Double Delta Function Research Articles

A BOSE-EINSTEIN CONDENSATE WITH PT-SYMMETRIC DOUBLE-DELTA FUNCTION LOSS AND GAIN IN A HARMONIC TRAP: A TEST OF RIGOROUS ESTIMATES

We consider the linear and nonlinear Schrödinger equation for a Bose-Einstein condensate in a harmonic trap with <em>PT</em>-symmetric double-delta function loss and gain terms. We verify that the conditions for the applicability of a recent proposition by Mityagin and Siegl on singular perturbations of harmonic oscillator type self-adjoint operators are fulfilled. In both the linear and nonlinear case we calculate numerically the shifts of the unperturbed levels with quantum numbers n of up to 89 in dependence on the strength of the non-Hermiticity and compare with rigorous estimates derived by those authors. We confirm that the predicted 1/<em>n</em><sup>1/2</sup> estimate provides a valid upper bound on the shrink rate of the numerical eigenvalues. Moreover, we find that a more recent estimate of log(<em>n</em>)/<em>n</em><sup>3/2</sup> is in excellent agreement with the numerical results. With nonlinearity the shrink rates are found to be smaller than without nonlinearity, and the rigorous estimates, derived only for the linear case, are no longer applicable.

Generalized Continuity Equation for Quasi-Hermitian Hamiltonians

The continuity relation is generalized to quasi-Hermitian one-dimensional Hamiltonians. As an application we show that the reflection and transmission coefficients computed with the generalized current obey the conventional unitarity relation for the continuous double delta function potential.

Eigenvalues of Particle Bound in Single and Double Delta Function Potentials through Numerical Analysis

Eigenvalues of Particle Bound in Single and Double Delta Function Potentials through Numerical Analysis

Eigenvalues of Particle Bound in Single and Double Delta Function Potentials through Numerical Analysis

This study employs the use of the fourth order Numerov scheme to determine the eigenstates and eigenvalues of particles, electrons in particular, in single and double delta function potentials. For the single delta potential, it is found that the eigenstates could only be attained by using specific potential depths. The depth of the delta potential well has a value that varies depending on the delta strength. These depths are used for each well on the double delta function potential and the eigenvalues are determined. There are two bound states found in the computation, one with a symmetric eigenstate and another one which is antisymmetric. Keywords—Double Delta Potential, Eigenstates, Eigenvalue, Numerov Method, Single Delta Potential

Bounded stochastic shell mixing model for turbulent mixing of multiple scalars with arbitrary diffusivities

In the paper by Xia et al. [Flow, Turbul. Combust. 85, 689 (2010)], we derived a stochastic shell mixing model (SSMM) for turbulent mixing of multiple scalars with arbitrary diffusivities. The framework uses a Monte Carlo scheme to advance notional particles that move with the fluid and has scalar concentrations that are subdivided into “shells” that loosely represent a spectral decomposition of the scalar fluctuations. The variance within each shell is equivalent to the scalar spectrum evaluated at the corresponding wavenumber. Small-scale phenomena such as differential diffusion and the dependence of mixing on the scalar length scale are captured by the inherently spectral nature of the model. Furthermore, the sum of the scalar over shells for each notional particle simultaneously provides a fine-grained representation of the joint scalar probability density function (PDF). This aspect of the model was introduced, but not fully explored in the earlier paper, which focused on scalar fluctuations arising from a uniform mean gradient. At the level of this model, the scalar PDF reduces to a joint Gaussian. In this study, we explore more complex mixing scenarios, including the initial double delta function, which arises when two streams are brought into sudden contact. This introduces two important complexities that are the focus of this paper. The first is the shape changes to the PDF that result as the scalar PDF relaxes into a Gaussian form. The second is due to the bounds that the scalar must satisfy. Monte Carlo schemes inherently violate bounds; we have modified the model so as to ensure the bounds are satisfied by nearly all of the particles (bounded SSMM or BSSMM). Extensive comparisons between the BSSMM predictions and direct numerical simulations (DNS) are made for two decaying scalars (with different diffusivities) in forced, isotropic turbulence. Overall the agreement is very good. Additionally, we test a conjecture made in deriving the BSSMM model, namely the rate of mixing of a scalar is assumed to be a function of its spectrum, but not its PDF. Using DNS, we contrast the mixing rate of two scalars with identical PDFs, but different spectra against two scalars with identical spectra, but different PDFs. We conclude from this comparison that the mixing rate is much more sensitive to the scalar spectrum than to its PDF, in support of the assumption used to derive the new mixing model.

