A Fleming–Viot Particle Representation¶of the Dirichlet Laplacian

We consider a model with a large number N of particles which move according to independent Brownian motions. A particle which leaves a domain D is killed; at the same time, a different particle splits into two particles. For large N, the particle distribution density converges to the normalized heat equation solution in D with Dirichlet boundary conditions. The stationary distributions converge as N→∞ to the first eigenfunction of the Laplacian in D with the same boundary conditions. Received: 11 November 1999 / Accepted: 19 May 2000     

The kinetic boundary layer for the Fokker-Planck equation with absorbing boundary

The Fokker-Planck equation for the distribution of position and velocity of a Brownian particle is a particularly simple linear transport equation. Its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly. By a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall, and that approach a linearly increasing position space density far from the wall. Various aspects of these approximate solutions are discussed. In particular we find that the extrapolated asymptotic density reaches zero at a distance x_M beyond the wall. We find x_M=1.46 in units of the velocity persistence length of the Brownian particle. This study was motivated by certain problems in the theory of diffusion-controlled reactions, and the results might be used to test approximate theories employed in that field.     

Mean first passage time of active Brownian particle in one dimension

We investigate the mean first passage time of an active Brownian particle in one dimension using numerical simulations. The activity in one dimension is modelled as a two state model; the particle moves with a constant propulsion strength but its orientation switches from one state to other as in a random telegraphic process. We study the influence of a finite resetting rate r on the mean first passage time to a fixed target of a single free active Brownian particle and map this result using an effective diffusion process. As in the case of a passive Brownian particle, we can find an optimal resetting rate r* for an active Brownian particle for which the target is found with the minimum average time. In the case of the presence of an external potential, we find good agreement between the theory and numerical simulations using an effective potential approach.     

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.     

Equilibrium Fluctuations for a Model of Coagulating-Fragmenting Planar Brownian Particles

We study a model of mass-bearing coagulating-fragmenting planar Brownian particles. Coagulation occurs when two particles are within a distance of order ε. Our model is macroscopically described by an inhomogeneous Smoluchowski’s equation in the low ε limit provided that the initial number of particles N is of order |log ε|. When a detailed balance condition is satisfied, we establish a central limit theorem by showing that in the low ε limit, the particle density fluctuation fields obey an Ornstein-Uhlenbeck stochastic differential equation.     

The kinetic boundary layer around an absorbing sphere and the growth of small droplets

Deviations from the classical Smoluchowski expression for the growth rate of a droplet in a supersaturated vapor can be expected when the droplet radius is not large compared to the mean free path of a vapor molecule. The growth rate then depends significantly on the structure of the kinetic boundary layer around a sphere. We consider this kinetic boundary layer for a dilute system of Brownian particles. For this system a large class of boundary layer problems for a planar wall have been solved. We show how the spherical boundary layer can be treated by a perturbation expansion in the reciprocal droplet radius. In each order one has to solve a finite number ofplanar boundary layer problems. The first two corrections to the planar problem are calculated explicitly. For radii down to about two velocity persistence lengths (the analog of the mean free path for a Brownian particle) the successive approximations for the growth rate agree to within a few percent. A reasonable estimate of the growth rate for all radii can be obtained by extrapolating toward the exactly known value at zero radius. Kinetic boundary layer effects increase the time needed for growth from 0 to 10 (or 2 1/2) velocity persistence lengths by roughly 35% (or 175%).     

An invariance principle for reversible Markov processes. Applications to random motions in random environments

We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for ad-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in ad-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in ad-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.     

A new thermal conductivity model for nanofluids

In a quiescent suspension, nanoparticles move randomly and thereby carry relatively large volumes of surrounding liquid with them. This micro-scale interaction may occur between hot and cold regions, resulting in a lower local temperature gradient for a given heat flux compared with the pure liquid case. Thus, as a result of Brownian motion, the effective thermal conductivity, k_eff, which is composed of the particles conventional static part and the Brownian motion part, increases to result in a lower temperature gradient for a given heat flux. To capture these transport phenomena, a new thermal conductivity model for nanofluids has been developed, which takes the effects of particle size, particle volume fraction and temperature dependence as well as properties of base liquid and particle phase into consideration by considering surrounding liquid traveling with randomly moving nanoparticles.The strong dependence of the effective thermal conductivity on temperature and material properties of both particle and carrier fluid was attributed to the long impact range of the interparticle potential, which influences the particle motion. In the new model, the impact of Brownian motion is more effective at higher temperatures, as also observed experimentally. Specifically, the new model was tested with simple thermal conduction cases, and demonstrated that for a given heat flux, the temperature gradient changes significantly due to a variable thermal conductivity which mainly depends on particle volume fraction, particle size, particle material and temperature. To improve the accuracy and versatility of the k_effmodel, more experimental data sets are needed.     

