Dirac Delta Type Research Articles

Exact results for nonequilibrium dynamics in Wigner phase space

We study time evolution of Wigner function of an initially interacting one-dimensional quantum gas following the switch-off of the interactions. For the scenario where at t=0 the interactions are suddenly suppressed, we derive a relationship between the dynamical Wigner function and its initial value. A two-particle system initially interacting through two different interactions of Dirac delta type is examined. For a system of particles that is suddenly let to move ballistically (without interactions) in a harmonic trap in d dimensions, and using time evolution of one-body density matrix, we derive a relationship between the time dependent Wigner function and its initial value. Using the inverse Wigner transform we obtain, for an initially harmonically trapped noninteracting particles in d dimensions, the scaling law satisfied by the density matrix at time t after a sudden change of the trapping frequency. Finally, the effects of interactions are analyzed in the dynamical Wigner function.

Insights into the Kramers’ flux‐over‐population rate for chemical reactions in liquid phases through the matrix transport equation

Kramers’ equation models a chemical reaction as a Brownian particle diffusing over a potential barrier under the influence of medium viscosity. In the case of high viscosity, the equation reduces to a simpler Smoluchowski equation. In this report, we have contrived an equivalent matrix‐transport equation that relates the ordered pair (activity, flux) of the output (activated complex) to that of the input (reactant). With an initial condition of the Dirac delta type placed at the location of the reactant, and a reflecting boundary condition set on the reactant state, and an absorbing boundary condition on the activated complex state, we are able to prove the equality relation between the mean first passage time, , for the diffusion and the inverse of the rate constant, k−1, for the reaction counterpart. We have also derived , where λi is the ith eigenvalue of the Smoluchowski differential operator stipulated with the above‐mentioned boundary conditions. We have also deduced that, in the long time limit, the number of particles remaining inside the diffusion domain decays exponentially with a relaxation time just the same as the concentration of the reactant does for a first‐order reaction system.

Quantum out-of-equilibrium cosmology

In this work, our prime focus is to study the one to one correspondence between the conduction phenomena in electrical wires with impurity and the scattering events responsible for particle production during stochastic inflation and reheating implemented under a closed quantum mechanical system in early universe cosmology. In this connection, we also present a derivation of quantum corrected version of the Fokker–Planck equation without dissipation and its fourth order corrected analytical solution for the probability distribution profile responsible for studying the dynamical features of the particle creation events in the stochastic inflation and reheating stage of the universe. It is explicitly shown from our computation that quantum corrected Fokker–Planck equation describe the particle creation phenomena better for Dirac delta type of scatterer. In this connection, we additionally discuss Itô, Stratonovich prescription and the explicit role of finite temperature effective potential for solving the probability distribution profile. Furthermore, we extend our discussion of particle production phenomena to describe the quantum description of randomness involved in the dynamics. We also present computation to derive the expression for the measure of the stochastic non-linearity (randomness or chaos) arising in the stochastic inflation and reheating epoch of the universe, often described by Lyapunov Exponent. Apart from that, we quantify the quantum chaos arising in a closed system by a more strong measure, commonly known as Spectral Form Factor using the principles of random matrix theory (RMT). Additionally, we discuss the role of out of time order correlation function (OTOC) to describe quantum chaos in the present non-equilibrium field theoretic setup and its consequences in early universe cosmology (stochastic inflation and reheating). Finally, for completeness, we also provide a bound on the measure of quantum chaos (i.e. on Lyapunov Exponent and Spectral Form Factor) arising due to the presence of stochastic non-linear dynamical interactions into the closed quantum system of the early universe in a completely model-independent way.

