Arbitrary Initial Distribution Research Articles

The Kullback–Leibler Divergence Rate Between Markov Sources

In this work, we provide a computable expression for the Kullback-Leibler divergence rate lim/sub n/spl rarr//spl infin//1/nD(p/sup (n)//spl par/q/sup (n)/) between two time-invariant finite-alphabet Markov sources of arbitrary order and arbitrary initial distributions described by the probability distributions p/sup (n)/ and q/sup (n)/, respectively. We illustrate it numerically and examine its rate of convergence. The main tools used to obtain the Kullback-Leibler divergence rate and its rate of convergence are the theory of nonnegative matrices and Perron-Frobenius theory. Similarly, we provide a formula for the Shannon entropy rate lim/sub n/spl rarr//spl infin//1/nH(p/sup (n)/) of Markov sources and examine its rate of convergence.

Dimension-exchange algorithms for token distribution on tree-connected architectures

Load balancing on a multi-processor system involves redistributing tasks among processors so that each processor has roughly the same amount of work to perform. The token-distribution problem is a static variant of the load balancing problem for the case in which the workloads in the system cannot be divided arbitrarily; that is, where each token represents an atomic element of work. A scalable method for distributing tokens over a parallel architecture is the so-called dimension-exchange approach. Our results include improved analysis of two existing dimension-exchange algorithms for token distribution on arbitrary graphs and on arbitrary trees, respectively. In particular, we establish a logarithmic upper bound on the discrepancy of the resulting distribution when the second algorithm is applied to an arbitrary initial distribution on a tree. We then present a new dimension-exchange algorithm for token distribution on trees, which assuming each node knows the number of nodes in the tree, determines a ‘perfectly balanced’ distribution. Furthermore, the rate of convergence is worst-case optimal for trees of bounded degree. Note that an algorithm for token-distribution on trees is applicable to arbitrary architectures, since the algorithm can be applied on a spanning tree of any given connected graph.

Development of particle dispersion characteristics from arbitrary initial conditions in isotropic turbulence

Discrete particles released into turbulent carrier flows respond to that turbulence over periods of time which depend on the particle inertia and the initial state of the particle. Due to the turbulence, the velocity of the particles becomes random and they will become dispersed throughout the carrier flow. The level of randomness of the particle velocities can be quantified by the particle kinetic stress and the spreading rate is characterized by the dispersion coefficient. In the idealized case of stationary isotropic turbulence, the kinetic stress and dispersion coefficients approach limiting values long after release. Previous analysis by the author has provided development times for dispersion coefficients and kinetic stress in cases where particles were released from a point source: (i) from an initial state of rest, and (ii) with kinetic stress identical to that found in the steady state. This paper generalizes the analysis by allowing for arbitrary initial particle velocity and displacement distributions. As with the previous analysis, the primary focus is on high-inertia particles. Unlike the previous work, however, the present analysis is applicable to the common practice in direct numerical simulations of setting initial particle velocities equal to local fluid velocities. Several different estimators of the particle dispersion coefficient are considered. Whereas each estimator leads to the same long-time value, the time taken to approach this value can vary dramatically. Expressions for development times are given for the particle kinetic stress and dispersion coefficients. Consequences of the analysis for experimental and computational studies are discussed.

On the theory of the modal expansion method for the EM fields inside a closed volume

The theory of the modal expansion method for the transient electromagnetic fields inside a closed region is reviewed. Although the concept of modal expansion is well known in the theory of electromagnetism, this formulation may make it more comprehensive where the scope of applicability, the effect of surface excitations, and the introduction of real modal fields to avoid complex notations in the time-domain formulation is concerned. It can be shown that, without resorting to the explicit expression for the quasi-static solution, which should be used to satisfy the inhomogeneous boundary conditions, the transient responses of EM fields inside a closed region could be expressed explicitly in terms of the natural modes of the system. This formulation assumes an arbitrarily inhomogeneous anisotropic material, volume and surface sources of arbitrary temporal and spatial dependence, and arbitrary initial electric and magnetic field distributions. For simplicity, the effects of losses are neglected in this formulation.

