High-precision Chebyshev series approximation to the exponential integral
Calculating exponential integral using Chebyshev series expansion of associated functions
Lee, K. L.↗
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Results for “EXPONENTIAL FUNCTION”
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The luminosity function of galaxy systems - From single galaxies and small groups to rich clusters
The luminosity function (LF) of galaxy systems is determined, from single galaxies and small groups to rich clusters. The LF of rich (Abell) clusters is found to be a steeply decreasing function of cluster number density with cluster luminosity, and the groupings LF (that is, groups to rich clusters) can be well represented by a Schechter-type (1976) analytic form. The general LF of all galaxy systems is also found to be a smooth and steeply decreasing function of number density with luminosity, exhibiting an exponential cutoff in the rich-cluster luminosity domain, a power law dependence over most of the intermediate luminosity range, and a flattening of slope below the characteristic galactic luminosity. Comparisons of the LF with the LF of galaxies and X-ray clusters of galaxies are also discussed.
Bahcall, N. A.↗
Effect of physical aging on stress relaxation of poly/methyl methacrylate/
A study was made on the stress relaxation behavior at 25 C of poly(methyl methacrylate) in uniaxial tension as a function of physical aging at both room temperature and 60 C. Test specimens were compression molded at 165 C, then quenched to room temperature and allowed to age for up to 30 days prior to testing. Stress relaxation curves measured after different aging times could be superposed to a single master curve for each aging temperature. Superposition was achieved by applying vertical and horizontal shifts. Hence, the shape of the response curves was not changed by aging. This is in accordance with observations made by Struik for tensile creep curves. Volume changes as a function of physical aging were also determined. Simple exponential relationships were observed between volume and both horizontal and vertical shifts. The horizontal shift implies a shift in the effective time scale caused by a change in free volume. The vertical shifts could be correlated with changes in Young's modulus caused by a change in density. For the range of aging studied, the response time scale varied over nearly two decades of log-time. For the same conditions modulus varied by 30 percent.
Cizmecioglu, M.↗
On time-dependent radiative transfer
An integral equation is developed for application to time-dependent laboratory experiments in which partial redistribution effects are important. The equation of transport with the Heasly-Kneer (1976) emission coefficient and the equation of statistical equilibrium lead to a time-dependent redistribution function containing an absorption - reemission term which decays exponentially in time and a scattering term which is instantaneous. This integral equation does not agree with an equation written by Payne et al. (1974) that has been used to compare theory with experiments. The difference between the Payne equation and the equation developed here needs to be examined in detail, since it might under some circumstances be on the same order as the difference between partial and complete redistribution.
Streater, A.↗
SGR J1550-5418 Bursts Detected with the Fermi Gamma-Ray Burst Monitor during Its Most Prolific Activity
We have performed detailed temporal and time-integrated spectral analysis of 286 bursts from SGR J1550-5418 detected with the Fermi Gamma-ray Burst Monitor (GBM) in 2009 January, resulting in the largest uniform sample of temporal and spectral properties of SGR J1550-5418 bursts. We have used the combination of broadband and high time-resolution data provided with GBM to perform statistical studies for the source properties.We determine the durations, emission times, duty cycles, and rise times for all bursts, and find that they are typical of SGR bursts. We explore various models in our spectral analysis, and conclude that the spectra of SGR J15505418 bursts in the 8-200 keV band are equally well described by optically thin thermal bremsstrahlung (OTTB), a power law (PL) with an exponential cutoff (Comptonized model), and two blackbody (BB) functions (BB+BB). In the spectral fits with the Comptonized model, we find a mean PL index of -0.92, close to the OTTB index of -1. We show that there is an anti-correlation between the Comptonized E(sub peak) and the burst fluence and average flux. For the BB+BBfits, we find that the fluences and emission areas of the two BB functions are correlated. The low-temperature BB has an emission area comparable to the neutron star surface area, independent of the temperature, while the high temperature BB has a much smaller area and shows an anti-correlation between emission area and temperature.We compare the properties of these bursts with bursts observed from other SGR sources during extreme activations, and discuss the implications of our results in the context of magnetar burst models.
bursts↗
Redshift modifications to HEAO A-1 cluster X-ray luminosities
New redshift measurements for the 24 Abell galactic clusters are presented. The optical observations are described and applied to the cluster X-ray luminosity function, and the results are used to delimit the cluster contribution to the diffuse X-ray background (DXRB). The evolution of the X-ray luminosity function is then investigated. It is found that while simple two-parameter analytic forms do not fit this function very well, three-parameter forms such as the exponential times power law do fit it very well. Future redshift observations will probably not change the volume emissivity significantly, but they may in a statistically significant way affect the parameters of analytic fits which define the shape of the luminosity function. The contribution of rich clusters to the DXRB in the 2-6 keV range is 5.2 percent. Future redshift observations will modify this result by no more than a factor of about 1.1.