Suppressing self-excited vibrations of mechanical systems by impulsive force excitation

In this contribution, self-excited mechanical systems subjected to force excitation of impulsive type are investigated. It is shown that applying force impulses which are equally spaced in time, but whose impulsive strength depends in a certain manner on the state-variables of the mechanical system, results in a periodic energy exchange between lower and higher modes of vibration. Moreover, in the theoretical case of Dirac delta impulses, it is possible that no energy crosses the system boundary while energy is transferred across modes, i.e. neither external energy is fed to the mechanical system, nor energy is extracted from the mechanical system. Shifting energy to higher modes of vibration, whose natural damping is larger compared to lower ones, results in a faster dissipation of energy. An analytical stability investigation is presented using the assumption of impulsive forcing of Dirac delta type, which allows deciding easily about the stability by evaluating the eigenvalues of the coefficient matrix of a corresponding set of difference equations. It is shown that the developed impulsive forcing concept is capable to suppress self-excited vibrations of mechanical systems. Some numerical results of a simple mechanical system with two degrees of freedom underline the presented approach.

Zitterbewegung, internal momentum and spin of the circular travelling-wave electromagnetic model electron

The study of this paper demonstrates that electron has Dirac delta like internal momentum (u,p_{{\theta}}), going round in a circle of radius equal to half the reduced Compton wavelength of electron with tangential velocity c. The circular momentum p_{{\theta}} and energy u emanate from circular Dirac delta type rotating monochromatic electromagnetic (EM) wave that itself travels in another circle having radius equal to the reduced Compton wavelength of electron. The phenomenon of Zitterbewegung and the spin of electron are the natural consequences of the model. The spin is associated with the internal circulating momentum of electron in terms of four component spinor, which leads to the Dirac equation linking the EM electron model with quantum mechanical theory. Our model accurately explains the experimental results of electron channelling experiment, [P. Catillon et al., Found.Phys. 38, 659 (2008)], in which the momentum resonance is observed at 161.784MeV/c corresponding to Zitterbewegung frequency of 80.874MeV/c electron beam.

The Infinite Square Well with a Point Interaction: A Discussion on the Different Parameterizations

The construction of Dirac delta type potentials has been achieved with the use of the theory of self adjoint extensions of non-self adjoint formally Hermitian (symmetric) operators. The application of this formalism to investigate the possible self adjoint extensions of the one dimensional kinematic operator $K=-d^2/dx^2$ on the infinite square well potential is quite illustrative and has been given elsewhere. This requires the definition and use of four independent real parameters, which relate the boundary values of the wave functions at the walls. By means of a different approach, that fixes matching conditions at the origin for the wave functions, it is possible to define a perturbation of the type $a\delta(x)+b\delta'(x)$, thus depending on two parameters, on the infinite square well. The objective of this paper is to investigate whether these two approaches are compatible in the sense that perturbations like $a\delta(x)+b\delta'(x)$ can be fixed and determined using the first approach.

A novel approximation method for multivariate data partitioning

PurposeThe plain High Dimensional Model Representation (HDMR) method needs Dirac delta type weights to partition the given multivariate data set for modelling an interpolation problem. Dirac delta type weight imposes a different importance level to each node of this set during the partitioning procedure which directly effects the performance of HDMR. The purpose of this paper is to develop a new method by using fluctuation free integration and HDMR methods to obtain optimized weight factors needed for identifying these importance levels for the multivariate data partitioning and modelling procedure.Design/methodology/approachA common problem in multivariate interpolation problems where the sought function values are given at the nodes of a rectangular prismatic grid is to determine an analytical structure for the function under consideration. As the multivariance of an interpolation problem increases, incompletenesses appear in standard numerical methods and memory limitations in computer‐based applications. To overcome the multivariance problems, it is better to deal with less‐variate structures. HDMR methods which are based on divide‐and‐conquer philosophy can be used for this purpose. This corresponds to multivariate data partitioning in which at most univariate components of the Plain HDMR are taken into consideration. To obtain these components there exist a number of integrals to be evaluated and the Fluctuation Free Integration method is used to obtain the results of these integrals. This new form of HDMR integrated with Fluctuation Free Integration also allows the Dirac delta type weight usage in multivariate data partitioning to be discarded and to optimize the weight factors corresponding to the importance level of each node of the given set.FindingsThe method developed in this study is applied to the six numerical examples in which there exist different structures and very encouraging results were obtained. In addition, the new method is compared with the other methods which include Dirac delta type weight function and the obtained results are given in the numerical implementations section.Originality/valueThe authors' new method allows an optimized weight structure in modelling to be determined in the given problem, instead of imposing the use of a certain weight function such as Dirac delta type weight. This allows the HDMR philosophy to have the chance of a flexible weight utilization in multivariate data modelling problems.

