Adiabatic approximation of the correlation function in the density-functional treatment of ionization processes

The ionization of a one-dimensional model helium atom in short laser pulses using time-dependent density-functional theory is investigated. We calculate ionization probabilities as a function of laser intensity by approximating the correlation function of the system adiabatically with an explicit dependence on the fractional number of bound electrons. For the correlation potential we take the derivative discontinuity at integer numbers of bound electrons explicitly into account. This approach reproduces ionization probabilities from the solution of the time-dependent Schr?dinger equation, in particular, the so-called knee due to nonsequential ionization.     

Local vertex corrections from exchange-correlation kernels with a discontinuity

The fundamental gap of an interacting many-electron system is given by the sum of the single-particle Kohn-Sham gap and the derivative discontinuity. The latter can be generated by advanced approximations to the exchange-correlation (XC) energy and is the key quantity to capture strong correlation with density functional theory (DFT). In this work we derive an expression for the derivative discontinuity in terms of the XC kernel of time-dependent density functional theory and demonstrate the crucial role of a discontinuity in the XC kernel itself. By relating approximate XC kernels to approximate local vertex corrections we then generate beyond-GW self-energies that include a discontinuity in the local vertex function. The quantitative importance of this result is illustrated with a numerical study of the local exchange vertex on model systems.     

Towards a description of the Kondo effect using time-dependent density-functional theory

We demonstrate that the zero-temperature conductance of the Anderson model can be calculated within the Landauer formalism combined with static density-functional theory. The proposed approximate functional is based on finite-temperature density-functional theory and yields the exact Kohn-Sham potential at the particle-hole symmetric point. Furthermore, in the limit of zero temperature it correctly exhibits a derivative discontinuity which is shown to be essential to reproduce the conductance plateau. On the other hand, at the Kondo temperature the exact Kohn-Sham conductance overestimates the real one by an order of magnitude. To understand the failure of density-functional theory, we resort to its time-dependent version and conclude that the suppression of the Kondo resonance must be attributed to dynamical exchange-correlation corrections.     

Time-dependent density-functional theory meets dynamical mean-field theory: real-time dynamics for the 3D Hubbard model

We introduce a new class of exchange-correlation potentials for a static and time-dependent density-functional theory of strongly correlated systems in 3D. The potentials are obtained via dynamical mean-field theory and, for strong enough interactions, exhibit a discontinuity at half-filling density, a signature of the Mott transition. For time-dependent perturbations, the dynamics is described in the adiabatic local density approximation. Results from the new scheme compare very favorably to exact ones in clusters. As an application, we study Bloch oscillations in the 3D Hubbard model.     

Time-dependent perturbation theory for nonequilibrium lattice models

We develop a time-dependent perturbation theory for nonequilibrium interacting particle systems. We focus on models such as the contact process which evolve via destruction and autocatalytic creation of particles. At a critical value of the destruction rate there is a continuous phase transition between an active steady state and the vacuum state, which is absorbing. We present several methods for deriving series for the evolution starting from a single seed particle, including expansions for the ultimate survival probability in the super- and subcritical regions, expansions for the average number of particles in the subcritical region, and short-time expansions. Algorithms for computer generation of the various expansions are presented. Rather long series (24 terms or more) and precise estimates of critical parameters are presented.     

旋转间断过渡层中的粒子轨道

用一维粒子Code研究了旋转间断过渡层中的粒子轨道。发现绝大多数粒子都能穿过过渡层到达下游区;但是,有少量的粒子被捕获在过渡层内;也有一定数量的粒子被反射回上游区。捕获粒子数和反射粒子数与过渡层厚度D有关。在稳定旋转间断中,反射粒子数和捕获粒子数分别为2％和3％左右。对于不稳定的旋转间断,这两个比值可上升2-4倍。反射粒子和捕获粒子的动力学效应可以很好地解释旋转间断,中间激波和多重激波模拟的一些重要结果.     

Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation

We construct the formal solution to the Cauchy problem for the dispersionless Kadomtsev-Petviashvili equation as an application of the inverse scattering transform for the vector field corresponding to a Newtonian particle in a time-dependent potential. This is in full analogy with the Cauchy problem for the Kadomtsev-Petviashvili equation, associated with the inverse scattering transform of the time-dependent Schrödinger operator for a quantum particle in a time-dependent potential.     

