Stochastic dynamics in a two-level model of disorder: comparison of mean-field and exact solutions

Stochastic dynamics in the presence of quenched disorder (e.g., diffusion in a random medium) is generally treated in a suitable mean-field or effective medium approximation. While numerical simulations may help determine the accuracy of such approximations in specific models, there are relatively few instances in which analytic solutions are possible, to enable a precise comparison to be made with the mean-field results. We consider in this paper a simple but general model of quenched disorder in which a system variablex jumps stochastically between two valuesx _a andx _b . However, in each level there occurs with a certain probability a branch (or internal) state into which the system may fall, and from which a jump to the other level is possible only after a return to the original (or ‘active’) state. Four different configurations of the states of the system are thus possible, and the transitions between the states are governed by Markovian transition probabilities. The moments ofx and its autocorrelation function are computed in each case, and then configuration-averaged over the four realizations. This represents the exact solution. Next, a mean-field theory of the dynamics is developed: this turns out to involve an effective waiting-time density at each of the two levels that is non-exponential in time, so that the mean-field dynamics is a non-Markovian alternating renewal process. The moments and autocorrelation ofx are again computed, and compared with the exact solutions. The extent of the differences at both short and long times is elucidated, and a numerical comparison is presented for the case of maximal disorder.     

A comparison of projection operator formalisms for the study of self-diffusion

The utility of projection operator formalisms for describing the dynamics of many-body systems is studied, and the compatibility of these formalisms with certain approximation schemes is evaluated in the light of known behavior of such systems. For simplicity the investigation is limited to the study of Brownian motion. Specifically, a memory kernel formalism and a kinetic equation formalism are compared for the calculation of the time evolution of the momentum autocorrelation function. Both perturbation expansions and averaged propagator approximations are investigated. The results from these studies suggest that the long-time behavior of the momentum autocorrelation function is sensitive to the long-range nature of the interparticle potential.This work was supported in part by the National Science Foundation under Grant GK-19360X.     

Anderson impurity model: Vertex corrections within the finite U non-crossing approximation

We revise the simplest possible approximations to solve numerically the vertex equations for the single impurity Anderson model (SIAM) within the finite U non-crossing approximation (UNCA), considering the self-energies at lowest order in the 1/N diagrammatic expansion. We introduce an approximation to the vertex corrections that includes the double energy dependence and compare it with an approximation (NCAf²v) that neglects a second energy argument. Finally, we analyse the influence of the different approximations on the estimated Kondo scale for simple electronic models.     

Generalized Malkmus line intensity distribution for CO2 infrared radiation in Doppler broadening regime

A new line intensity distribution function is introduced in order to improve the accuracy of statistical narrow-band (SNB) models in the Doppler line broadening regime and in a wide temperature range. This distribution function generalizes the Malkmus distribution through a parameter enabling to adjust the contribution of small line intensities. This new model is shown to enhance significantly SNB accuracy for CO₂ uniform column radiation, especially at low temperatures. For non-uniform columns, Lindquist-Simmons type approximations are derived for this new intensity distribution. Their results are in much closer agreement with line by line results than Curtis-Godson approximation ones when steep temperature gradients are considered.     

Angular Velocity Distribution of the Electric Microfield in Plasma

In theory of the spectral line shapes, the conventional scheme use two approximations for the local electric field (microfield) due to all charged particles of the plasma. The quasi‐static approximations for the ions and the impact approximation for the electrons. The first approximation consists to say that the electric field is constant during the characteristic time. In this work we shall transpose the idea of the first approximation, to the angular velocity of the microfield whereas its strength is kept constant and equal to its mean value. We shall use the Holtsmark approach and the independent particles model (due to Margeneau and Lewis) to compute the static distribution function of the angular velocity of the microfield. In the first approach (Holtsmark), the distribution shows a Lorentzian behavior, whereas the second approach (Margenau and Lewis) shows a gaussian behavior. Subsequently, we have applied the obtained static distribution to show the effect on the broadening of Lyman‐alpha line for a plasma composed of He⁺ ions. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Free energies of interfaces from quasi-harmonic theory: A critical appraisal

