A comment on early-time solutions of the Smoluchowski equation

We present a simple derivation of classes of early-time solutions of the Smoluchowski equation in the presence of boundaries, simplifying and generalizing an analysis by van Kampen.     

Eigenvalues of the one-dimensional Smoluchowski equation

The eigenvalues and eigenfunctions of the Smoluchowski equation are investigated for the case of potentials withN deep wells. The small parameter =kT/V, which measures the ratio of the thermal energy to a typical well depth, is used in connection with the method of matched asymptotic expansion to obtained asymptotic approximations to all the eigenvalues and eigenfunctions. It is found that the eigensolutions fall into two classes, namely (i) the top-of-the-well and (ii) the bottom-of-the-well eigensolutions. The eigenvalues for both classes of solutions are integer multiples of the squqres of the frequencies at the top or bottom of the various wells. The eigenfunctions are, in general, localized to the top or bottom of the corresponding well. The very small eigenvalues require special consideration because the asymptotic analysis is incapable of distinguishing them from the zero eigenvalue with multiplicityN. Another approximation reveals that, in addition to the true zero eigenvalue, there areN-1 eigenvalues of order exp(–). The case of other possible multiple eigenvalues is also examined.     

Multiscaling for classical nanosystems: Derivation of Smoluchowski & Fokker–Planck equations

Using multiscale analysis and methods of statistical physics, we show that a solution to the N-atom Liouville equation can be decomposed via an expansion in terms of a smallness parameter , wherein the long scale time behavior depends upon a reduced probability density that is a function of slow-evolving order parameters. This reduced probability density is shown to satisfy the Smoluchowski equation up to O(²) for a given range of initial conditions. Furthermore, under the additional assumption that the nanoparticle momentum evolves on a slow time scale, we show that this reduced probability density satisfies a Fokker–Planck equation up to O(²). This approach has applications to a broad range of problems in the nanosciences.     

On the derivation of Smoluchowski equations with corrections in the classical theory of Brownian motion

Differential equations governing the time evolution of distribution functions for Brownian motion in the full phase space were first derived independently by Klein and Kramers. From these so-called Fokker-Planck equations one may derive the reduced differential equations in coordinate space known as Smoluchowski equations. Many such derivations have previously been reported, but these either involved unnecessary assumptions or approximations, or were performed incompletely. We employ an iterative reduction scheme, free of assumptions, and calculate formally exact corrections to the Smoluchowski equations for many-particle systems with and without hydrodynamic interaction, and for a single particle in an external field. In the absence of hydrodynamic interaction, the lowest order corrections have been expressed explicitly in terms of the coordinate space distribution function. An additional application of the method is made to the reduction of the stress tensor used in evaluating the intrinsic viscosity of particles in solution. Most of the present work is based on classical Brownian motion theory, but brief consideration is given in an appendix to some recent developments regarding non-Markovian equations for Brownian motion.Supported by the National Science Foundation.     

Renormalization-group derivation of Navier-Stokes equation

The Navier-Stokes equation is proved from first principles (rotational symmetry and conservation of momentum, mass, and energy) using renormalization-group ideas. That is, we consider a system described by one (classical) conserved vector field and two conserved scalar fields, and demonstrate that on a large scale it obeys the Navier-Stokes equation. No assumptions about the physical meanings of the fields are required; in particular, no results from thermodynamics are used. The result comes about because the Euler equation is an exact fixed point of an appropriate scale-coarsening transformation, and the coefficients of the eigenvectors of the transformation with the largest (most relevant) eigenvalues include (in dimensiond>2) the thermal conductivity and the bulk and shear viscosities, leading to Navier-Stokes behavior on a large scale. Ford<2, the largest eigenvalue corresponds to a convection term, and the Navier-Stokes equation is incorrect. Our method differs from previous renormalization approaches in using time-coarsening as well as space-coarsening transformations. This allows renormalization trajectories to be determined exactly, and allows the determination of the macroscopic behavior of specific microscopic models. The Navier-Stokes equation we obtain is almost, but not exactly, the same as the conventional one; distinguishing between them experimentally would require measurement of the very small asymmetry of the Brillouin line in a simple fluid.     

Analytical Solution of Smoluchowski Equation in Harmonic Oscillator Potential

Non-equilibrium fission has been described by diffusion model. In order to describe the diffusion process analytically, the analytical solution of Smoluchowski equation in harmonic oscillator potential is obtained. This analytical solution is able to describe the probability distribution and the diffusive current with the variable x and t. The results indicate that the probability distribution and the diffusive current are relevant to the initial distribution shape, initial position, and the nuclear temperature T; the time to reach the quasi-stationary state is proportional to friction coefficient β, but is independent of the initial distribution status and the nuclear temperature T. The prerequisites of negative diffusive current are justified. This method provides an approach to describe the diffusion process for fissile process in complicated potentials analytically.     

