Time-dependent Moment Research Articles

Partial stochastic resetting with refractory periods

Abstract The effect of refractory periods in partial resetting processes is studied. Under Poissonian partial resets, a state variable jumps to a value closer to the origin by a fixed fraction at constant rate, $x\to a x$. Following each reset, a stationary refractory period of arbitrary duration takes place. We derive an exact closed-form expression for the propagator in Fourier-Laplace space, which shows rich dynamical features such as connections not only to other resetting schemes but also to intermittent motion. For diffusive processes, we use the propagator to derive exact expressions for time dependent moments of $x$ at all orders. At late times the system reaches a non-equilibrium steady state which takes the form of a mixture distribution that splits the system into two subpopulations; trajectories that at any given time in the stationary regime find themselves in the freely evolving phase, and those that are in the refractory phase. In contrast to conventional resetting, partial resets give rise to non-trivial steady states even for the refractory subpopulation. Moments and cumulants associated with the steady state density are studied, and we show that a universal optimum for the kurtosis can be found as a function of mean refractory time, determined solely by the strength of the resetting and the mean inter-reset time. The presented results could be of relevance to growth-collapse processes with periods of inactivity following a collapse.

Induced magnetization in anisotropic environment

The non-Markovian dynamics of a two-dimensional charged harmonic oscillator linearly coupled to a neutral bosonic heat bath is investigated in a linear-polarized electric field. The analytical expressions for the time-dependent and asymptotic magnetic moment are derived for the Markovian and non-Markovian dynamics. It is predicted that the liner-polarized electric field generates strong orbital currents and magnetization in a symmetric harmonic oscillator embedded into anisotropic heat bath. The role of the mixed dissipative kernel in magnetization is analyzed.

Electronic structure, absorption spectra and oxidation dynamics in polyynes and dicyanopolyynes.

The advent of femtosecond to attosecond experimental tools has made now possible to study such ultrafast carrier dynamics, e.g., the spatial and temporal charge density evolution, after an initial oxidation or reduction in molecules, candidates for atomic wires like polyynes and dicyanopolyynes. Here, we study the electronic structure and hole transfer in symmetric molecules containing carbon, nitrogen and hydrogen, the first members in the series of polyynic carbynes and dicyanopolyynes, using methods based on density functional theory (DFT): constrained DFT (CDFT), time-dependent DFT (TDDFT) and real-time TDDFT (RT-TDDFT), with Löwdin population analysis, comparing many levels of theory and obtaining convergence of the results. For the same purposes, we develop a tight binding (TB) variant using all valence orbitals of all atoms. This TB variant is applied here in linear molecules, but it is also adequate for electronic structure, charge transfer and charge transport of non-linear molecules and clusters of molecules. We calculate the electronic structure, the time-dependent dipole moment and the probabilities of finding the hole at each site, their mean over time values, the mean transfer rates from the oxidation site to other sites and the frequency content (using charge as well as dipole moment oscillations). We take into account zero-point motion. The initial conditions for RT-TDDFT are obtained by CDFT. For TB, we explore different initial conditions: we place the hole at a particular orbital or distribute it among a number of orbitals; it is also possible to include phase differences between orbitals. Finally, we compare with available experimental data.

Exact moments for trapped active particles: inertial impact on steady-state properties and re-entrance

In this study, we investigate the behavior of inertial active Brownian particles in a d-dimensional harmonic trap in the presence of translational diffusion. While the solution of the Fokker–Planck equation is generally challenging, it can be utilized to compute the exact time evolution of all time-dependent dynamical moments using a Laplace transform approach. We present the explicit form for several moments of position and velocity in d-dimensions. An interplay of time scales assures that the effective diffusivity and steady-state kinetic temperature depend on both inertia and trap strength, unlike passive systems. The distance from equilibrium, measured by the violation of equilibrium fluctuation-dissipation and the amount of entropy production, decreases with increasing inertia and trap strength. We present detailed ‘phase diagrams’ using kurtosis of velocity and position, showing possibilities of re-entrance to equilibrium.

