi-level Finite Difference Scheme for 1D Diffusion Equations Derived from the Lattice Boltzmann Method

An accurate and unconditionally stable explicit finite difference scheme for 1D diffusion equations is derived from the lattice Boltzmann method with rest particles. The system of the lattice Boltzmann equations for the distribution of the number of the fictitious particles is rewritten as a four-level explicit finite difference equation for the concentration of the diffused matter with two parameters. The consistency analysis of the four-level scheme shows that the two parameters which appear in the scheme, the relaxation parameter and the amount of rest particles, can be determined such that the scheme has the truncation error of fourth order. Numerical experiments demonstrate the fourth-order rate of convergence for various combinations of model parameters.     

Studies on aluminum powder combustion in detonation environment

The combustion mechanism of aluminum particles in a detonation environment characterized by high temperature (in unit 10³ K), high pressure (in unit GPa), and high-speed motion (in units km/s) was studied, and a combustion model of the aluminum particles in detonation environment was established. Based on this model, a combustion control equation for aluminum particles in detonation environment was obtained. It can be seen from the control equation that the burning time of aluminum particle is mainly affected by the particle size, system temperature, and diffusion coefficient. The calculation result shows that a higher system temperature, larger diffusion coefficient, and smaller particle size lead to a faster burn rate and shorter burning time for aluminum particles. After considering the particle size distribution characteristics of aluminum powder, the application of the combustion control equation was extended from single aluminum particles to nonuniform aluminum powder, and the calculated time corresponding to the peak burn rate of aluminum powder was in good agreement with the experimental electrical conductivity results. This equation can quantitatively describe the combustion behavior of aluminum powder in different detonation environments and provides technical means for quantitative calculation of the aluminum powder combustion process in detonation environment.     

Diffusion of tagged interacting spherically symmetric polymers

We consider the diffusion of two species of spherically symmetric macromolecules in solution under the influence of short range central pair potential interactions as well as two body hydrodynamic interactions. Starting from the N-particle Smoluchowski equation and using Felderhof's approach we derive, to linear order in densities, a pair of coupled diffusion equations for the single particle number densities. There are two independent diffusional modes each with an effective diffusion constant dependent in general upon both the interparticle potentials as well as the hydrodynamic model used for each type of macromolecule. However, in the limit that one species is present at very low density compared with the other species, one of the effective diffusion constants is dominated by hydrodynamic interactions. By tagging these tracer particles to observe their diffusion by light scattering, one can test both the mixed stick-slip boundary condition model and the permeable sphere model of the macromolecules.     

Hydrodynamics in two dimensions and vortex theory

We consider the Navier-Stokes equation for a viscous and incompressible fluid inR ². We show that such an equation may be interpreted as a mean field equation (Vlasov-like limit) for a system of particles, called vortices, interacting via a logarithmic potential, on which, in addition, a stochastic perturbation is acting. More precisely we prove that the solutions of the Navier-Stokes equation may be approximated, in a suitable way, by finite dimensional diffusion processes with the diffusion constant related to the viscosity. As a particular case, when the diffusion constant is zero, the finite dimensional theory reduces to the usual deterministic vortex theory, and the limiting equation reduces to the Euler equation.Partially supported by Italian CNR     

Large energy behavior of the velocity distribution for the hard-sphere gas

The initial-value problem for the Boltzmann-Lorentz equation for hard spheres at zero temperature is shown to be ill defined, the general solution depending on an arbitrary function. The uniqueness of the solution can be obtained by imposing the conservation of the number of particles (Carleman's type of condition does not suffice). The linearized Boltzmann equation for hard spheres is then analyzed, as it occurs in Enskog's method for calculating transport coefficients. It is demonstrated that in the case of viscosity and diffusion it is necessary to add supplementary conditions to obtain the uniqueness of the solution. The nonuniform character of Enskog's expansion and violation of positivity in the large velocity region are exhibited.     

Asymptotic of Grazing Collisions and Particle Approximation for the Kac Equation without Cutoff

The subject of this article is the Kac equation without cutoff. We first show that in the asymptotic of grazing collisions, the Kac equation can be approximated by a Fokker-Planck equation. The convergence is uniform in time and we give an explicit rate of convergence. Next, we replace the small collisions by a small diffusion term in order to approximate the solution of the Kac equation and study the resulting error. We finally build a system of stochastic particles undergoing collisions and diffusion, that we can easily simulate, which approximates the solution of the Kac equation without cutoff. We give some estimates on the rate of convergence.     