Stochastic Shell Model for Turbulent Mixing of Multiple Scalars with Mean Gradients and Differential Diffusion

In this paper, we develop a shell model for the velocity and scalar concentrations that, by design, is consistent with the eddy damped quasi-normal Markovian (EDQNM) model for multiple mixing scalars. We review the realizable form of the EDQNM model derived by Ulitsky and Collins (J Fluid Mech 412:303–329, 2000), which forms the basis for the shell model. The equations governing the velocity and scalar within each shell are stochastic ordinary differential equations with drift and diffusion terms chosen so that the velocity variance, velocity–scalar cross correlations, and scalar–scalar cross correlations within each shell precisely match the EDQNM model predictions. Consequently, shell averages can be thought of as a representation of the discrete three-dimensional spectrum. An advantage the shell model has over the original EDQNM equations is that the sum of each realization over the shells is a model for the fine-grained, joint velocity/scalar probability density function (PDF). Indeed, this provides some of the motivation for the development of the model. We cannot exploit this feature in the present study of the mixing of two scalars with uniform mean gradients, as the PDF is a joint Gaussian throughout (and hence the correlation matrix completely defines the distribution). The model is capable of predicting Lagrangian correlation functions for the scalar, scalar dissipation and velocity. We find the predictions of the model are in good qualitative agreement with direct numerical simulations by Yeung (J Fluid Mech 427:241–274, 2001). Eventually we will apply the shell model to scalars that are initially highly non-Gaussian (e.g., double delta function) and observe the relaxation towards a Gaussian. As the shell model contains information on the spectral distribution of the scalar field, the relaxation rate will depend upon the length and time scales of the turbulence and the scalar fields, as well as the molecular diffusivities of the species. The full capabilities of the PDF predictions of the model will be the subject of a future publication.

Assumed Probability Density Functions for Shallow and Deep Convection

The assumed joint probability density function (PDF) between vertical velocity and conserved temperature and total water scalars has been suggested to be a relatively computationally inexpensive and unified subgrid‐scale (SGS) parameterization for boundary layer clouds and turbulent moments. This paper analyzes the performance of five families of PDFs using large‐eddy simulations of deep convection, shallow convection, and a transition from stratocumulus to trade wind cumulus. Three of the PDF families are based on the double Gaussian form and the remaining two are the single Gaussian and a Double Delta Function (analogous to a mass flux model). The assumed PDF method is tested for grid sizes as small as 0.4 km to as large as 204.8 km. In addition, studies are performed for PDF sensitivity to errors in the input moments and for how well the PDFs diagnose some higher‐order moments. In general, the double Gaussian PDFs more accurately represent SGS cloud structure and turbulence moments in the boundary layer compared to the single Gaussian and Double Delta Function PDFs for the range of grid sizes tested. This is especially true for small SGS cloud fractions. While the most complex PDF, Lewellen‐Yoh, better represents shallow convective cloud properties (cloud fraction and liquid water mixing ratio) compared to the less complex Analytic Double Gaussian 1 PDF, there appears to be no advantage in implementing Lewellen‐Yoh for deep convection. However, the Analytic Double Gaussian 1 PDF better represents the liquid water flux, is less sensitive to errors in the input moments, and diagnoses higher order moments more accurately. Between the Lewellen‐Yoh and Analytic Double Gaussian 1 PDFs, it appears that neither family is distinctly better at representing cloudy layers. However, due to the reduced computational cost and fairly robust results, it appears that the Analytic Double Gaussian 1 PDF could be an ideal family for SGS cloud and turbulence representation in coarse‐grid CRMs, mesoscale models, and GCMs if the required input moments can be predicted or diagnosed accurately.