Geometric phases of nonlinear elastic [formula omitted]-rotors via Cartan’s moving frames

We study the geometric phases of nonlinear elastic N-rotors with continuous rotational symmetry. In the Hamiltonian framework, the geometric structure of the phase space is a principal fiber bundle, i.e., a base, or shape manifold B, and fibers F along the symmetry direction attached to it. The symplectic structure of the Hamiltonian dynamics determines the connection and curvature forms of the shape manifold. Using Cartan’s structural equations with zero torsion we find an intrinsic (pseudo) Riemannian metric for the shape manifold. Without lose of generality, we show that one has the freedom to define the rotation sign of the total angular momentum A of the elastic rotors as either positive or negative, e.g., counterclockwise or clockwise, respectively, or viceversa. This endows the base manifold B with two distinct metrics both compatible with the geometric phase. In particular, the metric is pseudo-Riemannian if A<0, and the shape manifold is a 2D Robertson-Walker spacetime with positive curvature. For A>0, the shape manifold is the hyperbolic plane H2 with negative curvature. We then generalize our results to free elastic N-rotors. We show that the associated shape manifold B is reducible to the product manifold of N−1 hyperbolic planes H2 (A>0), or 2D Robertson-Walker spacetimes (A<0) depending on the convection used to define the rotation sign of the total angular momentum. We then consider elastic N-rotors subject to time-dependent self-equilibrated moments. The N-dimensional shape manifold of the extended autonomous system has a structure similar to that of the (N−1)-dimensional shape manifold of free elastic rotors. The Riemannian structure of the shape manifold provides an intrinsic measure of the closeness of one shape to another in terms of curvature, or induced geometric phase.

Cost-Efficient High-Resolution Linear Absorption Spectra through Extrapolating the Dipole Moment from Real-Time Time-Dependent Electronic-Structure Theory.

We present a novel function fitting method for approximating the propagation of the time-dependent electric dipole moment from real-time electronic structure calculations. Real-time calculations of the electronic absorption spectrum require discrete Fourier transforms of the electric dipole moment. The spectral resolution is determined by the total propagation time, i.e., the trajectory length of the dipole moment, causing a high computational cost. Our developed method uses function fitting on shorter trajectories of the dipole moment, achieving arbitrary spectral resolution through extrapolation. Numerical testing shows that the fitting method can reproduce high-resolution spectra by using short dipole trajectories. The method converges with as little as 100 a.u. dipole trajectories for some systems, though the difficulty converging increases with the spectral density. We also introduce an error estimate of the fit, reliably assessing its convergence and hence the quality of the approximated spectrum.

Strong ultrafast demagnetization due to the intraband transitions

Demagnetization in ferromagnetic transition metals driven by a femtosecond laser pulse is a fundamental problem in solid state physics, and its understanding is essential to the development of spintronic devices. Ab initio calculation of time-dependent magnetic moment in the velocity gauge so far has not been successful in reproducing the large amount of demagnetization observed in experiments. In this work, we propose a method to incorporate intraband transitions within the velocity gauge through a convective derivative in the crystal momentum space. Our results for transition-element bulk crystals (bcc Fe, hcp Co and fcc Ni) based on the time-dependent quantum Liouville equation show a dramatic enhancement in the amount of demagnetization after the inclusion of an intraband term, in agreement with experiments. We also find that the effect of intraband transitions on each ferromagnetic material is distinctly different because of their band structure and spin property differences. Our finding has a far-reaching impact on understanding of ultrafast demagnetization.

Attosecond Real-Time Observation of Recolliding Electron Trajectories in Helium at Low Laser Intensities.

Laser-driven recollision physics is typically accessible only at field intensities high enough for tunnel ionization. Using an extreme ultraviolet pulse for ionization and a near-infrared (NIR) pulse for driving of the electron wave packet lifts this limitation. This allows us to study recollisions for a broad range of NIR intensities with transient absorption spectroscopy, making use of the reconstruction of the time-dependent dipole moment. Comparing recollision dynamics with linear vs circular NIR polarization, we find a parameter space, where the latter favors recollisions, providing evidence for the so far only theoretically predicted recolliding periodic orbits.