Cruddace, R. G.↗
Exponential acceleration of macroscopic quantum tunneling in a Floquet Ising model
The exponential suppression of macroscopic quantum tunneling (MQT) in the number of elements to be reconfigured is an essential element of broken symmetry phases. This suppression is also a core bottleneck in quantum algorithms, such as traversing an energy landscape in optimization, and adiabatic state preparation more generally. In this work, we demonstrate exponential acceleration of MQT through Floquet engineering with the application of a uniform, high frequency transverse drive field. Using the ferromagnetic phase of the transverse field Ising model in one and two dimensions as a prototypical example, we identify three phenomenological regimes as a function of drive strength. For weak drives, the system exhibits exponentially decaying tunneling rates but robust magnetic order; in the crossover regime at intermediate drive strength, we find polynomial decay of tunnelling alongside vanishing magnetic order; and at very strong drive strengths both the Rabi frequency and time-averaged magnetic order are approximately constant with increasing system size. We support these claims with extensive full wavefunction and tensor network numerical simulations, and theoretical analysis. An experimental test of these results presents a technologically important and novel scientific question accessible on NISQ-era quantum computers.
Grattan, George↗
Structure and dynamics of water confined in nanoporous carbon
We study the structure and dynamics of water confined in a nanoporous graphitic carbon structure using molecular-dynamics (MD) simulations. The carbon structure is generated by a reactive MD simulation of oxidation of a silicon carbide nanoparticle. We embed water molecules in the nanopores and study structural and dynamical properties of nanoconfined water as a function of temperature. MD simulation results indicate the presence of high-density water (HDW) and low-density water. Radial distribution functions and spatial density functions indicate that the second solvation shell of the HDW is broken. We calculate the self-diffusion coefficient of confined water molecules as a function of temperature and find a significant decrease in the diffusion of water molecules around T = 190K. The cage correlation function c(t) of confined water molecules at T = 200K exhibits stretched exponential decay, c(t) = exp[–(t/τ) β ], with β = 0.43, which matches exactly with the theoretical prediction β = 3/7. Finally, the self-intermediate scattering function at T = 200K indicates differences in small-scale and large-scale dynamics of water molecules.
36 MATERIALS SCIENCE↗
Stabilization of solitons under competing nonlinearities by external potentials
We report results of the analysis for families of one-dimensional (1D) trapped solitons, created by competing self-focusing (SF) quintic and self-defocusing (SDF) cubic nonlinear terms. Two trapping potentials are considered, the harmonic-oscillator (HO) and delta-functional ones. The models apply to optical solitons in colloidal waveguides and other photonic media, and to matter-wave solitons in Bose-Einstein condensates loaded into a quasi-1D trap. For the HO potential, the results are obtained in an approximate form, using the variational and Thomas-Fermi approximations, and in a full numerical form, including the ground state and the first antisymmetric excited one. For the delta-functional attractive potential, the results are produced in a fully analytical form, and verified by means of numerical methods. Both exponentially localized solitons and weakly localized trapped modes are found for the delta-functional potential. The most essential conclusions concern the applicability of competing Vakhitov-Kolokolov (VK) and anti-VK criteria to the identification of the stability of solitons created under the action of the competing SF and SDF terms.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Exponential approximation for daily average solar heating or photolysis
When incorporating formulations of instantaneous solar heating or photolytic rates as functions of altitude and sun angle into long range forecasting models, it may be desirable to replace the time integrals by daily average rates that are simple functions of latitude and season. This replacement is accomplished by approximating the integral over the solar day by a pure exponential. This gives a daily average rate as a multiplication factor times the instantaneous rate evaluated at an appropriate sun angle. The accuracy of the exponential approximation is investigated by a sample calculation using an instantaneous ozone heating formulation available in the literature.