Analytic solutions to the Vlasov equations for expanding plasmas.

The renormalization-group approach is applied to derive an exact solution to the self-consistent Vlasov kinetic equations for plasma particles in the quasineutral approximation. The solutions obtained describe three-dimensional adiabatic expansion of a plasma bunch with arbitrary initial velocity distributions of the electrons and ions. The solution found is illustrated by the examples on ion acceleration in a plasma with hot electrons and in a plasma with light impurity ions.

Transient well‐type flows in heterogenous formations

The problem of averaging transient flows by sources of a given head boundary condition in heterogeneous formations of random conductivity is investigated. The study generalizes the recently developed mathematical model of average transient nonuniform flow [Indelman, 1996;Tartakovsky and Neuman, 1998a]. The latter allows calculating the mean head for sources of flux boundary condition and, as such, is not applicable to modeling common well flows. To account for the head condition, the source term in the local flow model is modeled by a random function proportional to the hydraulic conductivity. The local flow equations are further averaged in order to determine the mean flow variables. It is shown that the effective conductivity cannot be defined for arbitrary initial head distributions. This precludes deriving the mathematical model of an average transient flow with the head boundary condition. However, the average flow equations are derived for the important particular case of a uniformly distributed initial head. The effective conductivity tensor is defined for an arbitrary heterogeneous formation and analyzed in detail for isotropic media. It is shown that at the initial stage of the transient flow, the effective conductivity is larger than the conductivity arithmetic mean. The fundamental solution of the average flow equations (mean Green function), corresponding to the mean head distribution due to the instantaneous injection of the unit mean amount of water through a point source, is derived at first order in the conductivity variance and is compared with the mean head for the flux boundary condition.

Interdiffusion under the Chemical Potential Gradient; Comparison of Onsager and Darken Models

The generalized Darken method of interdiffusion in multicomponent systems (GDM) enables obtaining an exact expression for the evolution of component distributions for arbitrary initial distributions and time dependent boundary conditions. In this work we studied the consequences of the more general formulae for the diffusional flux (Planck's equation) which permits to take into account thermodynamical driving forces. This paper is based on our studies of diffusion couples in the Cu-Fe-Ni system at 1273 K. The Cu-Fe-Ni system was chosen because it is a single phase in a wide range of compositions and because its thermodynamic properties are fairly well known. However, the solid solutions in this system are not ideal and consequently the diffusivities depend on composition. The driving force for the diffusion in such ternary system is the gradient of the chemical potential which can be calculated from the concentration profiles and using the known thermodynamical data of the system. Consequently the diffusional flux can be expressed as a function of the concentration gradients of all elements in the system, of the thermodynamical terms and of the mobilities. The diffusion paths are discussed in the light of the ternary interdiffusion coefficients and the self diffusivities with the use of generalized Darken method. A comparison of the Onsager and Darken models is presented. The results show the prospect for the future application of the GDM in the modelling of stress affected diffusion in ternary and higher solid solutions. The generalized Darken’s method A key problem in multicomponent diffusion is the prediction of the diffusion path between the two terminal alloys. For the predictive calculations, the data for the intrinsic diffusivities and/or self diffusion coefficients and their concentration dependence have to be known. Moreover the thermodynamic data for the system have to be known for the calculation of the chemical potentials for each component. The generalised Darken method (GDM) allows a complete quantitative description of the complex diffusional transport process and for the unlimited number of elements. It allows calculation of the diffusion paths when the interdiffusion coefficients are concentration dependent. The details of this model for the closed system [1] and the more general description of interdiffusion that incorporates the equation of motion can be found elsewhere [2-3]. In this paper the formulation of the initial-boundaryvalue problem for the interdiffusion in a closed system is presented. It differs from that presented recently [4,5] with the expression for the mass flux. Physical laws: 1) the law of the mass conservation of an i-th element:

A generalization of the Jablonski diagram to account for polarization and anisotropy effects in time-resolved experiments