A critical assessment is given of the quasi-harmonic approximation, and various approximations to the quasi-harmonic approximation, with regard to predicting the free energy and atomic structure of grain boundaries in silicon at elevated temperatures. The quasi-harmonic results are compared with those obtained by molecular dynamics and thermodynamic integration. It is found that the quasi-harmonic approximation yields accurate excess free energies and atomic structures of grain boundaries at 1,000 K. The anharmonic contribution to the free energy that is absent in the quasi-harmonic contribution is virtually the same at a grain boundary in Si and in the perfect crystal. The second-moment and Einstein approximations to the full quasi-harmonic theory yield unreliable free energies, but reasonably accurate atomic structures. However, excess free energies are quite well described by the Einstein model. It is concluded that the quasi-harmonic approximation works remarkably well in silicon. The simplest approximations to the phonon density of states lead to unreliable results for the free energy, but cancellation of errors occurs to a large extent when excess free energies are computed.     

Closure Approximations for Passive Scalar Turbulence: A Comparative Study on an Exactly Solvable Model with Complex Features

Some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep. This model has a general geometric structure of a jet flow with transverse disturbances, which occur in a number of contexts, and it encompasses a wide variety of possible spatio-temporal statistical structures for the velocity field, including strong long-range correlations. Even though the Eulerian and Lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader-than-Gaussian intermittency, this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations. Our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly. In particular it is shown that the direct interaction approximation (DIA), one of the most sophisticated closure approximations available, fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions, even though the equations it produces are rather complicated and difficult to analyze. Two alternative closure approximations, the Modified DIA (MDIA) and the Renormalized Lagrangian Approximation (RLA), with different levels of sophistication, both are simpler to use than the DIA and perform better. In particular, it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the MDIA is even able to capture intermittency effects.     

Existence, Stability, and Convergence of Solutions of Discrete Velocity Models to the Boltzmann Equation

We prove the convergence of finite-difference approximations to solutions of the Boltzmann equation. An essential step is the proof of convergence of discrete approximations to the collision integral. This proof relies on our previous results on the consistency of this approximation. For the space-homogeneous problem we prove strong convergence of our discrete approximation to the strong solution of the Boltzmann equation. In the space-dependent case we prove weak convergence to DiPerna–Lions solutions.     

Diffusion in lattice Lorentz gases : Nearest scatterers approximation

The expressions for diffusion coefficients and for velocity autocorrelation functions of lattice Lorentz gases are derived both in the nearest scatterers and Boltzmann approximations. The results are obtained for linear chain, square, triangular, simple cubic, body centred cubic, face centred cubic and face centred hyper cubic lattices. The diffusion coefficients are compared with those from the effective medium approximation for the square lattice and with computer simulation results for triangular, simple cubic and body centred cubic lattices.     

Local methods for constructing stationary distribution functions of systems of stochastic differential Langevin-type equations: Noise influence on simple bifurcation

The influence is considered of two additive correlated noise effects on a two-dimensional quadratic-nonlinear system describing the behavior of two hydrodynamic modes. Using the method of Gaussian approximation, local characteristics of the distribution function are calculated, which are used to construct the global distribution function with the aid of the method of fraction-rational approximations. It is shown that for a system at whose bifurcation point the asymptotic stability is lost, in an expanded space of parameters (bifurcation parameter in the absence of noise plus noise parameters) there appears an instability zone within which the stationary distribution function does not exist. The effect of noise correlation on the stationary characteristics of the system is studied.     

Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions

The phase function for solar light scattering by large particles such as cloud droplets is strongly anisotropic due to very strong peaking in the forward direction. This creates numerical difficulties when attempting to calculate accurate reflected and transmitted radiances, which are important for remote sensing of atmospheric and surface properties. A popular approach uses the delta function to approximate the forward-scattering peak in a fraction of energy and a limited number of polynomial terms or a geometrically truncated function for the remaining fraction (so-called truncation approximations). This article compares and discusses several methods for fast and accurate calculations using truncation approximations. When using a single truncation approximation for all scattering orders, large biases appear in directions near the solar and anti-solar points. As shown here, high accuracy can be obtained using different truncation approximations depending on the order of scattering. Of particular importance is the use of phase functions close to the exact phase functions for the first few orders of scattering. Applying the method in combination with the Monte Carlo (MC) method, in which the truncation fraction for a scattering order depends on the scattering angle at the previous scattering event, obtains accurate radiance calculations under almost all geometrical and optical conditions, including in directions near the solar point. Because the method also reduces computational noise due to the MC sampling of radiance, it is useful for fast and accurate radiance calculations for cloudy atmospheres.     

Evaluation of thesscf andpy approximations for quadrupolar fluids

Two approximations, the single super chainf-expansion (sscf), and Percus-Yevick (py) approximation, are evaluated for a molecular fluid in which the molecules interact with a pair potential, that is the sum of Lennard-Jones and quadrupole-quadrupole parts at two values of reduced quadrupole moment. These results are compared with Monte-Carlo results. Except for the harmonic coefficienth (222;r), thesscf approximation seems to be quite accurate for the lower value of quadrupole moment but at higher valuespy approximation produces much better results except forh(220;r).     

Structure and equilibrium optical properties of liquid CS2

Equilibrium optical properties of liquid carbon disulphide (CS₂), i.e. its refractive index, Kerr constant and depolarized light scattering intensity are calculated using two models of optical response of the fluid. The first one, the point polarizability approximation (PPA) assumes that a point dipole, proportional to the total polarizability, is induced in each molecule. The second one, the point atomic polarizability approximation (PAPA) assumes that point dipoles are induced in individual atomic sites.

The symmetry components of the intermolecular pair distribution function needed to calculate optical properties of the fluid are obtained by Monte Carlo computer simulation on a hard triatomic model of CS₂ as well as by two approximate approaches using this same model. The approximations are both based on the use of the site superposition approximation (SSA) for the intermolecular pair distribution function. In the first approach, the SSA pair distribution is obtained using the Monte Carlo site-site functions and in the second using the site-site functions calculated using the reference interaction site model (RISM) equations.

Extensive comparisons are carried out between the Monte Carlo results and the two approximations in order to examine the influence of the SSA and the RISM equations on optical properties of CS₂. We conclude that, while these approximations, especially the SSA, have a substantial effect on individual symmetry components of the pair distribution, they predict measurable optical properties with satisfactory accuracy.     

一般幂指数中心势作用下的波包回归和部分回归

利用WKB近似和自关联函数方法,我们研究了一般幂指数中心势V(r)=rk (-20)作用下波包的回归和部分回归。对于排斥势(>0, k>0), 势是一长程势,量子化能级结构中只有一个量子数,波包的回归结构和一维幂指数势的情况类似。这一结果表明能级结构相同的体系具有相同的波包回归结构。 对于吸引势,能级结构中有两个量子数, 当 k取不同的值时,波包的回归结构不同。对于库仑吸引势,波包回归和部分回归出现; 但是对于其它的k值, 经过一段时间后,波包出现坍塌。本文的研究对于探讨里德堡原子和分子中电子运动的经典极限提供了一个新的方法。     

一般幂指数中心势作用下的波包回归和部分回归

利用WKB近似和自关联函数方法,我们研究了一般幂指数中心势V(r)=γrk (-20)作用下波包的回归和部分回归.对于排斥势(γ>0, k>0), 势是一长程势,量子化能级结构中只有一个量子数,波包的回归结构和一维幂指数势的情况类似.这一结果表明能级结构相同的体系具有相同的波包回归结构.对于吸引势,能级结构中有两个量子数, 当 k取不同的值时,波包的回归结构不同.对于库仑吸引势,波包回归和部分回归出现;但是对于其它的k值, 经过一段时间后,波包出现坍塌.本文的研究对于探讨里德堡原子和分子中电子运动的经典极限提供了一个新的方法.     