Cogley, A. C.↗
Partitioned exponential methods for coupled multiphysics systems
Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible semi-discretization in space, the class of problems under consideration is modeled by a system of ordinary differential equations where the right-hand side is a summation of two component functions, each corresponding to a given set of physical processes. The partitioned-exponential methods proposed herein evolve each component of the system via an exponential integrator, and information between partitions is exchanged via coupling terms. Here, the traditional approach to constructing exponential methods, based on the variation-of-constants formula, is not directly applicable to partitioned systems. Rather, our approach to developing new partitioned-exponential families is based on a general-structure additive formulation of the schemes. Two method formulations are considered, one based on a linear-nonlinear splitting of the right hand component functions, and another based on approximate Jacobians. The paper develops classical (non-stiff) order conditions theory for partitioned exponential schemes based on particular families of T-trees and B-series theory. Several practical methods of third order are constructed that extend the Rosenbrock-type and EPIRK families of exponential integrators. Several implementation optimizations specific to the application of these methods to reaction-diffusion systems are also discussed. Numerical experiments reveal that the new partitioned-exponential methods can perform better than traditional unpartitioned exponential methods on some problems.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Adiabatic quantum decoherence in many non-interacting subsystems induced by the coupling with a common boson bath
Highlights: • System–environment quantum correlation: a main solid state NMR decoherence channel. • Non-separable system–environment model yields realistic spin decoherence rates. • New open quantum system approach explains irreversible decay of refocused NMR echoes. • Adiabatic quantum decoherence is inherently irreversible and eigen-selective. This work addresses adiabatic quantum decoherence of many-body spin systems coupled with a boson field in the framework of open quantum systems theory. We generalize the traditional spin-boson model by considering a system–environment interaction Hamiltonian that represents a partition of non-interacting subsystems and highlights the collective correlation that appears exclusively due to the coupling with a common environment. Remarkably, this simple, exactly solvable model encompasses relevant aspects of a many-body open quantum system and features the subtle quantum effects that arise when the size scales up to a macroscopic level. We derive an analytical expression for the time dependence of the density matrix elements (in the preferred basis) without assuming coarse-graining. The resulting decoherence function is eigen-selective and is a complex exponential whose exponent has a real part that introduces a decay similar to that in the spin-boson model. On the contrary, the imaginary part depends on the quantum numbers and geometry of the whole partition and does not reflect the system temperature. Motivated by decoherence in solid-state NMR, and in search of realistic numerical estimations, we apply the theoretical results to a partition of dipole-coupled spin pairs in contact with a common phonon bath, using typical parameters of hydrated salts. The proposal allows estimating the decoherence time scale in terms of the system physical constants: sound velocity and eigenvalue distribution width. As a significant novelty, the decoherence function phase depends on the eigenvalue distribution throughout the sample. It plays the leading role, overshadowing the mechanism associated with the bath thermal state. Finally, we apply the formalism to describe decoherence in the “magic echo” NMR reversal experiment. We find that the system–environment correlation explains the origin of irreversibility, and both the decoherence rate value and its dependence on the dipolar frequency, are remarkably similar to the experiment.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
A High Frequency Model of Cascade Noise
Closed form asymptotic expressions for computing high frequency noise generated by an annular cascade in an infinite duct containing a uniform flow are presented. There are two new elements in this work. First, the annular duct mode representation does not rely on the often-used Bessel function expansion resulting in simpler expressions for both the radial eigenvalues and eigenfunctions of the duct. In particular, the new representation provides an explicit approximate formula for the radial eigenvalues obviating the need for solutions of the transcendental annular duct eigenvalue equation. Also, the radial eigenfunctions are represented in terms of exponentials eliminating the numerical problems associated with generating the Bessel functions on a computer. The second new element is the construction of an unsteady response model for an annular cascade. The new construction satisfies the boundary conditions on both the cascade and duct walls simultaneously adding a new level of realism to the noise calculations. Preliminary results which demonstrate the effectiveness of the new elements are presented. A discussion of the utility of the asymptotic formulas for calculating cascade discrete tone as well as broadband noise is also included.
Envia, Edmane↗
Stability of large-scale systems with stable and unstable subsystems.
The purpose of this paper is to develop new methods for constructing vector Liapunov functions and broaden the application of Liapunov's theory to stability analysis of large-scale dynamic systems. The application, so far limited by the assumption that the large-scale systems are composed of exponentially stable subsystems, is extended via the general concept of comparison functions to systems which can be decomposed into asymptotically stable subsystems. Asymptotic stability of the composite system is tested by a simple algebraic criterion. With minor technical adjustments, the same criterion can be used to determine connective asymptotic stability of large-scale systems subject to structural perturbations. By redefining the constraints imposed on the interconnections among the subsystems, the considered class of systems is broadened in an essential way to include composite systems with unstable subsystems. In this way, the theory is brought substantially closer to reality since stability of all subsystems is no longer a necessary assumption in establishing stability of the overall composite system.
Grujic, Lj. T.↗
Asymptotic stability and instability of large-scale systems
The purpose of this paper is to develop new methods for constructing vector Lyapunov functions and broaden the application of Lyapunov's theory to stability analysis of large-scale dynamic systems. The application, so far limited by the assumption that the large-scale systems are composed of exponentially stable subsystems, is extended via the general concept of comparison functions to systems which can be decomposed into asymptotically stable subsystems. Asymptotic stability of the composite system is tested by a simple algebraic criterion. By redefining interconnection functions among the subsystems according to interconnection matrices, the same mathematical machinery can be used to determine connective asymptotic stability of large-scale systems under arbitrary structural perturbations.