Small-Scale and Mesoscale Variability in Cloudy Boundary Layers: Joint Probability Density Functions

The joint probability density function (PDF) of vertical velocity and conserved scalars is important for at least two reasons. First, the shape of the joint PDF determines the buoyancy flux in partly cloudy layers. Second, the PDF provides a wealth of information about subgrid variability and hence can serve as the foundation of a boundary layer cloud and turbulence parameterization. This paper analyzes PDFs of stratocumulus, cumulus, and clear boundary layers obtained from both aircraft observations and large eddy simulations. The data are used to fit five families of PDFs: a double delta function, a single Gaussian, and three PDF families based on the sum of two Gaussians. Overall, the double Gaussian, that is binormal, PDFs perform better than the single Gaussian or double delta function PDFs. In cumulus layers with low cloud fraction, the improvement occurs because typical PDFs are highly skewed, and it is crucial to accurately represent the tail of the distribution, which is where cloud occurs. Since the double delta function has been shown in prior work to be the PDF underlying mass-flux schemes, the data analysis herein hints that mass-flux simulations may be improved upon by using a parameterization built upon a more realistic PDF.

Small-Scale and Mesoscale Variability of Scalars in Cloudy Boundary Layers: One-Dimensional Probability Density Functions

A key to parameterization of subgrid-scale processes is the probability density function (PDF) of conserved scalars. If the appropriate PDF is known, then grid box average cloud fraction, liquid water content, temperature, and autoconversion can be diagnosed. Despite the fundamental role of PDFs in parameterization, there have been few observational studies of conserved-scalar PDFs in clouds. The present work analyzes PDFs from boundary layers containing stratocumulus, cumulus, and cumulus-rising-into-stratocumulus clouds. Using observational aircraft data, the authors test eight different parameterizations of PDFs, including double delta function, gamma function, Gaussian, and double Gaussian shapes. The Gaussian parameterization, which depends on two parameters, fits most observed PDFs well but fails for large-scale PDFs of cumulus legs. In contrast, three-parameter parameterizations appear to be sufficiently general to model PDFs from a variety of cloudy boundary layers. If a numerical model ignores subgrid variability, the model has biases in diagnoses of grid box average liquid water content, temperature, and Kessler autoconversion, relative to the values it would obtain if subgrid variability were taken into account. The magnitude of such biases is assessed using observational data. The biases can be largely eliminated by three-parameter PDF parameterizations. Prior authors have suggested that boundary layer PDFs from short segments are approximately Gaussian. The present authors find that the hypothesis that PDFs of total specific water content are Gaussian can almost always be rejected for segments as small as 1 km.

Macroscopic description of phase-coherent transport in quasi-one-dimensional superconducting structures

The crossover between ideal Josephson behavior and uniform superconducting flow is studied by solving exactly the Ginzburg-Landau equation for a one-dimensional superconductor in the presence of an effective delta function potential of arbitrary strength. The relation between self-consistency and charge conservation is pointed out. As the effective scattering is turned off, the pairs of Josephson solutions with equal current evolve into a uniform and a solitonic solution with nonzero phase offset. An experimental realization of the crossover is proposed in which the temperature is used as the driving parameter. We also study the case of a double delta function, which may quantitatively describe a SNSNS structure. In the Josephson limit of strong scattering, the passage of Cooper pairs through a double barrier structure may be viewed as the nonlinear counterpart of resonant tunneling. For interbarrier distances smaller than d = πξ( T) no supercurrent-carrying solutions exist. For distances between d 0 and 2 d 0, four solutions appear. The two symmetric solutions obey a current-phase relation of sin(Δϕ/2). The two asymmetric solutions satisfy Δϕ = π for all values of the current. As the distance exceeds nd 0, a new group of four solutions appears which contain n soliton-type oscillations between the barriers. It is shown that, for a SNSNS structure, the dependence of the critical current on the temperature differs qualitatively from that of the SNS case.

