Equation For Average Research Articles

A deep learning based nonlinear upscaling method for transport equations

We will develop a nonlinear upscaling method for nonlinear transport equations. The proposed scheme gives a coarse scale equation for the cell average of the solution. In order to compute the parameters in the coarse scale equation, a local downscaling operator is constructed. This downscaling operation recovers fine scale properties using cell averages. This is achieved by solving the equation on an oversampling region with the given cell average as constraint. Due to the nonlinearity, one needs to compute these downscaling operations on the fly and cannot pre-compute these quantities. In order to give an efficient downscaling operation, we apply a deep learning approach. We will use a deep neural network to approximate the downscaling operation. Our numerical results show that the proposed scheme can achieve good accuracy and efficiency.

Upstream swimming and Taylor dispersion of active Brownian particles

Locomotion of self-propelled particles such as motile bacteria or phoretic swimmers often takes place in the presence of applied flows and confining boundaries. Interactions of these active swimmers with the flow environment are important for the understanding of many biological processes, including infection by motile bacteria and the formation of biofilms. Recent experimental and theoretical works have shown that active particles in a Poiseuille flow exhibit interesting dynamics including accumulation at the wall and upstream swimming. Compared to the well-studied Taylor dispersion of passive Brownian particles, a theoretical understanding of the transport of active Brownian particles (ABPs) in a pressure-driven flow is relatively less developed. In this paper, employing a small wave-number expansion of the Smoluchowski equation describing the particle distribution, we explicitly derive an effective advection-diffusion equation for the cross-sectional average of the particle number density in Fourier space. We characterize the average drift (specifically upstream swimming) and effective longitudinal dispersion coefficient of active particles in relation to the flow speed, the intrinsic swimming speed of the active particles, their Brownian diffusion, and the degree of confinement. In contrast to passive Brownian particles, both the average drift and the longitudinal dispersivity of ABPs exhibit a nonmonotonic variation as a function of the flow speed. In particular, the dispersion of ABPs includes the classical shear-enhanced (Taylor) dispersion and an active contribution called the swim diffusivity. In the absence of translational diffusion, the classical Taylor dispersion is absent and we observe a giant longitudinal dispersion in the strong flow limit. Our continuum theory is corroborated by a direct Brownian dynamics simulation of the Langevin equations governing the motion of each ABP.

Asymptotics for the long-time evolution of kurtosis of narrow-band ocean waves

In this paper we highlight that extreme events such as freak waves are a transient phenomenon in keeping with the old fisherman tale that these extreme events seem to appear out of nowhere. Janssen (J. Phys. Oceanogr., vol. 33, 2003, pp. 863–884) obtained an evolution equation for the ensemble average of the excess kurtosis, which is a measure for the deviation from normality and an indicator for nonlinear focusing resulting in extreme events. In the limit of a narrow-band wave train, whose dynamics is governed by the two-dimensional nonlinear Schrödinger (NLS) equation, the excess kurtosis is under certain conditions seen to grow to a maximum after which it decays to zero for large times. This follows from a numerical solution of the problem and also from an analytical solution presented by Fedele (J. Fluid Mech., vol. 782, 2015, pp. 25–36). The analytical solution is not explicit because it involves an integral from initial time to actual time. We therefore study a number of properties of the integral expression in order to better understand some interesting features of the time-dependent excess kurtosis and the generation of extreme events.

Statistics of incremental averages of passive scalar fluctuations

Whereas statistical moments of differences of turbulent quantities measured over a given separation (viz., structure functions) have been extensively studied, statistics of incremental sums (or equivalently averages, e.g., Σθ≡[θ(x+r)+θ(x)]/2) of the same quantities have only been the subject of recent research. The present work investigates incremental averages of a turbulent passive scalar (temperature), measured in nearly homogeneous, and isotropic (passive and active), grid-generated turbulence, for turbulent Reynolds numbers in the range 94≤Rλ(≡urmsλ/ν)≤582. The scalar field is generated by the action of the turbulent velocity field against an imposed mean temperature gradient. Following the approach of Mouri and Hori [Phys. Fluids 22, 115110 (2010)] for the velocity field, we examine statistics of incremental averages of the passive scalar field as a function of separation (viz., incremental average structure functions) for different Reynolds numbers, comparing them with both the results of Mouri and Hori, as well as the corresponding incremental average structure functions for the velocity field for the flows studied herein. While the statistics of Σθ are primarily large-scale quantities, and would therefore be expected to be flow dependent, they exhibit certain similarities to the statistics of incremental averages of velocity (Σuα), measured both in the flow under consideration as well as the different classes of flows studied by Mouri and Hori. Finally, we derive a scale-dependent evolution equation for the incremental average of the scalar field fluctuations, Σθ. We discuss its relationship to Yaglom's four-thirds law for differences in passive scalar fluctuations and compare the results with the experimental data.