Multiple light scattering from a moving layer of Brownian particles: comparison with a rigid phase screen

Abstract

The temporal fluctuations in the intensity of light scattered by a moving layer of emulsions and suspensions containing Brownian particles are investigated experimentally, and a comparison is made with light scattered by a translating phase screen. The intensity fluctuations of the scattered light are detected through an imaging system, which collects the light emanating only from a limited volume in the medium. The effect of translational motion of the particle layer on the decay rate of the autocorrelation function of intensity fluctuations depends on the illuminating form of a laser beam and on the point spread function of the imaging system. The Brownian motion of the particles causes the scattered light to fluctuate more rapidly than that arising from the translating phase screen. In the multiple-scattering regime, the influence of this diffusional motion increases with an increase of the particle concentration in the layer.     

Fokker–Planck Equation,Molecular Friction,and Molecular Dynamics for Brownian Particle Transport near External Solid Surfaces

The Fokker–Planck (FP) equation describing the dynamics of a single Brownian particle near a fixed external surface is derived using the multiple-time-scales perturbation method, previously used by Cukier and Deutch and Nienhuis in the absence of any external surfaces, and Piasecki et al. for two Brownian spheres in a hard fluid. The FP equation includes an explicit expression for the (time-independent) particle friction tensor in terms of the force autocorrelation function and equilibrium average force on the particle by the surrounding fluid and in the presence of a fixed external surface, such as an adsorbate. The scaling and perturbation analysis given here also shows that the force autocorrelation function must decay rapidly on the zeroth-order time scale ₀, which physically requires N _Kn1, where N _Kn is the Knudsen number (ratio of the length scale for fluid intermolecular interactions to the Brownian particle length scale). This restricts the theory given here to liquid systems where N _Kn1. For a specified particle configuration with respect to the external surface, equilibrium canonical molecular dynamics (MD) calculations are conducted, as shown here, in order to obtain numerical values of the friction tensor from the force autocorrelation expression. Molecular dynamics computations of the friction tensor for a single spherical particle in the absence of a fixed external surface are shown to recover Stokes' law for various types of fluid molecule–particle interaction potentials. Analytical studies of the static force correlation function also demonstrate the remarkable principle of force-time parity whereby the particle friction coefficient is nearly independent of the fluid molecule–particle interaction potential. Molecular dynamics computations of the friction tensor for a single spherical particle near a fixed external spherical surface (adsorbate) demonstrate a breakdown in continuum hydrodynamic results at close particle–surface separation distances on the order of several molecular diameters.     

Lattice Boltzmann simulation of the dispersion of aggregated Brownian particles under shear flows

The deformation and breakup processes of a particle-cluster aggregate under shear flows are investigated by the two-phase lattice Boltzmann method. In the simulation the particle is modeled by a hard droplet with large viscosity and strong surface tension. The van der Waals attraction force is taken into account for the interaction between the particles. Also, the Brownian motion is considered for nano-particles. Two important dimensionless parameters are introduced in order to classify calculated results. One is the ratio of fluid force to the maximum inter-particle force, Y, and the other is the Péclet number which is the ratio of the rate of diffusion by a shear flow to the rate of diffusion by Brownian motion. It is found that Y is the key factor in dispersion and that the Brownian motion retards the dispersion.     

Quantum Brownian motion on potential surfaces coupled via tunneling in an external electric field[-2mm]

The present paper deals with the motion of a Brownian particle on two identical but shifted potential surfaces, coupled via a tunneling matrix element in an external electric field. Dissipation is induced by a heat bath represented by an infinite set of harmonic oscillators with a continuum range of frequencies. We derive a perturbative solution for the quantum coherence term of the particle system after performing a small-polaron-like transformation. This is subsequently necessary for the extraction of an equation that describes the reduced dynamics and the minimal action path of the Brownian particle. Finally we extract expressions for the population relaxation rate and the pure quantum-dephasing rate of the two-level system. Received 4 January 2001 and Received in final form 12 March 2001     