Weighted Flow Algorithms (WFA) for stochastic particle coagulation

Stochastic particle-resolved methods are a useful way to compute the time evolution of the multi-dimensional size distribution of atmospheric aerosol particles. An effective approach to improve the efficiency of such models is the use of weighted computational particles. Here we introduce particle weighting functions that are power laws in particle size to the recently-developed particle-resolved model PartMC-MOSAIC and present the mathematical formalism of these Weighted Flow Algorithms (WFA) for particle coagulation and growth. We apply this to an urban plume scenario that simulates a particle population undergoing emission of different particle types, dilution, coagulation and aerosol chemistry along a Lagrangian trajectory. We quantify the performance of the Weighted Flow Algorithm for number and mass-based quantities of relevance for atmospheric sciences applications.     

Stochastic differential equation derivation: Comparison of the Markov method versus the additive method

There are several methods of transforming an ordinary differential equation into a stochastic differential equation (SDE). The two most common are adding noise to a system parameter or variable and transforming to a SDE or deriving the SDE by assuming an underlying Markov process. Using simple one- and two-dimensional systems we investigate the differences in dynamics and bifurcations between SDE derived by each method from simple deterministic population models.     

Stochastic Liouville equation studies of FT-EPR spectra of singlet-triplet mixing photo-chemically generated radical pair system

A stochastic Liouville equation (SLE) was numerically solved to obtain pulsed Fourier-transform (FT) EPR spectra on a radical pair system created in a photo-induced chemical reaction. Numerical calculations were applied to the photo-chemical reaction of deuterated acetone and 2-propanol at low temperatures. In this reaction system, the antiphase structures of the EPR signals, so called spin-correlated radical pair (SCRP) signals of two identical isopropyl ketyl radicals and spin-polarized free isopropyl ketyl radicals were observed by FT-EPR and continuous wave time-resolved (CW TR) EPR techniques. In the present work, FT-EPR spectra of the antiphase structure signals of the radical pair themselves as well as spin-polarized free radical signals were simulated. Additionally, rising behavior of free radical signals polarized by the radical pair mechanism (RPM) was also clarified. Furthermore two-dimensional (2D) FT-EPR nutation spectra were simulated in the both cases with and without the radical pairs by the use of SLE. In these simulations, strong DC components in the nutation frequency dimension, were well reproduced as was obtained in experiments. It was shown that relaxation during the microwave pulse was essential for the appearance of the DC components.     

A note on the derivation of the Schrödinger-Langevin equation

It is shown that the dynamics of open systems interacting with a chaotic environment can be formulated in a quantum mechanical scheme by means of Nelson's stochastic quantization procedures. As a result, a wave equation for a particle in the chaotic environment is found to be of the Schrödinger-Langevin type associated with an additional nonlinear and random potential.     

A theorem allowing the derivation of deterministic evolution equations from stochastic evolution equations. III The Markovian-non-Markovian mix

The proof of a theorem that allows one to construct deterministic evolution equations from a set, with two subsets, containing two types of discrete stochastic evolution equation is developed. One subset evolves Markovianly and the other non-Markovianly. As an illustrative example, the deterministic evolution equations of quantum electrodynamics are derived from two sets of Markovian and non-Markovian stochastic evolution equations, of different type, after an average over realization, using the theorem. This example shows that deterministic differential equations that contain both first-order and second-order time derivatives can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations. Two explicit examples, the first containing updating rules that depend on one previous time step and the second containing updating rules that depend on two previous time steps, are given in detail in order to show step by step the linear transformations that allow one to obtain the deterministic differential equations.     

A derivation of the Christoffel equation with damping

This paper provides a derivation of the Christoffel eigenvalue equation for acoustic wave propagation in an acoustically damped piezoelectric medium. The damping tensor is shown to couple into both the stress and displacement constitutive equations. Application of the quasi-static approximation leads to an additional term in the Christoffel equation that generates a complex k-vector, due both to introduction of a complex term and to breaking of symmetry in the left-hand side of the eigenvalue equation, subsequently resulting in damping and a phase shift for a plane wave solution. Shown are the effects of damping on the eigenvalues of the piezoelectrically stiffened Christoffel equation for plane wave propagation in unconstrained quartz over a 1 MHz to 1 GHz frequency range.     