The Jablonski diagram describes the kinetics of electronic transitions of organic dyes by a rate constant model. A generalization of the Jablonski diagram is presented which allows one to account for polarization and anisotropy effects. To do this, the discontinuous distribution approach (DDA), which models rotational diffusion by discontinuous jumps of the molecule between the principal molecular orientations, is employed. Thus, both rotational diffusion and population dynamics can be modeled by a system of rate differential equations rather than by solving the equation of rotational diffusion, which considerable simplifies the numerics. The generalized Jablonski diagram fully describes the kinetics of any desired electronic and orientational relaxation processes. In particular, orientational relaxation can be calculated for an arbitrary initial distribution of excited molecules. Thus, the new model is especially suited for applications in non-linear spectroscopy since it is not restricted to the small signal limit as many other approaches are.

Non-linear stochastic control via stationary response design

This paper presents a strategy for controlling externally excited stochastic systems with uncertain parameters. The control objective is to drive the system response from an arbitrary initial distribution to a prescribed stationary probability density function (PDF). This problem can be interpreted as the stochastic stabilization about a PDF. The control consists of a non-linear feedback part and a switching term inspired by robust sliding control concepts. The control of a one-dimensional stochastic process with a Gaussian target-PDF is used to illustrate the approach. The control performance is evaluated by studying the time evolution of the first and second order moments of the response. The dependence of the response on the number of feedback terms and the rate of convergence to the stationary PDF are studied numerically. Motivated by the fast convergence observed, a feedback controller with time-varying gains is applied to the problem of tracking a moving PDF. Monte Carlo simulations validate very well the control for both stabilization and tracking problems.

Transport theory of granular swarms.

The transport of trace granular gas (swarm) in a carrier granular fluid is studied by means of the Boltzmann-Lorentz kinetic equation. Time-dependent perturbation theory is used to follow the evolution of the granular swarm from an arbitrary initial distribution. A nonhydrodynamic extension of the diffusion equation is derived, with transport coefficients that are time dependent and implicitly depend on the wave vector. Transport coefficients of any order are obtained as velocity moments of the solutions of the corresponding kinetic equations derived from the Boltzmann-Lorentz equation. For the special case of the initial distribution of swarm particles, transport coefficients are identified as time derivatives of the moments of the number density. Finally the granular particle transport theory is extended by the introduction of the concept of non-particle-conserving collisions.

Burgers' equation: A general nonlinear solution of infiltration and redistribution

The general one‐dimensional solution of Burgers equation is developed using a series of transformations and the Green's function method. The solution gives the distribution of the reduced water content in a semi‐infinite and in a finite domain for arbitrary initial moisture distributions. The boundary condition at the soil surface can be either a time‐dependent flux or constant water content while the bottom boundary condition for the finite case is a constant water content value. Explicit results for a uniform, discrete, hydrostatic and steady state initial moisture profiles are derived. An analytical solution that spans the preponding to the postponding period of rainfall is also obtained. Expressions for the time to ponding and infiltration equations are presented for the various cases of initial moisture profiles and a shallow water table. These results allowed the quantification of the error in the various approximations to the time to ponding and the cumulative infiltration in the postponding period.

Alternative analytical solutions of the diffusion (thermal conductivity) equation for an arbitrary initial concentration (temperature) distribution

Alternative analytical solutions of the diffusion (or thermal conductivity) equation are presented, which ensure rapid convergence even for small values of Dt/l2 (αt/l2), where D is the diffusion coefficient, α is the thermal diffusivity, t is the time, and l is the characteristic size. The solutions possess a general character and are valid for an arbitrary initial distribution of the concentration (temperature).

