The kinetic boundary layer for the Fokker-Planck equation: Selectively absorbing boundaries

We study the joint distribution function for position and velocity of a Brownian particle near a wall. The wall absorbs all particles that hit it with sufficiently high velocity and reflects all slower ones, either specularly or diffusely. We determine in particular stationary distributions in the absence of external forces. Appreciable deviations from local equilibrium occur in a kinetic boundary layer near the wall; its details depend strongly on the way in which the slow particles are reflected. The resulting effective absorption rate is calculated and compared with the result of approximations analogous to the transition state theory of chemical reactions. The method used is a generalization of the one used in an earlier paper for the case of a completely absorbing wall; a numerical algorithm based on an expansion of the distribution function in terms of a presumably complete set of boundary layer solutions.     

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.     

The kinetic boundary layer for the Klein-Kramers equation; A new numerical approach

We explore a numerical technique for determining the structure of the kinetic boundary layer of the Klein-Kramers equation for noninteracting Brownian particles in a fluid near a wall that absorbs the Brownian particles. The equation is of interest in the theory of diffusion-controlled reactions and of the coagulation of colloidal suspensions. By numerical simulation of the Langevin equation equivalent to the Klein-Kramers equation we amass statistics of the velocities at the first return to the wall and of the return times for particles injected into the fluid at the wall with given velocities. The data can be used to construct the solutions of the standard problems at an absorbing wall, the Milne and the albedo problem. We confirm and extend earlier results by Burschka and Titulaer, obtained by a variational method vexed by the slow convergence of the underlying eigenfunction expansion. We briefly discuss some further boundary layer problems that can be attacked by exploiting the results reported here.     

The moment method for boundary layer problems in Brownian motion theory

We apply Grad's moment method, with Hermite moments and Marshak-type boundary conditions, to several boundary layer problems for the Klein-Kramers equation, the kinetic equation for noninteracting Brownian particles, and study its convergence properties as the number of moments is increased. The errors in various quantities of physical interest decrease asymptotically as inverse powers of this number; the exponent is roughly three times as large as in an earlier variational method, based on an expansion in the exact boundary layer eigenfunctions. For the case of a fully absorbing wall (the Milne problem) we obtain full agreement with the recent exact solution of Marshall and Watson; the relevant slip coefficient, the Milne length, is reproduced with an accuracy better than 10^–6. We also consider partially absorbing walls, with specular or diffuse reflection of nonabsorbed particles. In the latter case we allow for a temperature difference between the wall and the medium in which the particles move. There is noa priori reason why our method should work only for Brownian dynamics; one may hope to extend it to a broad class of linear transport equations. As a first test, we looked at the Milne problem for the BGK equation. In spite of the completely different analytic structure of the boundary layer eigenfunctions, the agreement with the exact solution is almost as good as for the Klein-Kramers equation.     

Distribution of mean velocity in turbulent boundary layer over permeable surface with air blowing

In the present article, we investigate the possibility of using simple physical models for predicting properties of incompressible turbulent boundary layer on permeable wall at various values of air-microblowing mass flow rate. It is shown that the velocity scaling U _??*/?? ₉₉ can be successfully used to approximate the distribution of mean velocity in the outer region of the boundary layer. The use of this scaling makes the velocity profiles invariant with respect to Reynolds-number variation; this circumstance largely facilitates the analysis of experimental data, making it independent of upstream flow conditions. The distribution of mean velocity in the logarithmic flow region of the boundary layer over permeable surface can be described with a modified law of the wall involving a constant C ₀ equal to the same constant for canonical boundary layer, and a quantity K being a weak function of blowing ratio.     

