A GENERALIZED SCALAR AUXILIARY VARIABLE METHOD FOR THE TIME-DEPENDENT GINZBURG-LANDAU EQUATIONS

This paper develops a generalized scalar auxiliary variable(SAV)method for the time-dependent Ginzburg-Landau equations.The backward Euler method is used for discretizing the temporal derivative of the time-dependent Ginzburg-Landau equations.In this method,the system is decoupled and linearized to avoid solving the non-linear equation at each step.The theoretical analysis proves that the generalized SAV method can preserve the maximum bound principle and energy stability,and this is confirmed by the numerical result,and also shows that the numerical algorithm is stable.

Time-dependent Diffusion Coefficient and Conventional Diffusion Constant of Nanoparticles in Polymer Melts by Mode-coupling Theory

Time-dependent diffusion coefficient and conventional diffusion constant are calculated and analyzed to study diffusion of nanoparticles in polymer melts. A generalized Langevin equa- tion is adopted to describe the diffusion dynamics. Mode-coupling theory is employed to calculate the memory kernel of friction. For simplicity, only microscopic terms arising from binary collision and coupling to the solvent density fluctuation are included in the formalism. The equilibrium structural information functions of the polymer nanocomposites required by mode-coupling theory are calculated on the basis of polymer reference interaction site model with Percus-Yevick closure. The effect of nanoparticle size and that of the polymer size are clarified explicitly. The structural functions, the friction kernel, as well as the diffusion coefficient show a rich variety with varying nanoparticle radius and polymer chain length. We find that for small nanoparticles or short chain polymers, the characteristic short time non-Markov diffusion dynamics becomes more prominent, and the diffusion coefficient takes longer time to approach asymptotically the conventional diffusion constant. This constant due to the microscopic contributions will decrease with the increase of nanoparticle size, while increase with polymer size. Furthermore, our result of diffusion constant from mode- coupling theory is compared with the value predicted from the Stokes-Einstein relation. It shows that the microscopic contributions to the diffusion constant are dominant for small nanoparticles or long chain polymers. Inversely, when nanonparticle is big, or polymer chain is short, the hydrodynamic contribution might play a significant role.

A time-dependent density functional theory investigation of plasmon resonances of linear Au atomic chains

We report theoretical studies on the plasmon resonances in linear Au atomic chains by using ab initio time- dependent density functional theory. The dipole responses are investigated each as a function of chain length. They converge into a single resonance in the longitudinal mode but split into two transverse modes. As the chain length increases, the longitudinal plasmon mode is redshifted in energy while the transverse modes shift in the opposite direction （blueshifts）. In addition, the energy gap between the two transverse modes reduces with chain length increasing. We find that there are unique characteristics, different from those of other metallic chains. These characteristics are crucial to atomic-scale engineering of single-molecule sensing, optical spectroscopy, and so on.

A three-dimensional time-dependent theory for helix traveling wave tubes in beam-wave interaction

This paper presents a three-dimensional time-dependent nonlinear theory of helix traveling wave tubes for beam- wave interaction. The radio frequency electromagnetic fields are represented as the superposition of azimuthally sym- metric waves in a vacuum sheath helix. Coupling impedance is introduced to the electromagnetic field equations＇ stimulating sources, which makes the theory easier and more flexible to realize. The space charge fields are calculated by electron beam space-charge waves expressed as the superposition solutions of Helmholtz equations. The focusing forces due to either a solenoidal field or a periodic permanent magnetic field is also included. The dynamical equations of electrons are Lorentz equations associating with electromagnetic fields, focusing fields and space-charge fields. The numerically simulated results of a tube are presented.

Theoretical investigation on the excited state intramolecular proton transfer in Me_(2)N substituted flavonoid by the time-dependent density functional theory method

Time-dependent density functional theory(TDDFT)method is used to investigate the details of the excited state intramolecular proton transfer(ESIPT)process and the mechanism for temperature effect on the Enol^(*)/Keto^(*)emission ratio for the Me_(2)N-substited flavonoid(MNF)compound.The geometric structures of the S_(0) and S_(1) states are denoted as the Enol,Enol^(*),and Keto*.In addition,the absorption and fluorescence peaks are also calculated.It is noted that the calculated large Stokes shift is in good agreement with the experimental result.Furthermore,our results confirm that the ESIPT process happens upon photoexcitation,which is distinctly monitored by the formation and disappearance of the characteristic peaks of infrared(IR)spectra involved in the proton transfer and in the potential energy curves.Besides,the calculations of highest occupied molecular orbital(HOMO)and lowest unoccupied molecular orbital(LUMO)reveal that the electronegativity change of proton acceptor due to the intramolecular charge redistribution in the S_(1) state induces the ESIPT.Moreover,the thermodynamic calculation for the MNF shows that the Enol^(*)/Keto^(*)emission ratio decreasing with temperature increasing arises from the barrier lowering of ESIPT.