Cross-separatrix flux in time-aperiodic and time-impulsive flows

A theory for the fluid flux generated across heteroclinic separatrices under the influence of time-aperiodic perturbations is presented. The flux is explicitly defined as the amount of fluid transferred per unit time, and its detailed time-dependence monitored. The perturbations are allowed to be significantly discontinuous in time, including for example impulsive (Dirac delta type) discontinuities. The flux is characterized in terms of time-varying separatrices, with easily computable formulae (directly related to Melnikov functions) provided.

Impulsive Dirac‐delta forces in the rocking motion

AbstractIn this work the classical theory of one block rocking motion is revisited. A Dirac‐delta type interaction as impact mechanism is found to be an alternative for the traditional model. Numerical computations with this new formulation have shown that the agreement with the classical theory is excellent for the case of slender blocks and small displacements. Good agreement with experimental data has also been found for the case of arbitrary angles and slenderness.The approach presented in this paper opens new lines for further theoretical developments and computational applications. Copyright © 2004 John Wiley & Sons, Ltd.

Spectral determinant method for interactingN-body systems including impurities

A general expression for the Green's function of a system of $N$ particles (bosons/fermions) interacting by contact potentials, including impurities with Dirac-delta type potentials is derived. In one dimension for $N>2$ bosons from our {\it spectral determinant method} the numerically calculated energy levels agree very well with those obtained from the exact Bethe ansatz solutions while they are an order of magnitude more accurate than those found by direct diagonalization. For N=2 bosons the agreement is shown analytically. In the case of N=2 interacting bosons and one impurity, the energy levels are calculated numerically from the spectral determinant of the system. The spectral determinant method is applied to an interacting fermion system including an impurity to calculate the persistent current at the presence of magnetic field.

Semilinear Hyperbolic Systems with Singular Non-Local Boundary Conditions: Reflection of Singularities and Delta Waves

In this paper we study initial-boundary value problems for first-order semilinear hyperbolic systems where the boundary conditions are non-local. We focus on situations involving strong singularities, of the Dirac delta type, in the initial data as well as in the boundary conditions. In such cases we prove an existence and uniqueness result in an algebra of generalized functions. Furthermore, we investigate the existence and structure of delta waves, i.e., distributional limits of solutions to the regularized systems. Due to the additional singularities in the boundary data the search for delta waves requires a delicate splitting of the solution into a linearly evolving singular part and a regular part satisfying a nonlinear equation. A new feature in the splitting procedure used here, compared to delta waves in pure initial value problems, is the dependence of the singular part also on part of the regular part due to singularities enetering from the boundary. Finally, we include simple examples where the existence of delta waves breaks down.

Optimal feed distribution in isothermal fixed-bed reactors for parallel reactions

The optimal design of isothermal fixed-bed catalytic reactors for maximum outlet yield is studied by considering optimal feed distribution (OFD) along the reactor axis for parallel reactions. The activity distribution in pellets is assumed to have the Dirac delta type and the active location is the same for all pellets. The OFD function is derived analytically, indicating a uniform distribution along the reactor portion starting at the zero of the axial coordinate. The size of the feed distribution portion and the active location in pellets need to be found numerically, but in some particular cases, they can be found also analytically. A comparison of the OFD to the optimal catalyst activity distribution (OCAD), along the reactor axis shows that the OFD can give the same outlet yield as the OCAD does, while the former needs only one type of the Dirac delta catalyst activity distribution for all pellets in the reactor.