Simulating creep induced moment redistribution in prestressed concrete bridges constructed by the balanced cantilever method: ad hoc traditional formulae versus real time-dependent analysis

Assessing available numerical techniques adopted to determine the design time-dependent moment for prestressed concrete segmental bridges constructed by the balanced cantilever method is of utmost importance to the bridge design community. In essence, despite some apparent diversity, there are basically two key conventional approaches to compute the design time-dependent moment accounting for creep effects for this type of bridges. The first is a family of varied simplified methods typically known to practicing designers and with pre-consensus on their reliability and effectiveness. Time-dependent moments retrieved from these classical methods always reside in an intermediate state falling between the results from “two” time-independent analysis cases, namely, (a) sequentially adding all partial permanent loads and prestressing pertaining to various construction steps using the part-bridge structural system corresponding to each step, and (b) assuming all loads and prestressing forces to be applied at-once to the final completed bridge. The second approach is through performing real sophisticated step-by-step time-dependent analysis using a specialized software. The research primary objective is to assess the validity/reliability of commonly used ad-hoc approaches that evolved over the years relying on simplified analyses/formulae to cater for time-dependent creep effects for this type of bridges. Aiming at realistic conclusions, three case-study real-world segmental balanced cantilever bridges over the Nile River in Egypt are elected. Midas Civil commercial package is used to perform time-dependent finite element analyses for the three bridges. Main parameters considered are, inter alia, time-dependent effects of creep and shrinkage of concrete, relaxation of prestressed steel, losses due to friction and anchor setting of prestressing tendons, sequence of construction, and construction-driven temporary change of support conditions (where applicable). The study concludes that creep-induced moment redistribution from simplified traditional formulae typically adopted in the literature may lead to a considerable error in estimating the design time-dependent moment in balanced cantilever bridges.

Waveform Modulation of High-Order Harmonics Generated from an Atom Irradiated by a Laser Pulse and a Weak Orthogonal Electrostatic Field

The detailed characteristics of the harmonics emission of atoms driven via a linearly polarized laser field combined with an orthogonal, weaker electrostatic field were investigated by numerically solving the time-dependent Schrödinger equation. It was found that the direction of the laser polarization and the polarization of the attosecond light, which is synthesized from the harmonic, can be controlled by the amplitude of the electrostatic field. With the analysis of the spatial distribution of the time-dependent dipole moment and the time-dependent evolution of the electronic wave packet, the control mechanism for the harmonic characters was investigated. The generation of harmonics in the vertical direction of the laser electric field is caused by the breaking of the symmetry of the time wave packet distribution. With this mechanism, we obtained circularly polarized attosecond light.

Analysis of Single Event Transients in Arbitrary Waveforms Using Statistical Window Analysis

Window or taper functions are commonly used in data processing to detect transient events or for time-averaging of frequency spectra. A generalized window function is demonstrated using the Ionizing Radiation Effects Spectroscopy (IRES) technique to enhance the measurement of transient anomalies within arbitrary waveforms. The IRES filter is used to convolve time data with a sliding window consisting of a moment-generating function. The resulting time-dependent statistical moments can be used to eliminate any steady-state signatures, including noise, and extract transient behaviors. The IRES filter is used to analyze data from heavy-ion exposures of commercial off-the-shelf (COTS) operational amplifiers, laser-induced transients in CMOS phase-locked loops, and simulated transients in digital and analog circuits. The performance of the IRES filter in noisy environments shows that transients can be measured with higher fidelity than standard amplitude thresholding. This statistical window analysis technique may remove the need for complex triggering mechanisms on instrumentation and does not require <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a-priori</i> knowledge of transient characteristics. Potential applications of IRES include real-time measurement, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in-situ</i> data analysis, and machine learning.