Theoretical Study on the Optical Properties for 2,7- and 3,6-Linked Carbazole Trimers by Time-dependent Density Functional Theory

Electronic properties, such as HOMO and LUMO energies, band gaps, ionization potential （IP） and electron affinity （EA） of 2,7- and 3,6-1inked carbazole trimers, two conjugated oligomcrs with different linkages of carbazole, were studicd by the density functional theory with Becke-Lee-Young-Parr composite exchange correlation functional （B3LYP）. The absorption spectra of these compounds were also investigated by time-dependent density functional theory （TD-DFT） with 6-3 IG＊ basis set. The calculated results indicated that the HOMO and LUMO of the 2,7- and 3,6-1inked carbazole trimers are both slightly destabilized on going from methyl substitution to sec-butyl substitution. Both IP and EA exhibit their good hole-transporting but poor electronaccepting ability. The presence of alkyl groups on the nitrogen atoms does not affect the intra-chain electronic delocalization along the molecular frame. Thus no significant effect on the band gap and absorption spectra of compounds has been found.

Determination of safe mud window considering time-dependent variations of temperature and pore pressure:Analytical and numerical approaches

Wellbore stability is a key to have a successful drilling operation.Induced stresses are the main factors affecting wellbore instability and associated problems in drilling operations.These stresses are significantly impacted by pore pressure variation and thermal stresses in the field.In order to address wellbore instability problems,it is important to investigate the mechanisms of rockefluid interaction with respect to thermal and mechanical aspects.In order to understand the induced stresses,different mathematical models have been developed.In this study,the field equations governing the problem have been derived based on the thermo-poroelastic theory and solved analytically in Laplace domain.The results are transferred to time domain using Fourier inverse method.Finite difference method is also utilized to validate the results.Pore pressure and temperature distributions around the wellbore have been focused and simulated.Next,induced radial and tangential stresses for different cases of cooling and heating of formation are compared.In addition,the differences between thermo-poroelastic and poroelastic models in situation of permeable and impermeable wellbores are described.It is observed that cooling and pore pressure distribution reinforce the induced radial stress.Whereas cooling can be a tool to control and reduce tangential stress induced due to invasion of drilling fluid.In the next step,safe mud window is obtained using Mohr-Coulomb,Mogi-Coulomb,and modified Lade failure criteria for different inclinations.Temperature and pore pressure distributions do not change the minimum allowable wellbore pressure significantly.However,upper limit of mud window is sensitive to induced stresses and it seems vital to consider changes in temperature and pore pressure to avoid any failures.The widest and narrowest mud windows are proposed by modified Lade and Mohr-Coulomb failure criteria,respectively.

Influence of Auxiliary Equation on Wave Functions for Time-Dependent Pauli Equation in Presence of Aharonov-Bohm Effect

Invariant operator method for discrete or continuous spectrum eigenvalue and unitary transformation approach are employed to study the two-dimensional time-dependent Pauli equation in presence of the Aharonov-Bohm effect （AB） and external scalar potential. For the spin particles the problem with the magnetic field is that it introduces a singularity into wave equation at the origin. A physical motivation is to replace the zero radius flux tube by one of radius R, with the additional condition that the magnetic field be confined to the surface of the tube, and then taking the limit R → 0 at the end of the computations. We point that the invariant operator must contain the step function θ（r - R）. Consequently, the problem becomes more complicated. In order to avoid this dimculty, we replace the radius R by ρ（t）R, where ρ（t） is a positive time-dependent function. Then at the end of calculations we take the limit R →0. The qualitative properties for the invariant operator spectrum are described separately for the different values of the parameter C appearing in the nonlinear auxiliary equation satisfied by p（t）, i.e., C 〉 0, C = 0, and C 〈0. Following the C＇s values the spectrum of quantum states is discrete （C 〉 0） or continuous （C ≤ 0）.

Time Evolution Caused by Hamiltonian Composed of Quadratic Combination of Canonical Operators and Time-Dependent Two-Mode Fresnel Operator

We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU（1,1） algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis-Riesenfeld＇s invariant operator theory of treating timedependent harmonic oscillators.