Shape sensitivity analysis in thermal problems using BEM

The shape design sensitivity analysis of thermal problems based on the adjoint variable method is presented. The boundary element method (BEM) is employed for the numerical evaluation of thermal responses and sensitivity of the performance functions. The adjoint variable method and the material derivative concept are utilized to obtain the sensitivity expressions. The results are verified numerically for two simple example problems using the BEM. Unlike previous studies, this approach takes into account the Dirac delta type of performance function and varying boundary temperatures. The sensitivity of the performance functions can then be evaluated more accurately at discrete points. This is achieved by an efficient application of the concentrated adjoint temperature in the boundary element framework.

Optimization of a nonisothermal nonadiabatic fixed-bed reactor using dirac-type silver catalysts for ethylene epoxidation

A heterogeneous, nonadiabatic, nonisothermal plug-flow reactor where ethylene epoxidation over silver catalyst occurs is studied theoretically. The catalyst distribution in the pellets is of a Dirac-delta type. The location of the Dirac distribution is optimized both the ethylene oxide yield and selectivity maximazation. Single-pellet results indicate that yield can be maximized either by surface or subsurface catalyst locations, depending on the operating conditions. However, for all practical purposes, selectivity is maximized only by surface catalysts. Significant improvements in reactor performance are found by using the appropriate Dirac catalyst, as compared to a uniform catalyst distribution. Further improvements can be achieved by using a two-zone reactor, with different catalyst location in each zone. Reactor runaway behavior can also be avoided by the use of Dirac catalysts.

Optimal catalyst pellet activity distributions for deactivating systems

The optimal activity distribution in catalyst pellets for reacting systems which undergo deactivation is analysed. A general optimality criterion is developed, which allows one to conclude that, under quite general conditions, the optimal activity distribution is of the Dirac-delta type.

Some characteristic features of shock waves in a Kelvin medium due to an input pulse of Dirac delta type : Pal Roy, P; Gupta, RN; Singh, B Min Sci TechnolV7, N2, July 1988, P195–203

Some characteristic features of shock waves in a Kelvin medium due to an input pulse of Dirac delta type : Pal Roy, P; Gupta, RN; Singh, B Min Sci TechnolV7, N2, July 1988, P195–203

Some characteristic features of shock waves in a Kelvin medium due to an input pulse of dirac delta type

Assuming the rock mass as a homogeneous viscoelastic material of Kelvin type, the phenomenological characterisation that brings the specific features of shock waves propagation in rock mass is discussed. The shock wave is created by a sudden impulse at a depth of “h” units below the free surface. Stress distribution at different periods and distances are obtained and shown in the form of graphs. It is observed that the propagation of shock waves becomes more similar to the propagation of a viscous wave in a Newtonian fluid.

Radiation Modes of Circular Dielectric Waveguides

Radiation modes of circular dielectric waveguides are discussed. First, a general expression is derived by which the basic formula defining the orthogonality is expresse in terms of only far fields. Next, the orthogonality property of the Dirac's delta type is derived using a general functional form of radiation modes; the idea of distribution function is also used. The conditions that the pairs of mutually orthogonal radiation modes should satisfy are also derived. By using these conditions, several examples of pairs of mutually orthogonal modes are given. One of them is found to coincide with the pair of incident transverse electric (ITE) mode and incident transverse magnetic (ITM) mode named by Snyder

Well-posedness, control and computation of a one-phase Stefan problem with Neumann condition*

SynopsisA one-phase Stefan problem can be reduced to an equivalent variational inequality by using the Baiocchi-Duvaut transformation. In this paper, we study the variational inequality by formulating it as a set-valued partial differential equation. The existence of solutions is proved by applying a generalized Schauder fixed fixed point theorem for set-valued mappings. Uniqueness and regularity of solutions are also obtained. In §3, we regard the boundary value Neumann data as boundary controls and combine both the variational inequality and the classical approaches to study the effects of controls on the free boundary and the state (i.e. temperature). In §4, we further use the theory to study an optimal “ice-melting” problem. Our results show that if the controls have fixed total input heat flux and are constrained in magnitude, then the optimal control is “bang-bang”. If the admissible controls are not constrained in magnitude, then the optimal control is a Dirac delta type distribution which is no longer admissible. In the last section, our existence theory is combined with the finite difference method and non-linear programming techniques to obtain numerical solutions.