Surface termination dependent optical characteristics of MXene nanoflakes

Two-dimensional (2D) transition metal carbides, nitrides, and carbonitrides, known as MXenes, are an expanding class of 2D materials. With the compositional diversity and range of surface terminations, they have the potential to facilitate the development of novel devices. Herein, based on quantum-mechanical simulations via time-dependent density functional theory, we investigate how the M element (Ti or V) and surface terminations (Tx such as F and O) affect the optical and plasmonic characteristics of M2CTx MXene nanoflakes. We reveal the formation of localized surface plasmon resonances (LSPRs) in MXene nanoflakes. Interestingly, the Janus MXene nanoflake Ti2CFO (top surface terminated with F and bottom surface terminated with O) produces the strongest and narrowest infrared LSPR, which is attributable to its largest out-of-plane time-dependent dipole moment. The findings of this study may be helpful to those exploring on MXene plasmonics.

On the impact of differential diffusion between soot and gas phase species in turbulent flames

The molecular diffusivities of larger PAHs and soot particles approach zero leading to differential diffusion with gas-phase species. The present work systematically quantifies the impact on soot moments, soot related statistical correlations and Particle Size Distributions (PSDs) using a fully coupled transported joint probability density function (JPDF) method featuring a 78-dimensional joint-scalar space, including enthalpy, gas phase species with the PSD discretised using 62 size classes via a mass and number density preserving sectional method. Differential diffusion of soot (DDS) is treated via a gradual decline of diffusivity among soot sections maintaining realisability and the expected exponential decay of variance. The solution of the flow field features a time-dependent second moment closure and an elliptic solver. The turbulent non-premixed Sandia C2H4 flame from the International Sooting Flame (ISF) data base was selected as a target along with the KAUST (C2H4/N2) variant of the same flame. Results show that reduced soot diffusion leads to a significant increase in the soot volume fraction RMS and that the correlation coefficient between soot volume fraction and temperature is further reduced in a particle-size-dependent manner. Similar observations are made for correlations between the soot volume fraction and the mass fractions of gas-phase species such as CO, OH, H and C2H2. The results suggest that computational methods that presume explicit (e.g. flame structure related) correlations between such scalars and with soot face leading order modeling challenges. It is also shown that the correlation between CO and soot increases due to oxidation of soot and that DDS leads to a modest downstream shift of PSDs towards larger particles.

Time-Dependent Seismic Reliability of Coastal Bridge Piers Subjected to Nonuniform Corrosion.

Coastal bridge piers suffer random performance deterioration owing to the presence of complex nonuniform corrosion characteristics and material uncertainties. Some of these piers will also be threatened by random earthquakes during a long-term service period, and therefore, structural safety needs to be probabilistically assessed by the seismic reliability method. To deal with this problem, we present a method to calculate the time-dependent reliability of the coastal bridge pier, comprehensively considering the randomness of a seismic event, nonuniform corrosion, and material uncertainty. First, the time-dependent M-N interaction diagrams are established by using the Monte Carlo simulation method. On the basis of the interaction diagrams, the moment resistance reduction function and time-dependent moment resistance distribution are determined. Subsequently, the moment demand under the seismic load is determined using the Poisson model and the response acceleration spectrum. Then, the formulas to calculate the time-dependent reliability of a nonuniform corroded pier are derived on the basis of the theorem of total probability. The proposed method is illustrated with a case study of a coastal bridge pier. It was found that the increase in corrosion damage would obviously increase time-dependent reliability. Furthermore, the increase in submerged zone height delayed the year when the failure section shifts from the pier bottom to the bottom of the splash and tidal zone, and it reduces the failure probability of the coastal pier. The research results presented herein show that the nonuniform corrosion manifestations influence the failure mode-related time-dependent seismic reliability of the coastal bridge pier.

