Calculating coefficients for a system of moment equations used to describe the magnetization kinetics of a superparamagnetic particle in a fluctuating field

A system of equations is derived for moments [averages of spherical harmonics 〈Y _l,m 〉(t)] that determine the dynamics of the magnetization M of a superparamagnetic particle in a fluctuating field. The system is derived by representing the Gilbert equation in a fluctuating field, and the corresponding Fokker-Planck equation for the distribution function of M, in terms of angular momentum operators, which in turn makes it possible to express the coefficients of the system of moment equations in terms of Clebsch-Gordan coefficients. Fiz. Tverd. Tela (St. Petersburg) 41, 2020–2027 (November 1999)     

Lattice Kinetic Formulation for Ferrofluids

The lattice Boltzmann approach is used to solve continuum equations describing colloids of ferromagnetic particles (ferrofluids) in a regime, where the particle spins are in equilibrium with magnetic torques. This limit of rapid spin adjustment yields a symmetric total stress tensor that is essential for a kinetic formulation based on the Boltzmann equation. The magnetisation equation is solved using a vector-valued distribution function analogous to the earlier treatment (J. Comput. Phys. 179, 95) of the induction equation in magnetohydrodynamics, but the details are rather more complex because the magnetisation equation is not in conservation form except in a weakly magnetised limit.     

Development of stochastic oscillations in a one-dimensional dynamical system described by the Korteweg-de Vries equation

The behavior of the solution of the Korteweg-de Vries equation for large-scale oscillating periodic initial conditions prescribed on the entire x axis is considered. It is shown that the structure of small-scale oscillations arising in a Korteweg-de Vries system as t→∞ loses its dynamical properties as a consequence of phase mixing. This process can be called the generation of soliton turbulence. The infinite system of interacting solitons with random phases developing under these conditions leads to oscillations having a stochastic character. Such a system can be described using the terms applied to a continuous random process, the probability density and correlation function. It is shown that for this it suffices to determine from the prescribed initial conditions amplitude distribution function of the solitons and their mean spatial density. The limiting stochastic characteristics of the mixed state for problems with initial data in the form of an infinite sequence of isolated small-scale pulses are found. Also, the problem of stochastic mixing under arbitrary initial conditions in the dispersionless limit (the Hopf equation) is completely solved. Zh. éksp. Teor. Fiz. 115, 333–360 (January 1999)     

On the theory of multiple scattering

In this paper a kinetic equation is derived for the distribution function in the variable q=2 sin(ϑ/2) for the case of a scattering cross section of general form under the assumption that the region of multiple scattering (the diffusion region) is small. The limits of the kinetic equation are discussed, with no restrictions imposed on the scattering angles. It is found that the equation has a solution in the form of an integral. Finally, it is established that the solution is applicable over the entire range of angles, from 0 to 180°. Zh. éksp. Teor. Fiz. 116, 418–435 (August 1999)     

Symmetry Reductions of a Hamilton-Jacobi-Bellman Equation Arising in Financial Mathematics

Abstract

We determine the solutions of a nonlinear Hamilton-Jacobi-Bellman equation which arises in the modelling of mean-variance hedging subject to a terminal condition. Firstly we establish those forms of the equation which admit the maximal number of Lie point symmetries and then examine each in turn. We show that the Lie method is only suitable for an equation of maximal symmetry. We indicate the applicability of the method to cases in which the parametric function depends also upon the time.     

Perturbation theory for the square-well dimer fluid

An analytical equation of state is presented for the square-well dimer fluid of variable well width (1 ≤ λ ≥ 2) based on Barker-Henderson perturbation theory using the recently developed analytical expression for radial distribution function of hard dimers. The integral in the first- and the second-order perturbation terms utilizes the Tang, Y and Lu, B. C.-Y., 1994, J. chem. Phys., 100, 6665 formula for the Hilbert transform. To test the equation of state, NVT and Gibbs ensemble Monte Carlo simulations for square-well dimer fluids are performed for three different well widths (λ = 1.3, 1.5 and 1.8). The prediction of the perturbation theory is also compared with that of thermodynamic perturbation theory in which the equation of state for the square-well dimer is written in terms of that of square-well monomers and the contact value of the radial distribution function.     