A quasi-steady approach to the instability of time-dependent flows in pipes

Asymptotic solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe subject to rapid deceleration and/or acceleration are derived and their stability investigated using linear and weakly nonlinear analysis. In particular, base flow solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe are derived by the method of matched asymptotic expansions. The solutions are valid for short times and can be successfully applied to the case of an arbitrary (but unidirectional) axisymmetric initial velocity distribution. Excellent agreement between asymptotic and analytical solutions for the case of an instantaneous pipe blockage is found for small time intervals. Linear stability of the base flow solutions obtained from the asymptotic expansions to a three-dimensional perturbation is investigated and the results are used to re-interpret the experimental results of Das & Arakeri (1998). Comparison of the neutral stability curves computed with and without the planar channel assumption shows that this assumption is accurate when the ratio of the unsteady boundary layer thickness to radius (i.e. δ1/R) is small but becomes unacceptable when this ratio exceeds 0.3. Both the current analysis and the experiments show that the flow instability is non-axisymmetric for δ1/R = 0.55 and 0.85. In addition, when δ1/R = 0.18 and 0.39, the neutral stability curves for n = 0 and n = 1 are found to be close to one another at all times but the most unstable mode in these two cases is the axisymmetric mode. The accuracy of the quasi-steady assumption, employed both in this research and in that of Das & Arakeri (1998), is supported by the fact that the results obtained under this assumption show satisfactory agreement with the experimental features such as type of instability and spacing between vortices. In addition, the computations show that the ratio of the rate of growth of perturbations to the rate of change of the base flow is much larger than 1 for all cases considered, providing further support for the quasi-steady assumption. The neutral stability curves obtained from linear stability analysis suggest that a weakly nonlinear approach can be used in order to study further development of instability. Weakly nonlinear analysis shows that the amplitude of the most unstable mode is governed by the complex Ginzburg–Landau equation which reduces to the Landau equation if the amplitude is a function of time only. The coefficients of the Landau equation are calculated for two cases of the experimental data given by Das & Arakeri (1998). It is shown that the real part of the Landau constant is positive in both cases. Therefore, finite-amplitude equilibrium is possible. These results are in qualitative agreement with experimental data of Das & Arakeri (1998).

Particle dynamics during adiabatic expansion of a plasma bunch

The renormalization-group approach is used to obtain an exact solution to the self-consistent Vlasov kinetic equations for plasma particles in the quasi-neutral approximation. This solution describes the one-dimensional adiabatic expansion of a plasma bunch into a vacuum for arbitrary initial particle velocity distributions. Ion acceleration is studied for two-temperature Maxwellian and super-Gaussian initial electron distributions, which predetermine distinctly different ion spectra. The solution found is used to describe the acceleration of ions of two types. The relative acceleration efficiency of light and heavy ions as a function of atomic weights and number densities is analyzed. The solutions obtained are of practical importance in describing ion acceleration during the interaction of an ultrashort laser pulse with nanoplasma, for example, cluster plasma or plasma produced when thin foils are irradiated by a laser.

Velocity correlations, diffusion, and stochasticity in a one-dimensional system

We consider the motion of a test particle in a one-dimensional system of equal-mass point particles. The test particle plays the role of a microscopic "piston" that separates two hard-point gases with different concentrations and arbitrary initial velocity distributions. In the homogeneous case when the gases on either side of the piston are in the same macroscopic state, we compute and analyze the stationary velocity autocorrelation function C(t). Explicit expressions are obtained for certain typical velocity distributions, serving to elucidate in particular the asymptotic behavior of C(t). It is shown that the occurrence of a nonvanishing probability mass at zero velocity is necessary for the occurrence of a long-time tail in C(t). The conditions under which this is a t(-3) tail are determined. Turning to the inhomogeneous system with different macroscopic states on either side of the piston, we determine its effective diffusion coefficient from the asymptotic behavior of the variance of its position, as well as the leading behavior of the other moments about the mean. Finally, we present an interpretation of the effective noise arising from the dynamics of the two gases, and thence that of the stochastic process to which the position of any particle in the system reduces in the thermodynamic limit.

Multidimensional Infiltration with Arbitrary Surface Fluxes

A new solution to the multidimensional linearized Richards equation was derived using a Fourier integral transform. Exponential functional forms k = kseαψ and θ = θr + (θs − θr)eαψ were used to represent the hydraulic conductivity and pressure relation and the soil water release curve. The analytical solutions consider the conditions of time dependence and nonuniform distribution of rainfall intensity and arbitrary initial water content distribution with a water table. The analytical solutions can be used to predict the ponding time and to obtain the volumetric water content distribution over time and space.

