A TABLE OF INTEGRALS INVOLVING POWERS, EXPONENTIALS, LOGARITHMS, AND THE EXPONENTIAL INTEGRAL
Table integrals involving powers, exponentials, logarithms and the exponential integral
EXPONENTIAL FUNCTION↗
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Nyström type exponential integrators for strongly magnetized charged particle dynamics
Solving for charged particle motion in electromagnetic fields (i.e. the particle pushing problem) is a computationally intensive component of particle-in-cell (PIC) methods for plasma physics simulations. This task is especially challenging when the plasma is strongly magnetized due numerical stiffness arising from the wide range of time scales between highly oscillatory gyromotion and long term macroscopic behavior. A promising approach to solve these problems is by a class of methods known as exponential integrators that can solve linear problems exactly and are A-stable. This work extends the standard exponential integration framework to derive Nyström-type exponential integrators that integrates the Newtonian equations of motion as a second-order differential equation directly. In particular, we derive second-order and third-order Nyström-type exponential integrators for strongly magnetized particle pushing problems. Numerical experiments show that the Nyström-type exponential integrators exhibit significant improvement in computation speed over the standard exponential integrators.
general physics↗
Decoherence dynamics in molecular qubits: Exponential, Gaussian and beyond
In this article, we examine how the structure of system–bath interactions can determine commonly encountered temporal decoherence patterns, such as Gaussian and exponential decay, in molecular and other qubits coupled to a thermal bosonic bath. The analysis, based on a pure dephasing picture that admits analytical treatment, shows that decoherence, in general, is neither purely Gaussian nor exponential but rather the exponential of oscillatory functions, with periods determined by the bath’s frequencies. For initially unentangled qubit-bath states, Gaussian decay is always present at early times. It becomes increasingly dominant with increasing temperature, qubit–bath interaction strength, and bath correlation time. Initial system–bath entanglement that arises due to displacement in the position of the bath states preserves the Gaussian decay. By contrast, strict exponential decay arises only in very specific models that we isolate. However, it becomes dominant for times longer than the bath correlation time or for early times when there is initial entanglement due to momentum displacement of the bath states. For molecular electronic decoherence, the long-time exponential regime plays a limited role as it emerges after most coherence is lost. Thus, the Gaussian decay provides a more suitable (albeit imperfect) model of such decoherence. Furthermore, we discuss the connection between electronic decoherence dynamics and electronic spectroscopic line shape theory, where Gaussian spectral peaks correspond to Gaussian coherence decay and Lorentzian peaks correspond to exponential coherence decay. We find that Gaussian spectral peaks, usually associated with inhomogeneous broadening, can emerge from the entangling unitary system–bath dynamics even when there is no inhomogeneity in the initial conditions.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Theory, computation, and application of exponential splines
A generalization of the semiclassical cubic spline known in the literature as the exponential spline is discussed. In actuality, the exponential spline represents a continuum of interpolants ranging from the cubic spline to the linear spline. A particular member of this family is uniquely specified by the choice of certain tension parameters. The theoretical underpinnings of the exponential spline are outlined. This development roughly parallels the existing theory for cubic splines. The primary extension lies in the ability of the exponential spline to preserve convexity and monotonicity present in the data. Next, the numerical computation of the exponential spline is discussed. A variety of numerical devices are employed to produce a stable and robust algorithm. An algorithm for the selection of tension parameters that will produce a shape preserving approximant is developed. A sequence of selected curve-fitting examples are presented which clearly demonstrate the advantages of exponential splines over cubic splines.
Mccartin, B. J.↗
The matrix exponential in transient structural analysis
The primary usefulness of the presented theory is in the ability to represent the effects of high frequency linear response with accuracy, without requiring very small time steps in the analysis of dynamic response. The matrix exponential contains a series approximation to the dynamic model. However, unlike the usual analysis procedure which truncates the high frequency response, the approximation in the exponential matrix solution is in the time domain. By truncating the series solution to the matrix exponential short, the solution is made inaccurate after a certain time. Yet, up to that time the solution is extremely accurate, including all high frequency effects. By taking finite time increments, the exponential matrix solution can compute the response very accurately. Use of the exponential matrix in structural dynamics is demonstrated by simulating the free vibration response of multi degree of freedom models of cantilever beams.
Minnetyan, Levon↗
Simple quasi-exponential slope generator
Circuitry for digitally generating an exponentially decaying wave function permits discrete values to be sampled from the exponential waveform for comparison with a binary number of specified accuracy. This exponential-decay generator employs a simple binary counter to count in the sequence of exponential decay.