Testing the validity of the Ehrenfest theorem beyond simple static systems: Caldirola–Kanai oscillator driven by a time-dependent force

The relationship between quantum mechanics and classical mechanics is investigated by taking a Gaussian-type wave packet as a solution of the Schr o¨dinger equation for the Caldirola–Kanai oscillator driven by a sinusoidal force. For this time-dependent system, quantum properties are studied by using the invariant theory of Lewis and Riesenfeld. In particular,we analyze time behaviors of quantum expectation values of position and momentum variables and compare them to those of the counterpart classical ones. Based on this, we check whether the Ehrenfest theorem which was originally developed in static quantum systems can be extended to such time-varying systems without problems.

Time-dependent Density Functional-based Tight-bind Method Efficiently Implemented with OpenMP Parallel and GPU Acceleration

The time-dependent density functional-based tight-bind （TD-DFTB） method is implemented on the multi-core and the graphical processing unit （GPU） system for excited state calcu-lations of large system with hundreds or thousands of atoms. Sparse matrix and OpenMP multithreaded are used for building the Hamiltonian matrix. The diagonal of the eigenvalue problem in the ground state is implemented on the GPUs with double precision. The GPU- based acceleration fully preserves all the properties, and a considerable total speedup of 8.73 can be achieved. A Krylov-space-based algorithm with the OpenMP parallel and CPU acceleration is used for finding the lowest eigenvalue and eigenvector of the large TDDFT matrix, which greatly reduces the iterations taken and the time spent on the excited states eigenvalue problem. The Krylov solver with the GPU acceleration of matrix-vector product can converge quickly to obtain the final result and a notable speed-up of 206 times can be observed for system size of 812 atoms. The calculations on serials of small and large systems show that the fast TD-DFTB code can obtain reasonable result with a much cheaper computational requirement compared with the first-principle results of CIS and full TDDFT calculation.

Anisotropy of Critical Fields in MgB_2: Two-Band Ginzburg-Landau Theory for Layered Superconductors

The temperature dependence of the anisotropy parameter of upper critical field γHe2 （T） = Hc2^‖ （T） /Hc2^⊥ （T） and London penetration depth γλ （T） = λ‖（T）/λ⊥ （T） are calculated using two-band Ginzburg-Landau theory for layered superconductors. It is shown that, with decreasing temperature the anisotropy parameter γHc2 （T） is increased, while the London penetration depth anisotropy γλ（T） reveals an opposite behavior. Results of our calculations are in agreement with experimental data for single crystal MgB2 and with other calculations..Results of an analysis of magnetic field Hc1 in a single vortex between superconducting layers are also presented.

Photodetachment dynamics of H^- ion in a harmonic potential plus a time-dependent oscillating electric field

The photodetachment dynamics of H^- ion in a harmonic potential plus an oscillating electric field is studied using the time-dependent closed orbit theory. An analytical formula for calculating the photodetachment cross section of this system is put forward. It is found that the photodetachment cross section of this system is nearly unaffected for the weak oscillating electric field strength, but oscillates complicatedly when the oscillating electric field strength turns strong. In addition, the frequency of the harmonic potential and the oscillating electric field （the frequency of the harmonic potential and the frequency of the oscillating electric field are the same in the paper, unless otherwise stated.） can also affect the photodetachment dynamics of this system. With the increase of the frequency in the harmonic potential and the oscillating electric field, the number of the closed orbits for the detached electrons increased, which makes the oscillatory structure in the photodetachment cross section much more complex. Our study presents an intuitive understanding of the photodetachment dynamics driven by a harmonic potential plus an oscillating electric field from a space and time dependent viewpoint. This study is very useful in guiding the future experimental research for the photodetachment dynamics in the electric field both changing with space and time.

Time-dependent density functional theoretical studies on the photo-induced dynamics of an HCI molecule encapsulated in C60 under femtosecond laser pulses

By using first-principles simulations based on time-dependent density functional theory, the chemical reaction of an HCl molecule encapsulated in C60 induced by femtosecond laser pulses is observed. The H atom starts to leave the Cl atom and is reflected by the C60 wall. The coherent nuclear dynamic behaviors of bond breakage and recombination of the HCl molecule occurring in both polarized parallel and perpendicular to the H-Cl bond axis are investigated. The radial oscillation is also found in the two polarization directions of the laser pulse. The relaxation time of the H-Cl bond lengths in transverse polarization is slow in comparison with that in longitudinal polarization. Those results are important for studying the dynamics of the chemical bond at an atomic level.