Transient Dynamics and Propagation of Voltage and Frequency Fluctuations in Transmission Grids

The energy transition towards increased electric power production from renewable energy (RE) resources creates new challenges to ensure the stability of power grids. In conventional power grids voltage fluctuations can be controlled locally. Here, it is explored whether this may be changed by the energy transition. It is well established that the increase of RE resources in power grids increases the amplitude of frequency deviations and the velocity with which these deviations spread throughout the power grid. However, its effect on voltage dynamics and propagation has not been systematically studied. Here, a systematic study is carried out of the transients of voltage amplitude, phase and frequency deviations due to local contingencies in dependence on system inertia, heterogeneity and topology. The 3rd order dynamic power grid model is studied numerically and analytically and compared with real grid simulations for the Nigerian (330 kV) power grid and other grid models, using DigSILENT PowerFactory software. A quantitative analysis of the parametric dependence of the velocity with which a disturbance propagates throughout the grid and of the period of oscillations of the frequency and voltage transients is provided. Beating patterns are found in the transients and are identified as footprints of the location of the fault bus, as caused by multiple reflections of propagating disturbances from the grid boundary. These may result in interarea oscillations. It is confirmed that voltage deviations remain local for realistic ranges of parameters, but that it can propagate by literally surfing on the frequency deviation wave. However, it is found that this no longer holds true when the electrical power in the grid approaches its critical value beyond which no stationary solution exists. Furthermore, time dependent second moments of the geodesic distance weighted with frequency deviations <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{\delta \omega }(t)$ </tex-math></inline-formula> and voltage deviations <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{\delta V}(t)$ </tex-math></inline-formula> , respectively are evaluated, confirming a ballistic disturbance propagation in homogeneous model grids. However, in real grid simulations, a linear time dependence of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S(t)$ </tex-math></inline-formula> is observed, indicating a diffusive propagation due to multiple scattering from the inhomogeneities in these power grids.

Quantum-Compute Algorithm for Exact Laser-Driven Electron Dynamics in Molecules.

In this work, we investigate the capability of known quantum computing algorithms for fault-tolerant quantum computing to simulate the laser-driven electron dynamics of excitation and ionization processes in small molecules such as lithium hydride, which can be benchmarked against the most accurate time-dependent full configuration interaction (TD-FCI) calculations. The conventional TD-FCI wave packet propagation is reproduced using the Jordan-Wigner transformation for wave function and operators and the Trotter product formula for expressing the propagator. In addition, the time-dependent dipole moment, as an example of a time-dependent expectation value, is calculated using the Hadamard test. To include non-Hermitian operators in the ionization dynamics, a similar approach to the quantum imaginary time evolution (QITE) algorithm is employed to translate the propagator, including a complex absorption potential, into quantum gates. The computations are executed on a quantum computer simulator. By construction, all quantum computer algorithms, except for the QITE algorithm used only for ionization but not for excitation dynamics, would scale polynomially on a quantum computer with fully entangled qubits. In contrast, TD-FCI scales exponentially. Hence, quantum computation holds promises for substantial progress in the understanding of electron dynamics of excitation processes in increasingly large molecular systems, as has already been witnessed in electronic structure theory.

Time-Dependent Moments From the Heat Equation and a Transport Equation

Abstract We present a new connection between the classical theory of full and truncated moment problems and the theory of partial differential equations, as follows. For the classical heat equation $\partial _t u = {\nu } \Delta u$, with initial data $u_0 \in {\mathcal {S}}(\mathbb {R}^n)$, we first compute the moments $s_{\alpha }(t)$ of the unique solution $u \in {\mathcal {S}}(\mathbb {R}^n)$. These moments are polynomials in the time variable, of degree comparable to $\alpha $, and with coefficients satisfying a recursive relation. This allows us to define the polynomials for any sequence, and prove that they preserve some of the features of the heat kernel. In the case of moment sequences, the polynomials trace a curve (which we call the heat curve), which remains in the moment cone for positive time, but may wander outside the moment cone for negative time. This provides a description of the boundary points of the moment cone, which are also moment sequences. We also study how the determinacy of a moment sequence behaves along the heat curve. Next, we consider the transport equation $\partial _t u = ax \cdot \nabla u$ and conduct a similar analysis. Along the way we incorporate several illustrating examples. We show that while $\partial _t u = {\nu }\Delta u + ax\cdot \nabla u$ has no explicit solution, the time-dependent moments can be explicitly calculated.