Anderson, T. O.↗
Numerical computation of exponential matrices using the Cayley-Hamilton theorem
A method for computing exponential matrices, which often arise naturally in the solution of systems of linear differential equations, is developed. An exponential matrix is generated as a linear combination of a finite number (equal to the matrix order) of matrices, the coefficients of which are scalar infinite sums. The method can be generalized to apply to any formal power series of matrices. Attention is focused upon the exponential function, and the matrix exponent is assumed tri-diagonal in form. In such cases, the terms in the coefficient infinite sums can be extracted, as recursion relations, from the characteristic polynomial of the matrix exponent. Two numerical examples are presented in some detail: (1) the three dimensional infinitesimal rotation rate matrix, which is skew symmetric, and (2) an N-dimensional tri-diagonal and symmetric finite difference matrix which arises in the numerical solution of the heat conduction partial differential equation. In the second example, the known eigenvalues and eigenvectors of the finite difference matrix permit an analytical solution for the exponential matrix, through the theory of diagonalization and similarity transformations, which is used for independent verification. The convergence properties of the scalar infinite summations are investigated for finite difference matrices of various orders up to ten, and it is found that the number of terms required for convergence increases slowly with the order of the matrix.
Walden, H.↗
Exponential-fitted methods for integrating stiff systems of ordinary differential equations: Applications to homogeneous gas-phase chemical kinetics
Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.
Pratt, D. T.↗
Future Exponential Economic Growth in Space
The human population and economy are now limited by the land area, materials and energy available on Earth. As explained by Malthus, human effort can increase this population limit only gradually and linearly at any fixed level of technology. The unexpected exponential increase in population since Malthus time is due to the even greater exponential increase in agricultural and industrial productivity due to modern technology. Current opinion is divided between the limiters who see fast approaching unavoidable limits to growth and growthers who expect continuing innovation and expansion. Given the minimum energy and food needed to support each human, the population of Earth is necessarily bounded. The past human exponential expansions were based on the invention of agriculture and industry. Continued expansion could occur with future colonization of the solar system. The initial occupation of near-Earth space would probably be for immediate military and economic gains rather than to escape the limits to growth on Earth. Space habitats could exploit solar system energy, materials, and living space, first in the Earth-moon system and later throughout the solar system. Space habitats would be supported by robotic power stations, mining, manufacturing, and transportation. These could travel to the stars with the addition of propulsion and nuclear power. Human expansion into the galaxy would allow continuing exponential growth and could establish a vast interconnected galactic civilization.
Jones, Harry W.↗
Firewalls at exponentially late times
We consider a version of the typical state firewall setup recently reintroduced by Stanford and Yang, who found that wormholes may create firewalls. We examine a late-time scaling limit in JT gravity in which one can resum the expansion in the number of wormholes, and we use this to study the exact distribution of interior slices at times exponential in the entropy. We consider a thermofield double with and without early perturbations on a boundary. These perturbations can appear on interior slices as dangerous high energy shockwaves. For exponentially late times, wormholes tend to teleport the particles created by perturbations and render the interior more dangerous. In states with many perturbations separated by large times, the probability of a safe interior is exponentially small, even though these would be safe without wormholes. With perturbation, even in the safest state we conceive, the odds of encountering a shock are fifty-fifty. One interpretation of the phenomenon is that wormholes can change time-ordered contours into effective out-of-time-ordered folds, making shockwaves appear in unexpected places.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Stretched exponential magnetic relaxation dynamics in artificial square ice revealed through x-ray photon correlation spectroscopy
X-ray photon correlation scattering measurements are undertaken on a thermally active artificial spin ice based on the square lattice, referred to as artificial square ice, to probe the fluctuation timescales as a function of temperature as the system passes through the paramagnetic-antiferromagnetic phase transition, which belongs to the two-dimensional Ising universality class. In the paramagnetic regime, a single exponential timescale is seen, whereas at and below the critical temperature, a stretched exponential decorrelation is observed, with the stretching exponent decreasing from unity down to below one-half as the temperature reduces. This trend is confirmed by kinetic Monte Carlo simulations of a simplified point-dipolar square ice system, and is in agreement with past theoretical work on the kinetic Ising model where stretched exponential relaxation due to equilibrium domain wall dynamics below the critical temperature was found.
36 MATERIALS SCIENCE↗
The effect of Be/+/ ion implanted exponential and uniform impurity profiles on the electrical characteristics of GaAs solar cells
The high surface recombination velocity is the major deterrent to obtaining efficient GaAs solar cells. If, however, an electric field is built in at the surface, the carriers will be swept away from the surface thus minimizing the surface recombination velocity problem. It has been previously shown theoretically that an exponential impurity distribution in the doped region of the cell results in a built-in electric field. Ion implantation was used to produce solar cells with an exponential impurity profile and cells with uniform profiles. It is shown that cells with an exponential impurity profile have higher open-circuit voltage, fill factors, and spectral response than those with a uniform impurity profile.
