Showing papers on "Dirac delta function published in 2003"
TL;DR: In this article, an integral representation of the fractional white noise as generalized Wiener integral was proposed for fractional Brownian motions BtH with arbitrary Hurst coefficients 0
138 citations
TL;DR: In this paper, the scattering properties of regularizing finite-range potentials constructed in the form of squeezed rectangles, which approximate the first and second derivatives of the Dirac delta function δ(x), are studied in the zero-range limit.
Abstract: The scattering properties of regularizing finite-range potentials constructed in the form of squeezed rectangles, which approximate the first and second derivatives of the Dirac delta function δ(x), are studied in the zero-range limit. Particularly, for a countable set of interaction strength values, a non-zero transmission through the point potential δ'(x), defined as the weak limit (in the standard sense of distributions) of a special dipole-like sequence of rectangles, is shown to exist when the rectangles are squeezed to zero width. A tripole sequence of rectangles, which gives in the weak limit the distribution δ''(x), is demonstrated to exhibit the total transmission for a countable sequence of the rectangle's width that tends to zero. However, this tripole sequence does not admit a well-defined point interaction in the zero-range limit, making sense only for a finite range of the regularizing rectangular-like potentials.
93 citations
TL;DR: In this paper, the delta function of Dirac's delta function was compared to the Dirac delta function in terms of the number of possible references, and the results showed that Dirac was much better than Dirac in the sense that it yielded only 12,100 references.
Abstract: Now this is not as simple as it sounds. It does require some finesse possessed, no doubt, by the average ten-year old (but not necessarily by older, less capable surfers). If one is naive enough to enter delta function in Google and click on search, the magician takes only 0.16 seconds to produce a staggering 1,100,000 possible references. Several lifetimes would not suffice to check all of these out. To make the search more meaningful, we enter \"delta function\" in quotes. This produces a less stupendous 58,600 references. As even this is too much, we try Dirac delta function, to get 52,500 references not much of an improvement. Once again, \"Dirac delta function\" is much better, because Google then locates only 12,100 references. \"Dirac's delta function\" brings this down to 872, while \"the delta function of Dirac\" yields a comfortable (but not uniformly helpful) 19 references.
40 citations
TL;DR: In this article, the radial delta function contributions are avoided by the judicious introduction of a step function, which can be applied to confirm some subtle details about the fields of point electric and magnetic dipoles.
Abstract: In calculations involving the divergence, curl, or Laplacian operators in spherical polar coordinates, the radial delta function contributions are sometimes inadvertently lost. This loss can be avoided by the judicious introduction of a step function. We apply this trick to confirm some subtle details about the fields of point electric and magnetic dipoles.
38 citations
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01 Jan 2003
TL;DR: In this paper, a volume-to-surface integral transformation for 2D aniotropic elasticity with body forces was proposed. And the convergence of the dual reciprocity method for Poisson's equation was studied.
Abstract: On the treatment of domain integrals in BEM The use of Dirac delta functions for reducing domain loading integrals in plate problems The multiple-reciprocity method for elastic problems with arbitrary body force Exact volume-to-surface integral transformation for 2D aniotropic elasticity with body forces Treatment of body forces in single-domain boundary integral equation method for anisotropic elasticity Generalized body forces in multi-field problems with material anisotropy Transformation of domain integrals to boundary intregrals in BEM analysis of plate and shell structures Recent developments in dual reciprocity method using compactly supported radial basis functions On the convergence of the dual reciprocity method for Poisson's equation.
27 citations
TL;DR: In this article, the nonvanishing quark condensate as a signal of spontaneous chiral symmetry breaking of the QCD vacuum brings about a delta-function singularity at $x=0$ in the chirally odd twist-3 distribution of the nucleon.
Abstract: By making full use of the field theoretical nature of the chiral quark soliton model, we demonstrate that the nonvanishing quark condensate as a signal of the spontaneous chiral symmetry breaking of the QCD vacuum brings about a delta-function singularity at $x=0$ in the chirally odd twist-3 distribution $e(x)$ of the nucleon. This singularity in $e(x),$ which would be observed as a sizable violation of the 1st moment sum rule, is then interpreted as giving a very rare case that the nontrivial vacuum structure of QCD manifests in an observable of a localized QCD excitation, i.e., the nucleon.