Correction factors for a total scatter/backscatter nephelometer

Total aerosol scattering and backscattering atmospheric values are typically obtained with an integrating nephelometer. Due to design limitations, measurements do not cover the full (0°–180°) angular range, and correction factors are necessary. The effect of angle cutoff is examined for a number of particle size distributions and refractive indices. Scattering data for sub-micron particles can be corrected by the use of a modified Anderson approximation, while data for larger particle distributions can be approximated by a function of the effective size parameter. Correction factors for the hemispheric backscatter ratio are found to be small if nonsphericity is assumed. Such approximations will help more accurate corrections for angle range, particularly at large size parameter values.     

Radiation patterns of the HE11 mode and Gaussian approximations

The problem of the approximation of the HE₁₁ radiation pattern by a Gaussian distribution is discussed. A numerical comparison between the HE₁₁ far-field theoretical pattern, and the Gaussian approximations derived by Abrams and by Crenn, permits and evaluation of the precision of these approximations. A new optimized HE₁₁ Gaussian approximation is calculated: the value of r_o=0.421a (or w_o=0.596a) for the beam radius at the waist is demonstrated to give the best HE₁₁ Gaussian approximation in the far-field and is very close to the result given by Crenn, while the Abrams value is less precise. The calculations are extended to the near-field. Universal curves for intensity, amplitude and power distribution are given for the HE₁₁ radiated mode. These results are of interest for laser waveguide applications and for plasma ECRH transmission systems.     

Spectral representations for the memory kernel characterizing self-diffusion

Approximate spectral representations are developed for the memory kernel which characterizes self-diffusion. These spectral representations are based upon approximate eigenfunctions constructed via the Rayleigh variational principle. A heuristic model is developed first in an effort to provide physical insight into the nature of the approximations employed, and then a number of specific trial functions are examined. These trial functions include sums of identical one- and two-particle functions as well as linear combinations of hydrodynamical variables. The results from these spectral representations indicate that the long-time behavior of the memory kernel (and thereby of the momentum autocorrelation function) is sensitive to the long-range effects of the interparticle potential. In addition, the equivalence of most of these spectral representations to specific low-order perturbation approximations is demonstrated.     

Analytic Approximations for the Velocity of Field-Driven Ising Interfaces

We present analytic approximations for the field, temperature, and orientation dependences of the interface velocity in a two-dimensional kinetic Ising model in a nonzero field. The model, which has nonconserved order parameter, is useful for ferromagnets, ferroelectrics, and other systems undergoing order–disorder phase transformations driven by a bulk free-energy difference. The solid-on-solid (SOS) approximation for the microscopic surface structure is used to estimate mean spin-class populations, from which the mean interface velocity can be obtained for any specific single-spin-flip dynamic. This linear-response approximation remains accurate for higher temperatures than the single-step and polynuclear growth models, while it reduces to these in the appropriate low-temperature limits. The equilibrium SOS approximation is generalized by mean-field arguments to obtain field-dependent spin-class populations for moving interfaces, and thereby a nonlinear-response approximation for the velocity. The analytic results for the interface velocity and the spin-class populations are compared with Monte Carlo simulations. Excellent agreement is found in a wide range of field, temperature, and interface orientation.     

The possibility of describing a two-particle quantum system and interparticle correlations in the self-consistent field approximation

The possibility of determining the energy spectrum and the wave function structure of the stationary states of a model two-particle system is examined in the self-consistent field approximation. The results of exact calculations are compared with the data obtained in the self-consistent field approximation with the use of orthogonal and nonorthogonal orbitals. The possibility of using these approximations for describing the interparticle quantum correlations is discussed.