Characteristic Length in a Linear Acceleration

The role of the characteristic length that characterizes linear acceleration is studied, in order to find how does this length determine the characteristic wavelength of the radiation created by the accelerated charge. Unruh equation for the temperature observed by a detector accelerated relative to the vacuum is used to determine the wavelength distribution of the radiation emitted by a linearly accelerated charge, and it is found that this distribution is peaked close to the characteristic length that characterizes the linear acceleration, which is the radius of curvature of the curved electric field created by the acelerated charge. PACS numbers: 11.10; 41.60.m.     

Correlation functions in the theory of atomic collision cascades: Ion location and the distribution in depth and size of damage clusters

Abstract

Simple depth distribution functions of ion bombardment damage predict the spatial extension of the cumulative damage caused by a beam of ions. Correlation functions need to be considered when more detailed information is desirable, such as the average size or depth of individual damage clusters, the average location of an ion within its damage cluster, and the fluctuations of these quantities. In this paper we establish an integral equation for the pair correlation function, coupling the individual ion range with the deposited energy. This pair correlation function determines the damage caused by all those ions that come to rest at a specific penetration depth. Solutions of the integral equation are found by standard methods. Explicit results are presented for elastic scattering governed by power cross sections. The depth distribution of damage clusters turns out to be significantly narrower than the gross damage distribution at all mass ratios except for M ₁ ? M ₂, and the size distribution appears insensitive to depth when measured perpendicular to the ion beam, but varying with depth when measured parallel. Predictions on ion location suffer from a surprizingly sensitive dependence on the scattering cross section. A note on the fluctuation of the sputtering yield by individual ions concludes the paper.     

Self-consistent mechanism for sustaining ionization waves in a low-pressure discharge

The possibility of constructing a self-consistent model for the sustaining of ionization waves is demonstrated for a low-pressure discharge in an inert gas. The model is based on the combined solution of the kinetic equation of the electrons and the equation of motion of the ions in a spatially periodic field. The distribution function is constructed in an experimentally measured field and then used to calculate the spatial distributions of the plasma density and the ionization rate. The solution of the equation of motion of the ions makes it possible to reconstruct a field similar to the original one. One specific feature of the mechanism considered is the pronounced nonlocal character of the formation of the electron distribution function by the entire nonuniform potential profile of the ionization wave. Zh. Tekh. Fiz. 67, 24–30 (February 1997)     

Modelling of grain refinement driven by negative grain boundary energy

Abstract

Grain refinement can be described by the classical kinetic equation using a negative value of the specific grain boundary Gibbs energy. A respective overview is offered reporting according observations and simulations, particularly linked to grain boundary segregation. Classical grain growth model is used in the treatment of evolution of the distribution function during refinement. The adapted model requires defining nucleation rate of new grains, which significantly influences the kinetics of the system of grains. Moreover, a jump in the distribution function is allowed at a certain value of the grain radius R_J, which separates old grains from newly nucleated ones. Evolution equation for both the critical radius R_c and separation radius R_J (jump position) as well as for the dimension-free distribution (shape) function are derived. Illustrative examples for the evolution of the system parameters under various nucleation rates of newly generated grains are presented.     

Special solutions of the Fokker-Planck transport equation for arbitrary band structures; Mobility and diffusivity in a uniform field of force

A Fokker-Planck equation can be derived from a transition-type transport equation if the transition rates are nearly local in momentum space compared with the inhomogeneity length of the distribution. It is a second-order differential equation, whose coefficients depend on the band structureE(k), the viscosity tensor (k), and the temperatureT. Classical solutions of the Fokker-Planck equation deal with the parabolic band structure of free Brownian particles in a field of force. Mobility and diffusivity are then independent of the applied field. Here the explicit solution for the stationary state and the time-integrated conditional probability will be given in one dimension. This suffices to determine mobility and diffusivity. Assuming = 1, these quantities become independent of the field and the band structure, if the latter is nonperiodic, though the distribution still depends on it. This property even holds in three dimensions fork-independent viscosity tensors. Field-dependent mobility and diffusivity are obtained for ak-dependent viscosity or = 1 and periodic band structures. The latter is demonstrated for the caseE-cosk, which is also related to the noise problem in Josephson junctions.     