Markov renewal theory for stationary ( m+1)-block factors: convergence rate results

This article continues work by Alsmeyer and Hoefs (Markov Process Relat. Fields 7 (2001) 325–348) on random walks ( S n ) n⩾0 whose increments X n are ( m+1)-block factors of the form ϕ( Y n− m ,…, Y n ) for i.i.d. random variables Y − m , Y − m+1 ,… taking values in an arbitrary measurable space ( S, S) . Defining M n =( Y n− m ,…, Y n ) for n⩾0, which is a Harris ergodic Markov chain, the sequence ( M n , S n ) n⩾0 constitutes a Markov random walk with stationary drift μ= E F m+1 X 1 where F denotes the distribution of the Y n 's. Suppose μ>0, let ( σ n ) n⩾0 be the sequence of strictly ascending ladder epochs associated with ( M n , S n ) n⩾0 and let ( M σ n , S σ n ) n⩾0 , ( M σ n , σ n ) n⩾0 be the resulting Markov renewal processes whose common driving chain is again positive Harris recurrent. The Markov renewal measures associated with ( M n , S n ) n⩾0 and the former two sequences are denoted U λ , U λ > and V λ >, respectively, where λ is an arbitrary initial distribution for ( M 0, S 0). Given the basic sequence ( M n , S n ) n⩾0 is spread-out or 1-arithmetic with shift function 0, we provide convergence rate results for each of U λ , U λ > and V λ > under natural moment conditions. Proofs are based on a suitable reduction to standard renewal theory by finding an appropriate imbedded regeneration scheme and coupling. Considerable work is further spent on necessary moment results.

Numerical consideration of quasilinear electron cloud dynamics in plasma

The dynamics of a hot electron cloud in the solar corona-like plasma based on the numerical solution of kinetic equations of weak turbulence theory is considered. Different finite difference schemes are examined to fit the exact analytical solutions of quasilinear equations in hydrodynamic limit (gas-dynamic solution). It is shown that the scheme suggested demonstrates correct asymptotic behavior and can be employed to solve initial value problems for an arbitrary initial electron distribution function.

Ion acceleration during adiabatic plasma expansion: Renormalization group approach

The renormalization group approach is applied to derive an exact solution to self-consistent Vlasov kinetic equations for plasma particles in the quasineutral approximation. The solution obtained describes the one-dimensional adiabatic expansion into vacuum of a plasma bunch with arbitrary initial velocity distributions of the electrons and ions. The ion acceleration is investigated for both a Maxwellian two-temperature initial electron distribution and a super-Gaussian initial electron distribution.

Renyi's divergence and entropy rates for finite alphabet Markov sources

In this work, we examine the existence and the computation of the Renyi divergence rate, lim/sub n/spl rarr//spl infin// 1/n D/sub /spl alpha//(p/sup (n)//spl par/q/sup (n)/), between two time-invariant finite-alphabet Markov sources of arbitrary order and arbitrary initial distributions described by the probability distributions p/sup (n)/ and q/sup (n)/, respectively. This yields a generalization of a result of Nemetz (1974) where he assumed that the initial probabilities under p/sup (n)/ and q/sup (n)/ are strictly positive. The main tools used to obtain the Renyi divergence rate are the theory of nonnegative matrices and Perron-Frobenius theory. We also provide numerical examples and investigate the limits of the Renyi divergence rate as /spl alpha//spl rarr/1 and as /spl alpha//spl darr/0. Similarly, we provide a formula for the Renyi entropy rate lim/sub n/spl rarr//spl infin// 1/n H/sub /spl alpha//(p/sup (n)/) of Markov sources and examine its limits as /spl alpha//spl rarr/1 and as /spl alpha//spl darr/0. Finally, we briefly provide an application to source coding.