Decay and diffusion in hierarchically organized systems

The interplay of decay and diffusion in a hierarchical system is investigated studying the properties of the underlying master equation. The long time behaviour of the total number of particles in the system is analyzed and the regions in the parameter plane where the decay is diffusion-limited are determined. The influence of the decay on the diffusional spreading is also investigated.On leave from: Sektion Physik der Humboldt-Universität, Invalidenstrasse 42, DDR-1040, Berlin, German Democratic Republic     

对利用动态光散射法测量颗粒粒径和液体黏度的改进

光散射技术通过测量悬浮液中布朗运动颗粒的平移扩散系数,得到颗粒流体力学直径或液体黏度.本文由单参数模型入手,建立了低颗粒浓度下,单颗粒平移扩散系数与颗粒集体平移扩散系数和颗粒浓度之间的线性依存关系并将其引入光散射法中,从而对现有的测量方法进行了改进.改进后的测量方法可实现纳米尺度球型颗粒标称直径的测量和液体黏度的绝对法测量.以聚苯乙烯颗粒+水和二氧化硅颗粒+乙醇两个分散系为参考样本,通过实验,验证了改进后方法的可行性.此外,还针对上述两个分散系,实验探讨了温度和颗粒浓度对颗粒集体平移扩散系数的影响规律,发现聚苯乙烯颗粒+水分散系中,颗粒间相互作用表现为引力;二氧化硅颗粒+乙醇分散系中,颗粒间相互作用表现为斥力.讨论了颗粒集体平移扩散系数随颗粒浓度变化规律与第二渗透维里系数的关系.     

A gas of Brownian particles in statistical dynamics

We consider a large number of particles on a one-dimensional latticel Z in interaction with a heat particle; the latter is located on the bond linking the position of the particle to the point to which it jumps. The energy of a single particle is given by a potentialV(x), x∈Z. In the continuum limit, the classical version leads to Brownian motion with drift. A quantum version leads to a local drift velocity which is independent of the applied force. Both these models obey Einstein's relation between drift, diffusion, and applied force. The system obeys the first and second laws of thermodynamics, with the time evolution given by a pair of coupled non linear heat equations, one for the density of the Brownian particles and one for the heat occupation number; the equation for a tagged Brownian particle can be written as a stochastic differential equation.     

Kinetic Models for Topological Nearest-Neighbor Interactions

We consider systems of agents interacting through topological interactions. These have been shown to play an important part in animal and human behavior. Precisely, the system consists of a finite number of particles characterized by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of its closest neighbor, the leader. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit kinetic equation is a non-standard spatial diffusion equation for the particle distribution function. We also study the case wherein the particles interact with their K closest neighbors and show that the corresponding kinetic equation is the same. Finally, we prove that these models can be seen as a singular limit of the smooth rank-based model previously studied in Blanchet and Degond (J Stat Phys 163:41–60, 2016). The proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments.     

Impurity transport in percolation media

An equation describing the impurity transport in a percolation medium is obtained and the inferences drawn from this equation are analyzed based on the scale invariance concept. A determining part in this analysis is allowance for the sinks inherent in such media. At distances shorter than the correlation length, the particles are transferred in the regime of subdiffusion; at large distances, the concentration asymptotics exhibits a characteristic “tail” shape. In the medium occurring in the state above the percolation threshold, the impurity transport over time periods longer than the characteristic time related to the correlation length is well described by the classical equation with a renormalized diffusion coefficient. In this case, the concentration tail has a Gaussian shape at moderate distances and tends to subdiffusion asymptotics at very long distances. A relation is established between the factor determining renormalization of the diffusion coefficient and the factor determining a decrease in the number of active impurity particles at large times.     

Macroscopic Diffusion from a Hamilton-like Dynamics

We introduce and analyze a model for the transport of particles or energy in extended lattice systems. The dynamics of the model acts on a discrete phase space at discrete times but has nonetheless some of the characteristic properties of Hamiltonian dynamics in a confined phase space: it is deterministic, periodic, reversible and conservative. Randomness enters the model as a way to model ignorance about initial conditions and interactions between the components of the system. The orbits of the particles are non-intersecting random loops. We prove, by a weak law of large number, the validity of a diffusion equation for the macroscopic observables of interest for times that are arbitrary large, but small compared to the minimal recurrence time of the dynamics.