26 citations
TL;DR: In this paper, the authors studied bound states of the Schrodinger equation for an attractive potential with any finite number (P) of Dirac delta-functions in Rn where n = 1, 2, 3,.... The potential is radially symmetric for n ≥ 2.
Abstract: We have studied bound states of the Schrodinger equation for an attractive potential with any finite number (P) of Dirac delta-functions in Rn where n = 1, 2, 3, .... The potential is radially symmetric for n ≥ 2 and is given as V(r) = −2/2m ∑Pi = 1 σiδ(r − ri) where σi > 0, r1 2l + n − 2 and none otherwise. We have also proven that there are at most P positive roots for the equation X22(k) = 0 where X = (X11X21X12X22) = MPMP−1 ... M1 and Mi SL(2, R) are the particular transfer matrices mentioned above.
26 citations
TL;DR: In this paper, the authors construct specific examples in which the curvature corresponding gab(xi;λ) becomes a Dirac delta function and gets concentrated on the horizon when the limit λ→0 is taken, but the action remains finite.
Abstract: A class of metrics gab(xi) describing spacetimes with horizons (and associated thermodynamics) can be thought of as a limiting case of a family of metrics gab(xi;λ)without horizons when λ→0. We construct specific examples in which the curvature corresponding gab(xi;λ) becomes a Dirac delta function and gets concentrated on the horizon when the limit λ→0 is taken, but the action remains finite. When the horizon is interpreted in this manner, one needs to remove the corresponding surface from the Euclidean sector, leading to winding numbers and thermal behavior. In particular, the Rindler spacetime can be thought of as the limiting case of (horizon-free) metrics of the form [g00=e2+a2x2; gμν=-δμν] or [g00=-gxx=(e2+4a2x2)1/2, gyy=gzz=-1] when e→0. In the Euclidean sector, the curvature gets concentrated on the origin of tE-x plane in a manner analogous to Aharanov–Bohm effect (in which the vector potential is a pure gauge everywhere except at the origin) and the curvature at the origin leads to nontrivial topological features and winding number.
25 citations
TL;DR: In this paper, the authors used first-order perturbation theory near the fermionic limit of the delta-function Bose gas in one dimension (i.e., a system of weakly interacting fermions) to study three situations of physical interest.
Abstract: We use first-order perturbation theory near the fermionic limit of the delta-function Bose gas in one dimension (i.e., a system of weakly interacting fermions) to study three situations of physical interest. The calculation is done using a pseudopotential which takes the form of a two-body delta''-function interaction. The three cases considered are the behavior of the system with a hard wall, with a point where the strength of the pseudopotential changes discontinuously, and with a region of finite length where the pseudopotential strength is non-zero (this is sometimes used as a model for a quantum wire). In all cases, we obtain exact expressions for the density to first order in the pseudopotential strength. The asymptotic behaviors of the densities are in agreement with the results obtained from bosonization for a Tomonaga-Luttinger liquid, namely, an interaction dependent power-law decay of the density far from the hard wall, a reflection from the point of discontinuity, and transmission resonances for the interacting region of finite length. Our results provide a non-trivial verification of the Tomonaga-Luttinger liquid description of the delta-function Bose gas near the fermionic limit.
24 citations
TL;DR: In this article, the Dirac delta function and its derivatives are used to derive Gauss' law in a dielectric medium directly from the charge densities, without using the potentials.
Abstract: Explicit sequences that approach the Dirac delta function and its derivatives are often helpful in presenting generalized functions We present a method by which a finite difference formula may be easily converted into a sequence that approaches a derivative of the Dirac delta function in one dimension In three dimensions, we employ a sequence for the Dirac delta function based on a uniformly charged sphere of infinitesimal radius and infinite charge density and show that the charge density of an electric dipole is (in the sense of a generalized function) equal to −(∂/∂z)δ3(r) We use this result to derive Gauss’ law in a dielectric medium directly from the charge densities, without using the potentials
24 citations
TL;DR: In this paper, the structure of mesonic vacua of N = 1 U(Nc) supersymmetric gauge theory with Nf flavors from the matrix model was investigated.