Ionic screening of charged impurities in electrolytically gated graphene: A partially linearized Poisson-Boltzmann model.

We present a model describing the electrostatic interactions across a structure that consists of a single layer of graphene with large area, lying above an oxide substrate of finite thickness, with its surface exposed to a thick layer of liquid electrolyte containing salt ions. Our goal is to analyze the co-operative screening of the potential fluctuation in a doped graphene due to randomness in the positions of fixed charged impurities in the oxide by the charge carriers in graphene and by the mobile ions in the diffuse layer of the electrolyte. In order to account for a possibly large potential drop in the diffuse later that may arise in an electrolytically gated graphene, we use a partially linearized Poisson-Boltzmann (PB) model of the electrolyte, in which we solve a fully nonlinear PB equation for the surface average of the potential in one dimension, whereas the lateral fluctuations of the potential in graphene are tackled by linearizing the PB equation about the average potential. In this way, we are able to describe the regime of equilibrium doping of graphene to large densities for arbitrary values of the ion concentration without restrictions to the potential drop in the electrolyte. We evaluate the electrostatic Green's function for the partially linearized PB model, which is used to express the screening contributions of the graphene layer and the nearby electrolyte by means of an effective dielectric function. We find that, while the screened potential of a single charged impurity at large in-graphene distances exhibits a strong dependence on the ion concentration in the electrolyte and on the doping density in graphene, in the case of a spatially correlated two-dimensional ensemble of impurities, this dependence is largely suppressed in the autocovariance of the fluctuating potential.

Energy exchange in fast optical soliton collisions as a random cascade model

We study the dynamics of a probe soliton propagating in an optical fiber and exchanging energy in fast collisions with a random sequence of pump solitons. The energy exchange is induced by Raman scattering or by cubic nonlinear loss/gain. We show that the equation describing the dynamics of the probe soliton's amplitude has the same form as the equation for the local space average of energy dissipation in random cascade models in turbulence. We characterize the statistics of the probe soliton's amplitude by the τ q exponents from multifractal theory and by the Cramér function S ( x ) . We find that the nth moment of the two-time correlation function and the bit-error-rate contribution from amplitude decay exhibit power-law behavior as functions of propagation distance, where the exponents can be expressed in terms of τ q or S ( x ) .

Evaluation of Mean Values for a Forced Pendulum with a Projection Operator Method

Employing a projection operator method, an averaged equation is derived from the equations of motion for a periodically forced pendulum with a damping term. A model equation for the ensemble average is derived from the averaged equation under some assumptions. In addition, the ensemble average is obtained by direct numerical simulation of the pendulum equations. The solutions of the model equation exhibit fair agreement with the numerical solutions for chaotic motion.

Dynamics of Wilson observables in non-commutative gauge theory

An equation for the quantum average of the gauge invariant Wilson loop in non-commutative Yang–Mills theory with gauge group U( N) is obtained. In the 't Hooft limit, the equation reduces to the loop equation of ordinary Yang–Mills theory. At finite N, the equation involves the quantum averages of the additional gauge invariant observables of the non-commutative theory associated with open contours in space–time. We also derive equations for correlators of several gauge invariant (open or closed) Wilson lines. Finally, we discuss a perturbative check of our results.

Configurational entropy and the non-Newtonian rheology of homogeneous silicate liquids.

Recently, the non-Newtonian viscosity of silicate liquids at high stress levels has attracted a fair amount of attention. Different explanations for this interesting phenomenon have been proposed. The purpose of the present paper is to demonstrate that an extension of the configurational entropy theory for the occurrence of the glass transition by Adam and Gibbs [J. Chem. Phys. 43, 139 (1965)] suffices to explain the development of non-Newtonian viscosity in liquids under high stress before rupture occurs. Attention is drawn to the fact that under steady-state conditions the logarithmic values of reduced viscosities of silicate liquids under high stress show a linear dependence on the square of the applied stress. This relationship suggests that the elastic work done by the stress on the liquid is the cause of the generation of configurational entropy. Adding this contribution to the configurational entropy term in the Adam-Gibbs equation for the ensemble average of the configurational transition probability is all that is needed to explain the heretofore mentioned non-Newtonian viscosity in liquid silicates and a molecular-dynamics Lennard-Jones liquid.

Mathematical models with exact renormalization for turbulent transport

The advection-diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and/or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media. A variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts. In this paper, a simple model problem for the advection-diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor. Explicit formulas for the anomalous time scaling in various regimes as well as the Green's function for the large-scale, long-time, ensemble average are developed here. Formulas for the renormalized higher order statistics are also developed. The simple form of the model problem is deceptive; the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied. These phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter $$\tilde \varepsilon $$ is varied as well as explicit regimes in $$\tilde \varepsilon $$ where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity. We use Fourier analysis and the Feynman-Kac formula as main tools in the explicit exact renormalization theory developed here.