Numerical calculation of the heavy-ion energy spectrum in the cathode sheath of a glow discharge in a gas mixture

A numerical method is developed for solving the equation for the heavy-ion total-energy distribution function in the cathode sheath of a glow discharge in an inert-gas mixture which requires much less computer time than the Monte Carlo method. It is shown that it allows one to calculate with satisfactory accuracy the energy spectrum of the heavy ions bombarding the cathode in glow-discharge devices. Zh. Tekh. Fiz. 68, 56–59 (June 1998)     

Influence of potential fluctuations on the instrumental functions of electrostatic analyzers

An equation which relates the output signal of an electrostatic dispersion analyzer and the energy distribution function of the charged particles entering it is derived with the fluctuations of the potentials on the defecting electrodes taken into account. Solutions of this equation are obtained. The influence of noise on the instrumental functions of analyzers is considered. Zh. Tekh. Fiz. 67, 92–95 (June 1997)     

Transport Equation Evaluation of Coupled Continuous Time Random Walks

The transport behavior of a migrating particle in a disordered medium is exhibited in the solution of a transport equation derived from a coupled continuous time random walk (CTRW). A core aspect of CTRW is the spectrum of transitions in displacement s and time t, ψ(s,t), that characterizes the disordered system, which determine the transport. In many applications the CTRW approach has successfully accounted for the anomalous or non-Fickian nature of the particle plume propagation based on a power-law dependence ψ(t) in a decoupled p(s)ψ(t) approximation to ψ(s,t). For example, this power-law dependence in t derives from the complex Darcy flow fields in geological formations. Recently, the fully coupled CTRW was analyzed using a particle tracking approach, demonstrating that the decoupled approximation is valid only for a compact distribution of s. In this paper we solve the nonlocal-in-time transport equation with a ψ(s,t) containing a power-law dependence in both s (a Lévy-like distribution) and t, which necessitates the strong s,t coupling. We show enhanced transport behavior (relative to the plume propagation behavior reported in the literature) that derives from the rare large displacements in s (limited by the transition t). The interplay between the two coupled power laws is clearly shown in the changes in the breakthrough curves in the arrival times, dispersion and dependence on the velocity (v=s/t) distribution. Similar enhancements are exhibited in the particle tracking results.     

On the acoustic wave attenuation in a turbulent medium

Abstract

The equation for the mean acoustic field has been obtained for a random turbulent medium using the Green function approach. The correlation function was described by the Karman distribution with the index n=2 approximately?11/6. Applying Bourret's approximation, the exact expression for the mass operator has been calculated analytically. The frequency dependence of the scattering coefficient of the mean field has been derived. Conditions of Cherenkov radiation are discussed.     

Heating of two-dimensional excitons by nonequilibrium acoustic phonons

The energy distribution functions of two-dimensional excitons in the presence of nonequilibrium acoustic phonons have been calculated for the geometry used in heat-pulse experiments. The results were obtained by solving numerically the kinetic equation for the case where the exciton gas equilibrates with phonons during its lifetime. The cases of the low and high exciton-gas density limits are considered. It is shown that at low exciton-gas densities the distribution does not follow the Boltzmann function and depends on the quantum-well width. A comparison with earlier experimental data is made. Fiz. Tverd. Tela (St. Petersburg) 41, 1707–1711 (September 1999)     

Renormalization Group Method Applied to Kinetic Equations: Roles of Initial Values and Time

The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include the Boltzmann equation in classical mechanics, the Fokker-Planck equation, and a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method are clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time t₀ to be on the averaged distribution function to be determined. The averaged distribution function may be thought of as an integral constant of the solution of the microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time t₀, and thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of the Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in the Fokker-Planck equation are also performed in a unified way in the present method.