Abstract: We investigate in detail the structure of mesonic vacua of N = 1 U(Nc) supersymmetric gauge theory with Nf flavors from the matrix model. We show that the Witten index from the matrix model calculation agrees with a result from field theoretical analysis. We also discuss the relationship between a diagrammatic summation and direct matrix integration with insertion of a variable changing delta function. Using this formalism, we obtain the quantum moduli space and evidence of the Seiberg duality from the matrix models.
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TL;DR: In this paper, the authors discuss Donsker's delta function within the framework of white noise analysis, in particular its extension to complex arguments, with a view towards application in quantum physics.
Abstract: Universidade da Madeira, P 9000 Funchal, PortugalPublished in Soochow Journal of Mathematics 20 (1994) pp. 401-418Abstract:We discuss Donsker‘s delta function within the framework of White Noise Analysis,in particular its extension to complex arguments. With a view towards applicationsto quantum physics we also study sums and products of Donsker‘s delta functions.
TL;DR: In this article, the Schrodinger operator H was studied in terms of the spectrum σess of the scattering matrix S(λ) associated to the operator H in the space L 2 (mathbb{R}^{d}) with a magnetic potential A(x).
Abstract: We consider the Schrodinger operator H in the space $ L_{2}(\mathbb{R}^{d})$ with a magnetic potential A(x) decaying as $ \vert x\vert^{-1} $ at infinity and satisfying the transversal gauge condition = 0. Our goal is to study properties of the scattering matrix S(λ) associated to the operator H. In particular, we find the essential spectrum σess of S(λ) in terms of the behaviour of A(x) at infinity. It turns out that σess(S(λ)) is normally a rich subset of the unit circle $\mathbb{T}$ or even coincides with $\mathbb{T}$. We find also the diagonal singularity of the scattering amplitude (of the kernel of S(λ) regarded as an integral operator). In general, the singular part S0 of the scattering matrix is a sum of a multiplication operator and of a singular integral operator. However, if the magnetic field decreases faster than $ \vert x\vert^{-1} $ for d ≥ 3 (and the total magnetic flux is an integer times 2π for dd = 2), then this singular integral operator disappears. In this case the scattering amplitude has only a weak singularity (the diagonal Dirac function is neglected) in the forward direction and hence scattering is essentially of short-range nature. Moreover, we show that, under such assumptions, the absolutely continuous parts of the operators S(λ) and S0 are unitarily equivalent. An important point of our approach is that we consider S(λ) as a pseudodifferential operator on the unit sphere and find an explicit expression of its principal symbol in terms of A(x). Another ingredient is an extensive use (for d ≥ 3) of a special gauge adapted to a magnetic potential A(x).
01 Jan 2003
TL;DR: In this paper, the authors provide an overview of analytic aspects of singular diffusivity in nonlinear evolution equations, as well as their application in image processing and other applications. But the authors do not address the problem of finding a solution for a singular diffused version of an evolution equation.
Abstract: There is a class of nonlinear evolution equations with singular diffusivity, so that diffusion effect is nonlocal. A simplest one-dimensional example is a diffusion equation of the form u_t = \delta(u_x)u_{xx} for u = u(x; t), where \delta denotes Dirac s delta function. This lecture is intended to provide an overview of analytic aspects of such equations, as well as various applications. Equations with singular diffusivity are applied to describe several phenomena in the applied sciences, and to provide several devices in technology, especially image processing. A typical example is a gradient flow of the total variation of a function, which arises in image processing, as well as in material science to describe the motion of grain boundaries. In the theory of crystal growth the motion of a crystal surface is often described by an anisotropic curvature flow equation with a driving force term. At low temperature the equation includes a singular diffusivity, since the interfacial energy is not smooth. Another example is a crystalline algorithm to calculate curvature flow equations in the plane numerically, which is formally written as an equation with singular diffusivity. Because of singular diffusivity, the notion of solution is not a priori clear, even for the above one-dimensional example. It turns out that there are two systematic approaches. One is variational, and applies to divergence type equations. However, there are many equations like curvature flow equations which are not exactly of divergence type. Fortu-nately, our approach based on comparision principles turns out to be succesful in several interesting problems. It also asserts that a solution can be considered as a limit of solution of an approximate equation. Since the equation has a strong diffusivity at a particular slope of a solution, a flat portion with this slope is formed. In crystal growth ploblems this flat portion is called a facet. The discontinuity of a solution (called a shock) for a scalar conservation law is also considered as a result of singular diffusivity in the vertical direction.