Recursion relation of character expansion coefficient in SU(3) lattice gauge theory

In the SU(3) lattice gauge theory the recursion relations of coefficients of the character expansion are derived through the Schwinger–Dyson equation for the ensemble average containing multilink variables. The variables form two kinds of recursion relations which correspond to two independent integers λ and μ of the SU(3) (λμ) representation of the Young tableau. The property of the recursion relations in comparison with those of the U(1) and SU(2) groups is discussed.

Dispersion effects in capillary zone electrophoresis

We qauntitatively analyzed possible causes of the observed spreading of sample components which degrades separation compared with theoretical limits, and identified four potentially significant causes, electrokinetic dispersion, wall adsorption, enhanced diffusion due to Poiseuille flows driven by pressure gradients, and enhanced diffusion due to mobility variations associated with transverse temperature differences. Electrokinetic dispersion is caused by changes in the conductivity and pH distributions, proportional to the concentration of the sample relative to the buffer components, and independent of the tube diameter. One-dimensional numerical results for the separation of seven species in a sodium acetate buffer are presented to illustrate the effect. A detailed discussion of methods to control it is also presented. It is suggested that both wall adsorption of sample species and most coating methods used to control it or to reduce electroosmosis can be understood as aspects fo the Debye double-layer theory. Large molecules, with a high degree of ionization with the opposite sign to the wall charge, are preferentially attracted to the layer, excluding smaller molecules, and decreasing the wall potential and mobility. We demonstrate the importance of choosing a combination of pH and tube material such that the wall and the large proteins in the sample have the same charge sign and repel each other. The diffusion enhancement analysis is based on the analytic approximation of balancing the flow and mobility distrubance terms in the concentration equation for a sample species with the transverse diffusion term. This determines the fluctuating part of the concentration, with zero transverse average. The interaction of this concentration distribution with the fluctuations modifies the equation for the transverse average of the concentration, by adding diffusivity a 2 Y 2/48 D i· . Here a is the radius, D i is the diffusivity of the species, and Y is a disturbance speed and is a sum of contributions from the pressure-gradient-driven Poiseuille flow, the transverse mobility variation due to the heating in the tube, and other effects which we believe are smaller. The numerical factor of 48 is exact for the Poiseuille flow and the transverse mobility variations effects, and presumably it improves the estimate for the other effects. The Poiseuille flow profile is driven by variations in the electroosmotic slip velocity, which are mostly caused by conductivity variations along the tube and by changes in the Debye layer structure associated with composition changes along the tube. Contributions from temperature or radius variations along the tube are estimated, and are apparently smaller. Relatively insignificant causes of dispersion include sample dispersion by the difference between the electroosmotic flow and the slower flow in the Debye layer (the layer is to thin), variations in the tube radius, and convection and electrohydrodynamics flows. Molecular diffusion is important (and theoretical plate numbers are achieved) if other dispersion effects are small; it is larger for sample species with small molecules.

Initial correlations of the multiplicative process, driven by random jumps

The authors consider a vector-valued stochastic process, which is multiplicatively driven by the Markov jump process. They obtain a closed expression for the average of the vector process by solving the Burshtein equation for the marginal average. It is shown that the solution for t>t0 requires knowledge of an initial correlation operator, due to the finite correlation time of the jump process. They derive the equation for the steady-state solution, which is applied to evaluate the stationary correlation functions. Then some specific limits of the jump process, are discussed, which arise if one takes the correlation time to be zero or infinite. For the special case L(x)=A+xB they derive from the Burshtein equation a recurrence relation between the moments of the vector process. This relation is solved and it is shown how all initial moments at t0 determine the moments for t>or=t0. The recurrence relation is applied to solve the two-state and the three-state process more explicitly. It is pointed out that the occurring initial correlations cannot be neglected in general.