Grujic, L. T.↗
Transition-state correlations for predicting thermochemistry of adsorbates and surface reactions
Estimating thermochemical properties from linear correlations may provide a pathway to circumvent expensive density functional theory (DFT) calculations for quantities such as pre-exponentials and temperature corrections to DFT energies. Here, in this work, we construct thermochemical scaling relations between C 1 –C 6 n-alkanes in the gas phase and adsorbed alkyl chains extending from several transition metal surfaces, and examine changes in the slope and fit between metals and adsorption sites. We subsequently add –OH, –NH 2 , C$=$O, and C$=$C functional groups to the C 1 –C 6 molecules and demonstrate strong linear correlations for thermochemistry across all species. We broaden the correlations to incorporate transition states of C 1 –C 6 n-alkane dehydrogenation reactions, where thermochemistry for computationally prohibitive transition-state calculations can be quickly assessed. Additionally, we rationalize the linearity of thermochemical correlations based on the composition of the homologous series and theoretical assessments. As an application of the correlations, we estimate pre-exponentials for elementary surface reactions of ethane and propane hydrogenolysis on Ru(0001), which is of relevance to plastic hydrogenolysis. Depending on kinetically important steps, entropic contributions may be necessary to include in certain reaction mechanisms; in contrasting examples, entropies are found to be relatively insignificant for ethane hydrogenolysis but pertinent for propane hydrogenolysis.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Dark matter halo mass functions and density profiles from mass and energy cascade
Abstract Halo abundance and structure play a central role for modeling structure formation and evolution. Without relying on a spherical or ellipsoidal collapse model, we analytically derive the halo mass function and cuspy halo density (inner slope of −4/3) based on the mass and energy cascade theory in dark matter flow. The hierarchical halo structure formation leads to halo or particle random walk with a position-dependent waiting time $$\tau _g$$ τ g . First, the inverse mass cascade from small to large scales leads to the halo random walk in mass space with $$\tau _g\propto m_h^{-\lambda }$$ τ g ∝ m h - λ , where $$m_h$$ m h is the halo mass and $$\lambda$$ λ is a halo geometry parameter with predicted value of 2/3. The corresponding Fokker-Planck solution for halo random walk in mass space gives rise to the halo mass function with a power-law behavior on small scale and exponential decay on large scale. This can be further improved by considering two different $$\lambda$$ λ for haloes below and above a critical mass scale $$m_h^*$$ m h ∗ , i.e. a double- $$\lambda$$ λ halo mass function. Second, a double- $$\gamma$$ γ density profile can be derived based on the particle random walk in 3D space with a position-dependent waiting time $$\tau _g \propto \Phi (r)^{-1} \propto r^{-\gamma }$$ τ g ∝ Φ ( r ) - 1 ∝ r - γ , where $$\Phi$$ Φ is the gravitational potential and r is the particle distance to halo center. Theory predicts $$\gamma =2/3$$ γ = 2 / 3 that leads to a cuspy density profile with an inner slope of −4/3, consistent with the predicted scaling laws from energy cascade. The Press-Schechter mass function and Einasto density profile are just special cases of proposed models. The small scale permanence can be identified due to the scale-independent rate of mass and energy cascade, where density profiles of different halo masses and redshifts converge to the $$-4/3$$ - 4 / 3 scaling law ( $$\rho _h \propto r^{-4/3}$$ ρ h ∝ r - 4 / 3 ) on small scales. Theory predicts the halo number density scales with halo mass as $$\propto m_h^{-1.9}$$ ∝ m h - 1.9 , while the halo mass density scales as $$\propto m_h^{4/9}$$ ∝ m h 4 / 9 . Results were compared against the Illustris simulations. This new perspective provides a theory for nearly universal halo mass functions and density profiles.
79 ASTRONOMY AND ASTROPHYSICS↗
Observable measurement-induced transitions
One of the main postulates of quantum mechanics is that measurements destroy quantum coherence (wave function collapse). Recently it was discovered that in a many-body system dilute local measurements still preserve some coherence across the entire system. As the measurement density is increased, a phase transition occurs that is characterized by the disentanglement of different parts of the system. Unfortunately, this transition is impossible to observe experimentally for macroscopic systems because it requires an exponentially costly full tomography of the many-body wave function or a comparison with the simulation on an oracle classical computer. In this work we report the discovery of another measurement-induced phase transition that can be observed experimentally if quantum dynamics can be reversed. On one side of this phase transition the quantum information encoded in some part of the Hilbert space is fully recovered after the time inversion. On the other side, all quantum information is corrupted. This transition also manifests itself as the change in the behavior of the probability to observe the same measurement outcome in the process that consists of identical blocks repeated many times. In each block the unitary evolution is followed by the measurement. On one side of the transition the probability decreases exponentially with the number of repetitions, on the other it tends to a constant as the number of repetitions is increased. We confirm the existence of the proposed phase transition through numerical simulations of realistic quantum circuits and analytical calculations using an effective random-matrix theory model.
Measurement-induced phase transitions↗