TL;DR: It is proved that the Lieb-Liniger cusp condition implementing the delta function interaction in one-dimensional Bose gases is dynamically conserved under phase imprinting by pulses of arbitrary spatial form.
Abstract: It is proved that the Lieb-Liniger cusp condition implementing the delta function interaction in one-dimensional Bose gases is dynamically conserved under phase imprinting by pulses of arbitrary spatial form, and the subsequent many-body dynamics in the thermodynamic limit is expressed approximately in terms of solutions of the time-dependent single-particle Schrodinger equation for a set of time-dependent orbitals evolving from an initial Lieb-Liniger-Fermi sea. As an illustrative application, a generation of gray solitons in a Lieb-Liniger gas on a ring by a phase-imprinting pulse is studied.
TL;DR: In this paper, a finite-limit-of-integration Wigner function, with oscillatory behavior and negative values for free particles, is proposed, which is related to the quantum dielectric function derived from the quantum Vlasov equation (QVE).
Abstract: The Wigner function is shown related to the quantum dielectric function derived from the quantum Vlasov equation (QVE), with and without a magnetic field, using a standard method in plasma physics with linear perturbations and a self-consistent mean field interaction via Poisson's equation. A finite-limit-of-integration Wigner function, with oscillatory behavior and negative values for free particles, is proposed. In the classical regimes, where the problem size is huge compared to the particle wavelength, these limits go to infinity, and for free particles, the Wigner function becomes a positive delta function as expected. For the harmonic oscillator potential, there is no distinction between finite and infinite limits of integration when these are larger than the eigenfunction localization length.
TL;DR: The paper uses results from the discretized spectral approximation in neutron transport theory and concludes that nature supports complexity to attain efficiently a multiplicity of states that otherwise would remain unavailable to it.
Abstract: This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of functions in an extended multifunction space [Sengupta & Ray, 2000] and the topological theory of convergence. Order, chaos and complexity are described as distinct components of this unified mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical convergence being the region in (Multi(X)) that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [Sengupta, 1988, 1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In (Multi(X)), the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of latent chaotic states to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain efficiently a multiplicity of states that otherwise would remain unavailable to it.
05 Oct 2003
TL;DR: In this paper, a closed form causal function describing absorption, dispersion and delay as a function of frequency and increasing distance is presented for broadband pulse echoes from a scanning acoustic microscope, which can be combined with the spatial impulse response for fast computation of combined diffraction and absorption effects.
Abstract: Effects of loss are most often handled in the frequency domain. An example of this approach is the material transfer function, MTF, a closed form causal function describing absorption, dispersion and delay as a function of frequency and increasing distance z. For power law absorption, /spl alpha/=/spl alpha//sub 0/f/sup y/ this function can be written in terms of a parameter a=/sup y//spl radic//spl alpha//sub 0/z, MTF(f)=M(af). The material impulse response function, mirf, is the inverse Fourier transform of this function, mirf(t)=m(t/a)/a. In time, loss can be seen as a decay of amplitude according to 1/a and a broadening of the response by a time stretching factor of 1/a with increasing distance z. This function encodes all absorption, dispersion and delay effects into a compact time domain expression. Properties of the material impulse response include delta function like behavior at z=0 and similarity in shape with distance. This real time function convolved with an input waveform at z=0 describes the changes of pulse shape for propagation in a lossy medium. Application of this approach to broadband pulse echoes from a scanning acoustic microscope are presented. The mirf can be combined with the spatial impulse response for fast computation of combined diffraction and absorption effects.