Quenching Evolution in a Quantum-Classical Hybrid System

The adiabatic theorem, an important theory in quantum mechanics, tells that a quantum system subjected to gradually changing external conditions remains to the same instantaneous eigenstate of its Hamiltonian as it initially in. In this paper, we study the quench evolution that is another extreme circumstance where the external conditions vary rapidly such that the quantum system can not follow the change and remains in its initial state (or wavefunction). We examine the matter-wave pressure and derive the requirement for such an evolution. The study is conducted by considering a quantum particle in an infinitely deep potential, the potential width Q is assumed to be change rapidly. We show that the total energy of the quantum subsystem decreases as Q increases, and this rapidly change exerts a force on the wall which plays the role of boundary of the potential. For Q < Q₀ (Q₀ is the initial width of the potential), the force is repulsive, and for Q > Q₀, the force is positive. The condition for the quenching evolution evolution is given via a spin-\( \frac{1}{2} \) in a rotating magnetic field.

     

Effect of the Noise Correlation Time on the Stationary Current of Brownian Particle in a Spatially Symmetric Periodic Potential

We investigate the noise-induced transport of Brownian particle in a deterministic spatial symmetrical periodic potential driven by colored cross correlation between a multiplicative white noise and an additive white noise. We derive the general formula of the stationary current. Based on numerical computation, we found that directed motion of the Brownian particles can be induced by the correlation time τ of cross correlation between the multiplicative noise and the additive noise and the current reversal and the direction of the current is controlled by the τ.     

Reliability of fluctuation-induced transport in a Maxwell-demon-type engine

We study the transport properties of an overdamped Brownian particle which is simultaneously in contact with two thermal baths. The first bath is modeled by an additive thermal noise at temperature T _A . The second bath is associated with a multiplicative thermal noise at temperature T _B . The analytical expressions for the particle velocity and diffusion constant are derived for this system, and the reliability or coherence of transport is analyzed by means of their ratio in terms of a dimensionless Péclet number. We find that the transport is not very coherent, though one can get significantly higher currents.     

Fundamental Solutions of the Axial Symmetric Goursat Problem

Abstract

Fundamental solutions (FS) with a given boundary condition on the characteristics of relativistic problems with axial symmetry are considered. This is so-called the Goursat problem (GP) or zero plane formalism in Dirac’s terminology, or modification of the proper time method in the Fock-Nambu-Schwinger formalism (FNS).

Closed FS for the Volkov problem from the point of view of GP can be found. This means that integration over proper time in a FNS integral transformation can be performed. Using the special chosen dynamic symmetry of the initial state, FS for a particle in constant magnetic or constant electric field may also be calculated.     

Improvement of the pressure algorithm of the stand-alone PDF method to treat unsteady three-dimensional turbulent reacting flows

This paper deals with the particle-mesh probability density function (PDF) method. It shows how an existing but less precise pressure algorithm for the stand-alone method can be improved. The present algorithm is able to handle the general case of an unsteady three-dimensional turbulent reacting flow. The transport equation of the joint PDF of velocity and composition is solved with a particle method. Open boundary conditions are realized and for statistical reasons a simple but effective particle splitting procedure is applied.

Based on a simple configuration, the properties of the presented improved pressure algorithm are analysed. It is shown which numerical condition must be taken care of so that the algorithm is able to correct the particle positions such that the normalization condition is fulfilled as accurately as specified.

To verify the algorithm the combustion of a methane–air mixture enclosed in an open simulation volume is calculated. It is shown that the simple particle splitting algorithm works very effectively in the studied case. The behaviour of the improved pressure algorithm is examined by different calculations. To analyse the convergence of the algorithm, the particle number per cell and the grid spacing are varied. To demonstrate the accuracy, a statistically stationary inflow/outflow configuration is used and the numerical solution is compared to an analytical one. For a less symmetric test case, the previous unsteady combustion problem is simulated, including an additional mean velocity in one direction.

The presented improved pressure algorithm provides the opportunity to calculate unsteady three-dimensional turbulent reacting flows with a stand-alone method, and offers an alternative to the complex hybrid finite-volume/particle PDF method.     

Note on the fractal dimension of hard sphere trajectories

Using Monte Carlo molecular dynamics, a new, careful study is made of the approach of the trajectory of a typical particle in a hard sphere fluid to that of a Brownian particle, discussed before by Powles and Quirke and Rapaport. The apparent fractal dimension of the trajectory, as a function of reduced length scale,(), characterizes the transition from mechanical to Brownian motion and differs markedly from 2 in all present computer simulations.