Probability and complex quantum trajectories

It is shown that in the complex trajectory representation of quantum mechanics, the Born’s ΨΨ probability density can be obtained from the imaginary part of the velocity field of particles on the real axis. Extending this probability axiom to the complex plane, we first attempt to find a probability density by solving an appropriate conservation equation. The characteristic curves of this conservation equation are found to be the same as the complex paths of particles in the new representation. The boundary condition in this case is that the extended probability density should agree with the quantum probability rule along the real line. For the simple, time-independent, one-dimensional problems worked out here, we find that a conserved probability density can be derived from the velocity field of particles, except in regions where the trajectories were previously suspected to be nonviable. An alternative method to find this probability density in terms of a trajectory integral, which is easier to implement on a computer and useful for single particle solutions, is also presented. Most importantly, we show, by using the complex extension of Schrodinger equation, that the desired conservation equation can be derived from this definition of probability density.     

Two-dimensional wave equation with degeneration on the curvilinear boundary of the domain and asymptotic solutions with localized initial data

In a two-dimensional domain Ω ? R ², we consider the wave equation with variable velocity c(x ₁, x ₂) degenerating on the boundary Γ = ?Ω as the square root of the distance to the boundary, and construct an asymptotic solution of the Cauchy problem with localized initial data. This problem is related to the so-called “run-up problem” in tsunami wave theory. One main idea (also used by the authors in earlier papers in the one-dimensional case and the two-dimensional case with c ²(x ₁, x ₂) = x ₁) is that the (singular) curve Γ is a caustic of special type. We use this idea to introduce a generalization of the Maslov canonical operator covering the problem with degeneration and obtain efficient formulas for the asymptotic solutions.     

Hydrodynamic interactions in Brownian dynamics: I. Periodic boundary conditions for computer simulations

Because of the long range nature of hydrodynamic interactions, the problem of boundary conditions on a finite simulation cell of a hydrodynamically dense suspension of particles in Brownian motion is quite as complicated as the analogous problem in simulation of the statistical mechanics of charged and dipolar systems. One resolution of this problem is to use periodic boundary conditions and to view them as a way of describing a physical system composed of a large spherical array of periodic replicas of the simulation cell. The hydrodynamic interactions are calculated using the quasi-static linearized Navier-Stokes equation. This requires that the suspending fluid velocity remains small throughout the array. That the sum of the particle velocities in the simulation cell be zero is insufficient to force boundedness of the fluid velocity as the array becomes large. Boundedness in the array of the suspending fluid velocity is achieved if a rigid wall boundary condition is applied at the outer edge of the array as the array becomes large. With this condition the net particle velocity equals zero condition is not needed. The condition allows lattice sum representations for the suspending fluid velocity to be derived. These lattice sums are absolutely and rapidly convergent and periodic. Representations of the velocity in the array with boundary condition allow calculation of mobility tensors which are also periodic and can be evaluated numerically in tolerable amounts of computer time. A major effect of these calculations is to identify the physical model system corresponding to a truly periodic fluid velocity and mobility tensor as a large array with rigid wall boundary condition.     

Thermal boundary layer for non-Newtonian fluid

Summary Analytical and numerical solutions for the momentum and thermal boundary layer equations of a non-Newtonian power law fluid are presented. The flow is assumed to be under the influence of an external magnetic fieldB (x) applied perpendicular to the surface and an electric fieldE(x) perpendicular toB(x) and the direction of the longitudinal velocity in the boundary layer. For the power law fluid it is assumed that the shear stress is proportional to then-th power of the velocity gradient andn is called the flow index. The variations of the velocity fieldf′, the temperature field θ, the shear stress on the surfaceτ _W , the displacement thicknessδ ₁ and the momentum thicknessδ ₂ with the magnetic-field parameter γ, the flow indexn, the heat transfer indexS and the Prandtl number Pr are studied. It is found that, if the outer flow velocityU(x) (potential flow) is proportional to the arc lengthx raised to a powerm, then the similarity solution for the thermal boundary layer equation is possible only whenm=1/3, which represents flow past a wedge of included angle π/2. It is established that the temperature of the wedge increases with the increase of γ, Pr,S and the decrease ofn. In general the magnetic field can be used as a heater for the surface of the wedge.     