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TL;DR: In this paper, a generalized Gaussian process with mean functional zero and covariance functional that can be interpreted as a fractional integral or fractional derivative of the delta-function is presented.
Abstract: Brownian motion can be characterized as a generalized random process and, as such, has a generalized derivative whose covariance functional is the delta function. In a similar fashion, fractional Brownian motion can be interpreted as a generalized random process and shown to possess a generalized derivative. The resulting process is a generalized Gaussian process with mean functional zero and covariance functional that can be interpreted as a fractional integral or fractional derivative of the delta-function.
TL;DR: In this paper, the Bethe-ansatz method was used to solve one dimensional SU(3) bosons with repulsive δ-function interaction by means of Bethe ansatz method and the features of ground state and low-lying excited states were studied by both numerical and analytic methods.
Abstract: In this paper we solve one dimensional SU(3) bosons with repulsive $\delta$-function interaction by means of Bethe ansatz method. The features of ground state and low-lying excited states are studied by both numerical and analytic methods. We show that the ground state is a SU(3) color ferromagnetic state. The configurations of quantum numbers for the ground state are given explicitly. For finite $N$ system the spectra of low-lying excitations and the dispersion relations of four possible elementary particles (holon, antiholon, $\sigma$-coloron and $\omega$-coloron) are obtained by solving Bethe-ansatz equation numerically. The thermodynamic equilibrium of the system at finite temperature is studied by using the strategy of thermodynamic Bethe ansatz, a revised Gaudin-Takahashi equation which is useful for numerical method are given . The thermodynamic quantities, such as specific heat, are obtain for some special cases. We find that the magnetic property of the model in high temperature regime is dominated by Curie's law: $\chi\propto 1/T$ and the system has Fermi-liquid like specific heat in the strong coupling limit at low temperature.
TL;DR: It is found that the RBNM is as accurate as or even more precise than the BEM, which suggests that the current RBNm could be robust and applicable.
Abstract: The boundary node method (BNM) takes the advantages of both the boundary integral equation in dimension reduction and the moving least-square (MLS) approximation in elements elimination. However, the BNM inherits the deficiency of the MLS approximation, in which the shape functions lack the delta function property. As a result boundary conditions could not be exactly implemented for the BNM. In this paper, a radial boundary node method (RBNM) is proposed. The RBNM uses radial basis functions (RBFs) instead of the MLS to construct its interpolation. Consequently, the interpolation function could pass through nodes exactly, and the shape functions are of the delta function property. The exponential (EXP) and the multiquadric (MQ) RBFs are used in the current RBNM, and their shape parameters are studied in detail through some analyses of two-dimensional elastic problems. The suitable ranges of the shape parameters are proposed for both the EXP and the MQ basis functions. It is found that the RBNM is as accurate as or even more precise than the BEM. This suggests that the current RBNM could be robust and applicable. q 2003 Published by Elsevier Ltd.
TL;DR: In this paper, the exact transfer matrix approach used in studying sectionally constant potentials in one dimension is generalized to cylindrical and spherical geometries, where the potential depends only on radius.
Abstract: The exact transfer matrix approach used in studying sectionally constant potentials in one dimension is generalized to cylindrical and spherical geometries, where the potential depends only on radius. In each geometry two transfer matrices suffice to completely describe the wave function: one for handling a discontinuity in potential and one for handling a delta-function potential barrier. This method is then applied to the problem of confining a wave function in a cylindrical configuration using only a series of carefully placed delta function potential barriers. It is found that confinement can be made to increase nearly exponentially with the number of barriers if placed correctly, but that this arrangement has an exponentially sharp dependence on both barrier position and energy.
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TL;DR: In this paper, a method for perturbatively expanding the three-body bound-state equation in inverse powers of the cutoff is developed, which allows us to extract some analytical results concerning the behavior of the system.