Vaidyanathan, K. V.↗
An Exponential Finite Difference Technique for Solving Partial Differential Equations
An exponential finite difference algorithm, as first presented by Bhattacharya for one-dimensianal steady-state, heat conduction in Cartesian coordinates, has been extended. The finite difference algorithm developed was used to solve the diffusion equation in one-dimensional cylindrical coordinates and applied to two- and three-dimensional problems in Cartesian coordinates. The method was also used to solve nonlinear partial differential equations in one (Burger's equation) and two (Boundary Layer equations) dimensional Cartesian coordinates. Predicted results were compared to exact solutions where available, or to results obtained by other numerical methods. It was found that the exponential finite difference method produced results that were more accurate than those obtained by other numerical methods, especially during the initial transient portion of the solution. Other applications made using the exponential finite difference technique included unsteady one-dimensional heat transfer with temperature varying thermal conductivity and the development of the temperature field in a laminar Couette flow.
Handschuh, Robert F.↗
Exponential ground impedance models and their interpretation
The authors compare the results of Donato's exponentially varying ground model, Attenborough's exponentially varying ground model and the rigid backed thin layer model. They show that these models produce similar results for slow variations. For rapid variations the results are quite different, but the basic theory used is only correct for the thin layer model. These results suggest that the exponentially varying models are not necessary for fitting ground impedance data.
Raspet, Richard↗
Quasi-random narrow-band model fits to near-infrared low-temperature laboratory methane spectra and derived exponential-sum absorption coefficients
Near-infrared 10/cm resolution spectra of methane obtained at various temperatures, pressures, and abundances are fit to a quasi-random narrow-band model. Exponential-sum absorption coefficients for three temperatures (112, 188, and 295 K), and 20 pressures from 0.0001 to 5.6 bars, applicable to the cold environments of the major planets, are then derived from the band model for the 230 wavelengths measured from 1.6 to 2.5 microns. RMS deviations between the laboratory and the exponential-sum synthetic transmissions are reported for the best fitting 50 wavelengths. Deviations relevant to broadband, 1-percent spectral resolution observations are also presented. The validity of exponential-sum coefficients derived from broadband (10/cm) transmission data is demonstrated via direct comparison with line-by-line calculations. The complete atlas of coefficients is available from the Planetary Data System-Planetary Atmospheres Discipline Node.
Baines, Kevin H.↗
Decoherence and Exponential Law: A Solvable Model
We analyze a modified version of the 'AgBr' Hamiltonian, solve exactly the equations of motion in terms of SU(2) coherent states, and study the weak-coupling, macroscopic limit of the model, obtaining an exponential behavior at all times. The asymptotic dominance of the exponential behavior is representative of a purely stochastic evolution and can be derived quantum mechanically in the so-called van Hove's limit (which is a weak-coupling, macroscopic limit). At the same time, a temporal behavior of the exponential type, yielding a 'probability dissipation' is closely related to dephasing ('decoherence') effects and one can expect a close connection with a dissipative and irreversible behavior. We stress the central relevance of the problem of dissipation to the quantum measurement theory and to the general topic of decoherence.
Pascazio, Saverio↗
Modeling the Role of Dislocation Substructure During Class M and Exponential Creep
The different substructures that form in the power-law and exponential creep regimes for single phase crystalline materials under various conditions of stress, temperature and strain are reviewed. The microstructure is correlated both qualitatively and quantitatively with power-law and exponential creep as well as with steady state and non-steady state deformation behavior. These observations suggest that creep is influenced by a complex interaction between several elements of the microstructure, such as dislocations, cells and subgrains. The stability of the creep substructure is examined in both of these creep regimes during stress and temperature change experiments. These observations are rationalized on the basis of a phenomenological model, where normal primary creep is interpreted as a series of constant structure exponential creep rate-stress relationships. The implications of this viewpoint on the magnitude of the stress exponent and steady state behavior are discussed. A theory is developed to predict the macroscopic creep behavior of a single phase material using quantitative microstructural data. In this technique the thermally activated deformation mechanisms proposed by dislocation physics are interlinked with a previously developed multiphase, three-dimensional. dislocation substructure creep model. This procedure leads to several coupled differential equations interrelating macroscopic creep plasticity with microstructural evolution.
Raj, S. V.↗
Slow Crack Growth of Brittle Materials With Exponential Crack-Velocity Formulation: Analysis - Part 1
Extensive slow-crack-growth (SCG) analysis was made using a primary exponential crack-velocity formulation under three widely used load configurations: constant stress rate, constant stress, and cyclic stress. Although the use of the exponential formulation in determining SCG parameters of a material requires somewhat inconvenient numerical procedures, the resulting solutions presented gave almost the same degree of simplicity in both data analysis and experiments as did the power-law formulation. However, the fact that the inert strength of a material should be known in advance to determine the corresponding SCG parameters was a major drawback of the exponential formulation as compared with the power-law formulation.
Choi, Sung R.↗