Direct numerical simulations of the turbulent mixing of a passive scalar

The evolution of scalar fields, of different initial integral length scales, in statistically stationary, homogeneous, isotropic turbulence is studied. The initial scalar fields conform, approximately, to ‘‘double-delta function’’ probability density functions (pdf ’s). The initial scalar-to-velocity integral length-scale ratio is found to influence the rate of the subsequent evolution of the scalar fields, in accord with experimental observations of Warhaft and Lumley [J. Fluid Mech. 88, 659 (1978)]. On the other hand, the pdf of the scalar is found to evolve in a similar fashion for all the scalar fields studied; and, as expected, it tends to a Gaussian. The pdf of the logarithm of the scalar-dissipation rate reaches an approximately Gaussian self-similar state. The scalar-dissipation spectrum function also becomes self-similar. The evolution of the conditional scalar-dissipation rate is also studied. The consequences of these results for closure models for the scalar pdf equation are discussed.

General theory of polycrystalline ENDOR patterns. Effects of finite EPR and ENDOR component linewidths

Recently (I) we presented a general method for calculating ENDOR spectra of a randomly oriented polycrystalline (powder) S = 4 paramagnet (2, 3) that has g and hyperfme tensors of arbitrary symmetry and relative orientation. That study (Paper I) laid a foundation for the detailed analysis of polycrystalline ENDOR spectra, as well as for the heuristic categorization of such spectra, much as EPR spectra long have been classified (isotropic, axial, rhombic (4)). To this end the calculations in I used a “double delta function” approach, employing a-function EPR and ENDOR component lineshapes. They showed that polycrystalline ENDOR spectra can exhibit a rich array of features when the static field is set to a g value where the EPR signal arises from a well-defined set of orientations of the paramagnet, rather than a single crystal-type setting. The principal values and relative orientation of a hyperline tensor can be obtained by examining such angle-selected spectra obtained as the observing g value (static field) is moved across the EPR envelope. We now have generalized the analysis to permit full simulations of polycrystalline ENDOR spectra by including finite EPR and ENDOR component linewidths. The calculations presented here show that the two types of features that arise in the double delta-function limit, divergences and maxima, both persist as absorption peaks in the full simulations. These ENDOR peaks broaden equally as the ENDOR component linewidth, W,,, increases, but in general they do not broaden equally as the EPR component linewidth, W,, increases. Instead, the rate at which a peak broadens with W, is related to (&/dg), the rate of change in its ENDOR frequency with observing g value. The essential features of the approach developed in I are first summarized, and then are extended to include EPR and ENDOR component linewidths. When the external field is set to an arbitrary value, Ho, within a a-function EPR envelope, the EPR signal intensity, and thus an ENDOR signal, arises from the selected molecular orientations associated with the curve on the unit sphere, S, comprising points for which the orientation dependent spectroscopic splitting factor, g = g(B, @), has a fixed value defined by the strength of the observing field: gmi, < g = hv/@& < g,,,. H’owever, although g is constant over the locus of points, S, in general the angledependent hyperfine coupling, A = A(& 4), is not, and the ENDOR pattern in general is more complex than a single crystal-like spectrum. The net ENDOR intensity at radiofrequency, Y, is determined by the sum of the probabilities that the

Model-Based Restoration Procedure For Small, Low Resolution Optical Images

A deterministic model-based restoration procedure is presented. The algorithm is effective for the restoration of coarsely sampled images degraded by diffraction and noise. The a priori information, an assumed parametric object model, is used to arrive at the solution. The object model is convolved with the optical system and sensing mechanism degradations and then matched against the limited number of available samples. The unknown parameters are then estimated using a numerical least mean square error optimization procedure. The method has been tested with a double delta function model via digital simulations. A zero-mean, additive, white Gaussian background noise process was assumed. The technique requires a high signal-to-noise ratio (SNR) to resolve the doublet with a separation distance smaller than the detector width. With increasing separation, good restoration can be achieved at low SNR. The procedure is applicable to restore generalized objects.

Susceptibilities of spin glass models: a Monte Carlo study

The Monte Carlo method has been used to study the behaviour of Ising model spin glasses. Both the double delta function and Gaussian distributions of interaction strengths have been considered. No evidence for cusp-like behaviour is obtained, but the susceptibilities appear to diverge at low temperature.