Abstract: We have investigated S-wave bound states composed of three identical bosons interacting via regulated delta function potentials in non-relativistic quantum mechanics. For low-energy systems, these short-range potentials serve as an approximation to the underlying physics, leading to an effective field theory.
A method for perturbatively expanding the three-body bound-state equation in inverse powers of the cutoff is developed. This allows us to extract some analytical results concerning the behavior of the system. Further results are obtained by solving the leading order equations numerically to 11 or 12 digits of accuracy. The limit-cycle behavior of the required three-body contact interaction is computed, and the cutoff-independence of bound-state energies is shown. By studying the relationship between the two- and three-body binding energies, we obtain a high accuracy numerical calculation of Efimov's universal function.
Equations for the first order corrections, necessary for the study of cutoff dependence, are derived. However, a numerical solution of these equations is not attempted.
TL;DR: In this paper, the spatial dipole delta function (SDF) is introduced and its properties are investigated from the viewpoint of analyzing the characteristics of piezoelectric transducers.
Abstract: `Spatial dipole delta function' is introduced and its properties are investigated from the viewpoint of analyzing the characteristics of piezoelectric transducers. This function can express a transmitter and a receiver with sufficiently small volume in a transducer, and is utilized in the framework of `complex series dynamics' to calculate the driving-point and transfer frequency response functions and the spatial distribution of vibrational modes with regard to the stored energy in the transducer. The calculation can be performed without solving the conventional complicated eigenvalue problem and without being restricted by the geometric configuration of the transmitter and receiver, and gives quantitatively reasonable results.
TL;DR: In this article, a piecewise linear collocation method for the solution of a pseudo-differential equation of order r=0, −1 over a closed and smooth boundary manifold is considered.
Abstract: In this paper, we consider a piecewise linear collocation method for the solution of a pseudo-differential equation of order r=0, −1 over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three-point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. For the corresponding wavelet algorithm, we show that the parametrization can be approximated by low-order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are composite rules of simple low-order quadratures. The whole algorithm for the assembling of the matrix requires no more than O(N [logN]3) arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than O(N−(2−r)/2). Note that, in contrast to well-known algorithms by Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required. In contrast to an algorithm of Ehrich and Rathsfeld, no multiplicative splitting of the kernel function is required. Beside the usual mapping properties of the integral operator in low order Sobolev spaces, estimates of Calderon–Zygmund type are the only assumptions on the kernel function. Copyright © 2003 John Wiley & Sons, Ltd.
TL;DR: In this paper, a generalized Gaussian process with mean functional zero and covariance functional that can be interpreted as a fractional integral or fractional derivative of the deltafunction is presented.
Abstract: Brownian motion can be characterized as a generalized random process and, as such, has a generalized derivative whose covariance functional is the delta function. In a similar fashion, fractional Brownian motion can be interpreted as a generalized random process and shown to possess a generalized derivative. The resulting process is a generalized Gaussian process with mean functional zero and covariance functional that can be interpreted as a fractional integral or fractional derivative of the deltafunction.
TL;DR: In this paper, the free energy functional of disordered dielectrics with dipole impurities is derived, where dipoles, being randomly positioned and oriented, create in the host dielectric lattice random electric field E in the points where other impurity dipoles are situated.
Abstract: We derive the free energy functional of disordered dielectrics with dipole impurities. These dipoles, being randomly positioned and oriented, create in the host dielectric lattice random electric field E in the points where other impurity dipoles are situated. The distribution function f(E) of this random field, defined as an average (over spatial and orientational disorder) value of Dirac delta contributions of each dipole, enables us to obtain equations for order parameters. Latter equations permit to derive the free energy functional which substitutes (within limitations of the proposed model) the initial Hamiltonian of disordered system.
01 Jan 2003
TL;DR: In this paper, it was shown that drop size distributions at the saturation or coarse scales maximize the spectral entropy and thus convey minimum information about drop distribution variation in space, and that drop distributions at these scales convey a considerable amount of information on drop variability.