Nonlinear fluctuations in transport equations studied for the Rayleigh piston

We treat the master equation in one dimension for the Rayleigh-piston problem of hard rods, i.e., a single heavy tagged particle with finite mass ${m}_{A}$ moving in an ensemble of light bath particles with mass ${m}_{B}$ and a time-independent velocity distribution. The backward form of the master equation is used to obtain a transport equation for a conditional average of a time-dependent physical quantity. It is shown how this integro-differential equation can be solved successively by transforming it into a closed set of first-order linear partial differential equations. The solution of the $l$th linear partial differential equation is completely determined by the solutions of the $l\ensuremath{-}1$ linear partial differential equations, meaning that higher approximations do not change lower ones. It is shown how the motion of the tagged particle can be separated into a deterministic (nonfluctuating) part and into fluctuating contributions. It turns out that the $l=0$ term in the successive approximation scheme satisfies a homogeneous linear partial differential equation and describes the nonfluctuating motion, whereas the higher approximations ($l\ensuremath{\ge}1$), which are solutions of nonhomogeneous linear partial differential equations, describe the fluctuating contributions. The calculations for arbitrary time-dependent conditional averages are performed explicitly up to order 2 in the expansion parameter ${\ensuremath{\Omega}}^{\ensuremath{-}1}=\frac{{m}_{B}}{({m}_{A}+{m}_{B})}$. This new method is employed to calculate the conditional averages for the time-dependent mean velocity, the mean-square velocity, the velocity-autocorrelation function, and the self-diffusion coefficient. In addition, the results show that conditional averages calculated via a Gaussian distribution function deviate considerably from the exact results obtained for mass ratios of order 1.

Brownian motion in phase space

In most treatments Brownian motion is considered either in position or in velocity space. While the description of Brownian motion as a Markov process via a master equation in phase space can, in principle, deal with the full problem, assumptions are usually introduced which lead to the formulation of the generalized Fokker-Planck equation. In the following we want to show how, in the field-free one-dimensional case, the transport equation for a time-dependent conditional average of an arbitrary physical quantity (depending on position and velocity coordinates) of a tagged particle in phase space can be derived and solved successively if the time- and position-independent moments of the transition probability can be expanded in terms of a parameter ${\ensuremath{\Omega}}^{\ensuremath{-}1}$. The transport equation can then be transformed into a set of $l$ linear partial differential equations, whose solutions provide the $l\mathrm{th}$ approximation in the expansion parameter ${\ensuremath{\Omega}}^{\ensuremath{-}1}$ to the full solution of the transport equation. The system of linear partial differential equations is closed in the sense that the solution of the $l\mathrm{th}$ partial differential equation is determined by the solutions of the $l\ensuremath{-}1$ partial differential equations. It turns out that the $l=0$ partial differential equation describes the macroscopic (nonfluctuating) motion of the tagged particle. It is discussed in which sense a bivariate Gaussian distribution describes the full Markov process and how the full phase-space treatment provides more insight into Brownian motion. It is shown how the special cases described by the Langevin equation and the Fokker-Planck equation are contained in the new method presented.

A Stochastic Formulation of Soil Erosion Caused by Wind

ABSTRACT DEVELOPMENT of a new equation to predict soil erosion requires the integration of the surface soil flux vector across the eroding area and for some interval of time. In this paper we analyze the time integration and show that both an arithmetic and statistical average must be considered for prediction. The arithmetic average, which is called the soil erosion, is shown to be a measure which may or may not be random. When random, its statistical mean is called the average soil erosion. For this average, consideration must be given to two time intervals, a soil loss accounting interval and a periodicity associated with the deterministic independent variables. This latter average is shown to be identical with the present measure of wind erosion when the two time intervals are 1 year. A general equation for the statistical average is developed and its use is illustrated by developing specific equations for two different soil loss accounting intervals.

Systematic method for solving transport equations derived from master equations

It is shown how a one-dimensional transport equation (derived from a master equation) for a time-dependent conditional average of an arbitrary function can be solved successively, if the moments of the transition probability can be expanded in a series of a special parameter. The transport equation for the time-dependent conditional average value can be transformed into a set of l linear partial differential equations of first order for the lth approximation of the conditional average value. The system of equations is closed in the sense that the lth partial differential equation is determined by the solutions of the l-1 partial differential equations. It is shown how the results obtained compare with the linear noise approximation. Two examples are chosen to illustrate the way to use the method. One is the classical example of the diffusion of a Brownian particle, the other is Alkemade's diode.

Baryons as solitons in loop space

The mass of the baryon in QCD at large N is expressed in terms of certain collective fields in the loop space, satisfying the nonlinear path integral equation. This equation is of the same kind as the MM equation for the loop average, and involves the loop average in the kernel.

Analytical solution of a new integral equation for triplet correlations in hard sphere fluids

Use is made of an equivalence between the calculation of triplet correlation functions and a molecular problem. The molecular problem is then tackled within the framework of the reference interaction site model. As a result, an integral equation for an angular average of the triplet correlation function is obtained. This equation is then analytically solved for a hard sphere fluid. A closed form expression for the result in the case in which two of the atoms in the triplet are in contact is given. The theory is compared with computer experiments and it is shown to be a significant improvement over the superposition approximation at high densities.

Non-linear strings in two-dimensional U(∞) gauge theory

A non-singular version of the Makeenko-Migdal equation for the Wilson loop average in two-dimensional U( N) gauge theory is derived. In the limit N→∞ the exact solution is obtained for an arbitrary (with any self-intersections) closed loop.