A statistical derivation of the hydrodynamic equations of change for a system of ionized molecules. I. General equation of change and the Maxwell equations

Nonrelativistic, classical statistical mechanics is used to describe a dense fluid of molecules composed of nuclei and electrons with purely Coulomb interaction potentials. A general equation of change is derived for the time rate of change of any macroscopic (ensemble averaged) dynamical variable. From this general equation, Maxwell's equations in a medium are derived and expressed in terms of molecular properties, e.g., polarization and magnetization densities.This research was carried out in part under Grant NsG-275-62 from the National Aeronautics and Space Administration and in part under a grant from the National Science Foundation. This paper is based on a thesis submitted by R. J. B. to the Graduate School of the University of Wisconsin in partial fulfillment of the requirements for the Ph.D. degree.     

Approximate solutions of the Liouville equation. III. Variational principles and projection operators

Aspects of stationary variational principles for the Laplace-transformed Liouville equation are discussed. Projection techniques are used to derive new stationary principles applicable to the space orthogonal to the space spanned by functions occurring in the conservation laws. As a result, any trial function automatically leads to results satisfying the conservation laws. The procedure is also applied to the parity-even and parity-odd distributions which obey equations governed by the square of the Liouville operator. The technique is extended to eliminate the one-body additive contribution to the solution exactly. Finally, the ideas of the moment method, which leads to the continued-fraction representation of autocorrelation functions, are applied to variational principles. We find continued-fraction variational principles such that a zero trial function yields the usual representation. However, a trial function representing noninteracting particles contains the results of the moment method and in addition yields the exact analytic behavior for free particles.Work supported by a grant from the National Science Foundation.     

Approximate solutions of the Liouville equation. II. Stationary variational principles

Stationary variational functionals for the Laplace transform of the Liouville distribution are constructed. The value of the functional is the autocorrelation function that one wishes to compute. It is shown that the functionals may be transformed to a renormalized form. Trial functions not involving the potential explicitly give rise to time-dependent autocorrelation functions determined only by equilibrium spatial correlation functions. Another class of functionals is constructed by independently varying the parity symmetric and antisymmetric parts of the distribution function. Trial functions need only be assumed for one of these—the optimum value of the other one is given exactly. This procedure is used to improve the simplest known theories for velocity and density autocorrelation functions.Work supported by a grant from the National Science Foundation.     

Probabilistic derivation of van der Waals' equation

A probabilistic derivation of van der Waals' equation is given where the non-ideal contributions to the entropy coincide with the log-likelihood ratios, at their maxima, of the binomial distribution and the upper or lower bounds on the hypergeometric and Pólya distributions. The hypergeometric distribution accounts for the repulsive interactions of a hard sphere gas while the Pólya distribution accounts for the attractive interactions.1. Work supported in part by contributions from the Ministry of Public Education (M.P.I.) and the National Science Council (C.N.R.).     

Formal structure of kinetic theory

We study the Liouville equation in the domain of small deviations from absolute equilibrium. The solution is expressed in terms of amplitudes ofn-body additive functions which are orthogonal with respect to the Gibbs weight factor. In the memory operator approach the memory operators are formally exact continued fractions. We show that with the isolation in the Liouville operator of a one-body additive operatorL _o, any memory operator can be written alternatively as an exact infinite series, each term of which can be calculated exactly. This yields improvements of the dressed particle approximation. We discuss the choice ofL _o, which is in general time dependent. The theory is developed both for smooth potentials and for hard spheres, where we use pseudo-Liouville operators. The theory can be formulated in an equivalent manner by introducing modified cumulant distributions, which are closely related to the amplitudes. The modified cumulants have the same spatial asymptotic properties as ordinary cumulants, but have superior short-time and small-distance behavior.Work supported by the National Science Foundation.     

时间分数阶Boussinesq方程的李对称分析

本文利用李群分析方法研究了时间分数阶Boussinesq方程,得到了该方程的李点对称,并把该方程约化为Erdelyi-Kobe分数阶常微分方程. 本文的行文过程也说明了李群分析方法对于约化分数阶非线性发展方程是有效的. 关键词： 李对称分析方法 时间分数阶Boussinesq方程 广义Riemann-Liouville导数 Erdelyi-Kober微分算子     

A kinetic derivation of extended irreversible thermodynamics

The basic postulates of the extended irreversible thermodynamics are derived from the kinetic model for a dilute monoatomic gas. Using the Grad 13-moment method to solve the full nonlinear Boltzmann equation for molecules conceived as soft spheres we obtain the microscopic expressions for the entropy flux, the entropy production, and the generalized Pfaffian for the extended definition of entropy as required by such a theory. Some of the physical implications of these results are discussed.Member, Colegio Nacional.