Abstract: Drop size and drop spatial distribution determine photon-cloud interaction. The classical approach assumes that the number of drops of a given radius is proportional to volume with a drop-size-dependent coefficient of proportionality, the drop concentration, which is a volume-independent function of the spatial point. This assumption underlies the derivation of the drop size density distribution function, from which one can derive the extinction coefficient and scattering phase function, which are in turn input to the classical radiative transfer equation. In general, drop size distributions depend on spatial scale. Liu et al. (2002) pointed out that there is a “saturation scale,” beyond which observed size distributions do not change much with further increases in averaging scale. The drop size distributions at the saturation or coarse scales maximize the spectral entropy and thus convey minimum information about drop distribution variation in space (Ash 1965). The use of such distributions in the radiative transfer equation, therefore, makes it insensitive to small-scale cloud heterogeneity. Another extreme is the case when drops are sampled at an infinitesimal scale. The drop distribution is then given by Dirac delta functions, which account for the total number of drops with specific radius and correspond to the minimum spectral entropy. Although information about drops conveyed by these distributions is maximized, the drop concentration can not be defined at this scale and thus can not be used in the radiative transfer equation. A Monte-Carlo technique seems to be the only way to account for the full amount of information conveyed by such distributions. Drop size distributions are strongly scale dependent when the sampling scale is between the infinitesimal and saturation scales (Liu et al. 2002). This suggests that drop distributions at these scales convey a considerable amount of information on drop variability. The radiative transfer equation, in turn, aims to relate small-scale properties of the medium to the photon distribution in the entire medium. The question then arises, how essential is this information? The objective of this paper is to address this question.
02 Jun 2003
TL;DR: In this article, an approach is presented to compute the force on a spherical particle in a rarefied flow of a monatomic gas. This approach relies on the development of a Green's function that describes the forces on the spherical particles in a delta function molecular velocity distribution function.
Abstract: An approach is presented to compute the force on a spherical particle in a rarefied flow of a monatomic gas. This approach relies on the development of a Green’s function that describes the force on a spherical particle in a delta‐function molecular velocity distribution function. The gas‐surface interaction model in this development allows incomplete accommodation of energy and tangential momentum. The force from an arbitrary molecular velocity distribution is calculated by computing the moment of the force Green’s function in the same way that other macroscopic variables are determined. Since the molecular velocity distribution function is directly determined in the DSMC method, the force Green’s function approach can be implemented straightforwardly in DSMC codes. A similar approach yields the heat transfer to a spherical particle in a rarefied gas flow. The force Green’s function is demonstrated by application to two problems. First, the drag force on a spherical particle at arbitrary temperature and ...
TL;DR: In this paper, a simple extension of Marcella's analysis of the double-slit experiment to two dimensions is presented. But the main difference is that the detection pattern at the detection screen is actually a measurement of the momentum distribution of the diffracted particles.
Abstract: Following Marcella’s approach to the double-slit experiment (Marcella T V 2002 Eur. J. Phys. 23 615–21), diffraction patterns for two-dimensional masks are calculated by Fourier transform of the Mask geometry into momentum space. Iw ish to describe a simple extension of Marcella’s [1] recent analysis of the double-slit experiment to two dimensions. The essential point Marcella makes in his unique treatment of this well-known experiment is that the diffraction pattern at the detection screen is actually a measurement of the momentum distribution of the diffracted particles. Therefore the calculate dd iffraction pattern is simply obtained from the Fourier transform of the coordinate space wavefunction (the doubleslit geometry) into momentum space. Marcella considered two spatial models: (model 1) infinitesimally thin slits represented by Dirac delta functions; and (model 2) slits of finite width. About 60 years ago Sir Lawrence Bragg [2] proposed the optical transform as an aid in the interpretation of the x-ray diffraction patterns of crystals. This required the fabrication of two-dimensional masks of various crystal or molecular geometries and the generation of the diffraction pattern using visible electromagnetic radiation. Present-day laser technology has made the generation of suc hd iffraction patterns routine, even in the classroom. In addition, Marcella’s computational approach makes calculating the diffraction patterns conceptually and mathematically straightforward. If one considers the mask as consisting of point scatterers (model 1), the coordinate space wavefunction is a linear superposition of the scattering positions: |� �= 1 √ N