Rayleigh wave at the boundary of an inhomogeneous nonlinear viscoelastic half-space

In the approximation of weak nonlinearity and weak viscosity of the medium, we obtain an equation describing the spectral density of the particle horizontal velocity for a Rayleigh wave propagating along the boundary of a half-space. The coefficients of nonlinear interaction between the wave harmonics are found on the assumption that the third-order elastic moduli arbitrarily depend on the depth. We find expressions for the complex correction to the wave frequency due to small relaxation corrections to the elastic moduli and small variations in the medium density, which arbitrarily depend on the depth as well. The imaginary part of this correction to the frequency determines the decay of the linear Rayleigh wave due to small relaxation corrections to the elastic moduli arbitrarily dependent on the depth. Using numerical simulation (with allowance for the interaction of 500 harmonics), we study distortions of an initially harmonic Rayleigh wave for a particular dependence of variations in the nonlinear moduli on the depth. An integral equation is derived for the nonlinear elastic moduli as functions of the depth. It is shown that for independent spatio-temporal distributions of the viscous moduli, functions determining the dependence of the viscosity on the depth are described by an analogous integral equation. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 50, No. 3, pp. 212–226, March 2007.     

Performance bounds on single-particle tracking by fluorescence modulation

We consider fundamental bounds on the performance of single-particle tracking schemes based on non-imaging, fluorescence modulation methods. We calculate the noise density of a linearized position estimate arising from photon-counting statistics and find the optimal estimate of a freely diffusing particle’s position in the presence of this noise. For the experimentally relevant case of a Gaussian laser rapidly translated in a circular pattern, explicit expressions are derived for the noise density. Tracking performance limits are obtained by considering the variance in the estimated position of a Brownian particle with diffusion coefficient D in the presence of a noise density n_m, which we find scales generically as (Dn_m ²)^1/2. For reasonable experimental parameters, a particle with diffusion coefficient D=1 μm²/s cannot be tracked with accuracy better than approximately 100 nm in three dimensions or 80 nm in two dimensions. Using a combination of exact results and numerical simulation, we construct a ‘phase diagram’ for determining parameter regimes in which a particle can be tracked in the presence of measurement noise. PACS 87.64.Tt; 87.64.Ni; 87.15.Vv     

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%).     

Trend to equilibrium of weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions

Recently R. Illner and the author proved that, under a physically realistic truncation on the collision kernel, the Boltzmann equation in the one-dimensional slab [0, 1] with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense. Here it is proved that when the Maxwellians associated with the boundary conditions atx=0 andx=1 are the same MaxwellianM _w , then the solution is uniformly bounded and tends toM _w fort.     

Statistical Properties of the Burgers Equation with Brownian Initial Velocity

We study the one-dimensional Burgers equation in the inviscid limit for Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian motion that starts from the origin x=0). We obtain the one-point distribution of the velocity field in closed analytical form. In the limit where we are far from the origin, we also obtain the two-point and higher-order distributions. We show how they factorize and recover the statistical invariance through translations for the distributions of velocity increments and Lagrangian increments. We also derive the velocity structure functions and we recover the bifractality of the inverse Lagrangian map. Then, for the case where the initial density is uniform, we obtain the distribution of the density field and its n-point correlations. In the same limit, we derive the n-point distributions of the Lagrangian displacement field and the properties of shocks. We note that both the stable-clustering ansatz and the Press-Schechter mass function, that are widely used in the cosmological context, happen to be exact for this one-dimensional version of the adhesion model.     

Heat and mass transfer of MHD convective boundary layer flow past a stretching porous wall embedded in a porous medium

The hydromagnetic convective boundary layer flow past a stretching porous wall embedded in a porous medium with heat and mass transfer in the presence of a heat source and under the influence of a uniform magnetic field is studied. Exact solutions of the basic equations of motion, heat and mass transfer are obtained after reducing them to nonlinear ordinary differential equations. The reduced equations of heat and mass transfer are solved using a confluent hypergeometric function. The effects of the flow parameters such as a suction parameter (N), magnetic parameter (M), permeability parameter (K _p ), wall temperature parameter (r), wall concentration parameter (n), and heat source/sink parameter (Q) on the dynamics are discussed. It is observed that the suction parameter appears in the boundary condition ensuring the variable suction at the surface. Transverse component of the velocity increases only when magnetic field strength exceeds certain value, but the thermal boundary layer thickness and concentration distribution increase for all values. Results presented in this paper are in good agreement with the work of the previous author and also in conformity with the established theory.     