Vibrations of graphene platelet reinforced composite doubly curved shells subjected to thermal shock

An investigation is performed in this research to study the response of doubly curved shells subjected to rapid surface heating. Doubly curved shell is made from a composite laminated media where each layer is reinforced with graphene platelets (GPLs). Also each layer may have different amount of GPLs which results in a piecewise functionally graded graphene platelet reinforced composite (FG-GPLRC) shell. The elasticity modulus of the shell is estimated by means of the Halpin-Tsai rule where the size of the reinforcements is also included. It should be noted that mass density, thermal expansion coefficient, Poisson’s ratio and specific heat are evaluated using the simple rule of mixture approach. The one-dimension transient heat conduction equation is established and solved for each layer of the composite media. The continuity of the temperature change and heat flux are also considered between the layers. After that, time dependent thermal moment and thermal force are evaluated and inserted into the equations of motion of the structure. First order shear deformation theory of shells and Donnell kinematics are applied to establish the basic equations of the shell. With the aid of the Ritz method and Chebyshev polynomials as the basic functions, the matrix representation of the motion equations of the doubly curved shallow shells subjected to rapid surface heating are established. These equations are traced in time using the Newmark time marching scheme. Results of this study are compared with the available data in the open literature. After that novel results are provided for vibrations induced by rapid surface heating in FG-GPLRC doubly curved shells.

Accelerating Real-Time Coupled Cluster Methods with Single-Precision Arithmetic and Adaptive Numerical Integration.

We explore the framework of a real-time coupled cluster method with a focus on improving its computational efficiency. Propagation of the wave function via the time-dependent Schrödinger equation places high demands on computing resources, particularly for high level theories such as coupled cluster with polynomial scaling. Similar to earlier investigations of coupled cluster properties, we demonstrate that the use of single-precision arithmetic reduces both the storage and multiplicative costs of the real-time simulation by approximately a factor of 2 with no significant impact on the resulting UV/vis absorption spectrum computed via the Fourier transform of the time-dependent dipole moment. Additional speedups─of up to a factor of 14 in test simulations of water clusters─are obtained via a straightforward GPU-based implementation as compared to conventional CPU calculations. We also find that further performance optimization is accessible through sagacious selection of numerical integration algorithms, and the adaptive methods, such as the Cash-Karp integrator, provide an effective balance between computing costs and numerical stability. Finally, we demonstrate that a simple mixed-step integrator based on the conventional fourth-order Runge-Kutta approach is capable of stable propagations even for strong external fields, provided the time step is appropriately adapted to the duration of the laser pulse with only minimal computational overhead.

Self-propulsion with speed and orientation fluctuation: Exact computation of moments and dynamical bistabilities in displacement.

We consider the influence of active speed fluctuations on the dynamics of a d-dimensional active Brownian particle performing a persistent stochastic motion. The speed fluctuation brings about a dynamical anisotropy even in the absence of shape anisotropy. We use the Laplace transform of the Fokker-Planck equationto obtain exact expressions for time-dependent dynamical moments. Our results agree with direct numerical simulations and show several dynamical crossovers determined by the activity, persistence, and speed fluctuation. The dynamical anisotropy leads to a subdiffusive scaling in the parallel component of displacement fluctuation at intermediate times. The kurtosis remains positive at short times determined by the speed fluctuation, crossing over to a negative minimum at intermediate times governed by the persistence before vanishing asymptotically. The probability distribution of particle displacement obtained from numerical simulations in two dimensions shows two crossovers between compact and extended trajectories via two bimodal distributions at intervening times. While the speed fluctuation dominates the first crossover, the second crossover is controlled by persistence like in the wormlike chain model of semiflexible polymers.