Extra Time Dimension: Deriving Relativistic Space-Time Transformations, Kinematics, and Example of Dimensional Compactification Using Time-Dependent Non-Relativistic Quantum Mechanics

We consider a five-dimensional Minkowski space with two time dimensions characterized by distinct speeds of causality and three space dimensions. Formulas for relativistic coordinate and velocity transformations are derived, leading to a new expression for the speed limit. Extending the ideas of Einstein’s Theory of Special Relativity, concepts of five-velocity and five-momenta are introduced. We get a new formula for the rest energy of a massive object. Based on a non-relativistic limit, a two-time dependent Schrödinger-like equation for infinite square-well potential is developed and solved. The extra time dimension is compactified on a closed loop topology with a period matching the Planck time. It generates interference of additional quantum states with an ultra-small period of oscillation. Some cosmological implications of the concept of four-dimensional versus five-dimensional masses are briefly discussed, too.

Case Studies on Time-Dependent Ginzburg-Landau Simulations for Superconducting Applications

The macroscopic electromagnetic properties of type-II superconductors are mainly influenced by the behavior of micro-scopic superconducting flux quantum units.Time-dependent Ginzburg-Landau(TDGL)theory is a well-known tool for describing and examining both the statics and dynamics of these superconducting entities.It have been instrumental in replicating and elucidat-ing numerous experimental results over the past decades.This paper provides a comprehensive overview of the progress in TDGL simulations,focusing on three key aspects of superconductor applications.We delve first into vortex rectification in supercon-ductors described within the TDGL framework,specifically highlighting the achievement of superconducting diode effect through asymmetric pinning landscapes and the reversible manipulation of vortex ratchets with dynamic pinning landscapes.In terms of the achievements of TDGL simulations concerning the critical current density of superconductors,we emphasize particularly on the optimization of pinning sites,including vortex pinning and dynamics in polycrystalline Nb3Sn with grain boundaries.In the third aspect,we concentrate on numerical modeling of vortex penetration and dynamics in superconducting radio-frequency cavities,including a discussion on superconductor-insulator-superconductor multilayer structures.Finally,we present key findings,insights,and perspectives derived from the discussed simulations.

CLASSICAL SOLUTIONS OF GENERAL GINZBURG-LANDAU EQUATIONS

In this paper, we prove the existence of global classical solutions to time-dependent Ginzburg-Landau（TDGL） equations. By the properties of Besov and Sobolev spaces, together with the energy method, we establish the global existence and uniqueness of classical solutions to the initial boundary value problem for time-dependent Ginzburg-Landau equations.

Finite dimensional global and exponential attractors for a class of coupled time-dependent Ginzburg-Landau equations

We study a coupled nonlinear evolution system arising from the Ginzburg-Landau theory for atomic Fermi gases near the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover.First,we prove that the initial boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses a global attractor.Then we establish the existence of an exponential attractor.As a consequence,we show that the global attractor is of finite fractal dimension.

Microscopic mechanism of plasmon-mediated photocatalytic H_(2) splitting on Ag-Au alloy chain

Alloy nanostructures supporting localized surface plasmon resonances has been widely used as efficient photocatalysts,but the microscopic mechanism of alloy compositions enhancing the catalytic efficiency is still unclear.By using time-dependent density functional theory(TDDFT),we analyze the real-time reaction processes of plasmon-mediated H_(2) splitting on linear Ag-Au alloy chains when exposed to femtosecond laser pulses.It is found that H_(2) splitting rate depends on the position and proportion of Au atoms in alloy chains,which indicates that specially designed Ag-Au alloy is more likely to induce the reaction than pure Ag chain.Especially,more electrons directly transfer from the alloy chain to the anti-bonding state of H_(2),thereby accelerating the H_(2) splitting reaction.These results establish a theoretical foundation for comprehending the microscopic mechanism of plasmon-induced chemical reaction on the alloy nanostructures.

Plasmon-induced nonlinear response on gold nanoclusters

The plasmon-induced nonlinear response has attracted great attention in micro-nano optics and optoelectronics applications,yet the underlying microscopic mechanism remains elusive.In this study,the nonlinear response of gold nanoclusters when exposed to a femtosecond laser pulse was investigated using time-dependent density functional theory.It was observed that the third-order tunneling current was augmented in plasmonic dimers,owing to a greater number of electrons in the dimer being excited from occupied to unoccupied states.These findings provide profound theoretical insights and enable the realization of accurate regulation and control of nonlinear effects induced by plasmons at the atomic level.