On the quantum-mechanical Fokker-Planck and Kramers-Chandrasekhar equation

In the classical theory of Brownian motion we can consider the Langevin equation as an infinitesimal transformation between the coordinates and momenta of a Brownian particle, given probabilistically, since the impulse appearing is characterized by a Gaussian random process. This probabilistic infinitesimal transformation generates a streaming on the distribution function, expressed by the classical Fokker-Planck and Kramers-Chandrasekhar equations. If the laws obeyed by the Brownian particle are quantum mechanical, we can reinterpret the Langevin equation as an operator relation expressing an infinitesimal transformation of these operators. Since the impulses are independent of the coordinates and momenta we can think of them as c numbers described by a Gaussian random process. The so resulting infinitesimal operator transformation induces a streaming on the density matrix. We may associate, according to Weyl functions with operators. The function associated with the density matrix is the Wigner function. Expressing, then, these operator relations in terms of these functions we can express the streaming as a continuity equation of the Wigner function. We find that in this parametrization the extra terms which appear are the same as in the classical theory, augmenting the usual Wigner equation.     

Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations

We developed a theory of fluctuations for Brownian systems with weak long-range interactions. For these systems, there exists a critical point separating a homogeneous phase from an inhomogeneous phase. Starting from the stochastic Smoluchowski equation governing the evolution of the fluctuating density field of Brownian particles, we determine the expression of the correlation function of the density fluctuations around a spatially homogeneous equilibrium distribution. In the stable regime, we find that the temporal correlation function of the Fourier components of density fluctuations decays exponentially rapidly, with the same rate as the one characterizing the damping of a perturbation governed by the deterministic mean field Smoluchowski equation (without noise). On the other hand, the amplitude of the spatial correlation function in Fourier space diverges at the critical point T=T_c (or at the instability threshold k=k_m) implying that the mean field approximation breaks down close to the critical point, and that the phase transition from the homogeneous phase to the inhomogeneous phase occurs sooner. By contrast, the correlations of the velocity fluctuations remain finite at the critical point (or at the instability threshold). We give explicit examples for the Brownian Mean Field (BMF) model and for Brownian particles interacting via the gravitational potential and via the attractive Yukawa potential. We also introduce a stochastic model of chemotaxis for bacterial populations generalizing the deterministic mean field Keller-Segel model by taking into account fluctuations and memory effects.     

Presheath in Fully Ionized Collisional Plasma in a Magnetic Field

The quasineutral presheath layer at the boundary of fully ionized, collisional, and magnetized plasma with an ambipolar flow to an adjacent absorbing wall was analyzed using a two fluid magneto‐hydrodynamic model. The plasma is magnetized by a uniform magnetic field B , imposed parallel to the wall. The analysis did not assume that the dependence of the particle density on the electric potential in the presheath is according to the Boltzmann equilibrium, and the dependence of the mean collision time τ on the varying plasma density within the presheath was not neglected. Based on the model equations, algebraic expressions were derived for the dependence of the plasma density, electron and ion velocities, and the electrostatic potential on the position within the presheath. The solutions of the model equations depended on two parameters: Hall parameter (β ), and the ratio (γ ), where γ = ZT_e /(ZT_e + T_i ), and T_e , T_i and Z are the electron and ion temperatures and ionicity, respectively. The characteristic scale of the presheath extension is several times r_i /β , where r_i is the ion radius at the ion sound velocity. The electric potential could have a non monotonic distribution in the presheath. The ions are accelerated to the Bohm velocity (sound velocity) in the presheath mainly near the presheath‐sheath boundary, in a layer of thickness ～r_i /β . The electric field accelerates the ions in the whole presheath if their velocity in the wall direction exceeds their thermal velocity. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)