Time-Dependent Gaussian Solution for the Kostin Equation Around Classical Trajectories

The structure of time-dependent Gaussian solutions for the Kostin equation in dissipative quantum mechanics is analyzed. Expanding the generic external potential near the center of mass of the wave packet, one conclude that: the center of mass follows the dynamics of a classical particle under the external potential and a damping proportional to the velocity; the width of the wave packet satisfy a non-conservative Pinney equation. An appropriate perturbation theory is developed for the free particle case, solving the long standing problem of finding analytic expressions for square integrable solutions of the free Kostin equation. The associated Wigner function is also studied.     

Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations

The generalized Riemann problem (GRP) scheme for the Euler equations and gas-kinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a slightly better computational efficiency, and has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solvers. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an equilibrium one through particle collisions. The different mechanism in the flux evaluation deviates their numerical performance. Through this study, we may conclude scientifically that it may NOT be valid to use the Euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution. To adapt the Navier–Stokes (NS) equations is NOT valid either because the NS equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity. This fact alludes to the consistency of the Euler and Navier–Stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high Mach number flows. The development of numerical algorithm is similar to the modeling process in deriving the governing equations, but the control volume here cannot be shrunk to zero.     

Brownian motion under a time-dependent periodic potential proportional to the square of the position of a particle and classical fluctuation squeezing

A general stationary case of a Brownian particle with a time-dependent periodic potential proportional to the square of the position of the particle is treated. Even though the vigorous change of the time-dependent proportionality coefficient is applied, there are cases where the fluctuation of the particle decreases in contrary to our intuition, which is called classical fluctuation squeezing. We obtain time-average variances analytically for general cases of an arbitrary change in the coefficient and find conditions favorable for classical fluctuation squeezing. We introduce an asymmetric function behaves like trigonometric cosine one and consider its behavior explicitly.     

Large Deviations in the Superstable Weakly Imperfect Bose-Gas

The Superstable Weakly Imperfect Bose-Gas (Sup-WIBG) was originally proposed to solve some inconsistencies of the Bogoliubov theory based on the WIBG. The grand-canonical thermodynamics of the Sup-WIBG has been recently studied in details but only out of the point of the (first order) phase transition. The present paper closes this gap. The key technical tools are the Large Deviations (LD) formalism and in particular the analysis of the Kac distribution function. It turns out that the condensate fraction discontinuity as a function of the chemical potential (that occurs at the phase transition point) disappears if one considers it as a function of the total particle density. We prove that at this point the equilibrium state of the Sup-WIBG is a mixture of two (low- and high-density) pure phases related to two critical particle densities. Non-zero Bose-Einstein condensate starts at the smaller critical density and continuously grows (for a constant chemical potential) until the second critical density. For higher particle densities, the Bose condensate fraction as well as the chemical potential both increase monotonously.     

Physalis method for heterogeneous mixtures of dielectrics and conductors: Accurately simulating one million particles using a PC

Prosperetti’s seminal Physalis method, an Immersed Boundary/spectral method, had been used extensively to investigate fluid flows with suspended solid particles. Its underlying idea of creating a cage and using a spectral general analytical solution around a discontinuity in a surrounding field as a computational mechanism to enable the accommodation of physical and geometric discontinuities is a general concept, and can be applied to other problems of importance to physics, mechanics, and chemistry. In this paper we provide a foundation for the application of this approach to the determination of the distribution of electric charge in heterogeneous mixtures of dielectrics and conductors. The proposed Physalis method is remarkably accurate and efficient. In the method, a spectral analytical solution is used to tackle the discontinuity and thus the discontinuous boundary conditions at the interface of two media are satisfied exactly. Owing to the hybrid finite difference and spectral schemes, the method is spectrally accurate if the modes are not sufficiently resolved, while higher than second-order accurate if the modes are sufficiently resolved, for the solved potential field. Because of the features of the analytical solutions, the derivative quantities of importance, such as electric field, charge distribution, and force, have the same order of accuracy as the solved potential field during postprocessing. This is an important advantage of the Physalis method over other numerical methods involving interpolation, differentiation, and integration during postprocessing, which may significantly degrade the accuracy of the derivative quantities of importance. The analytical solutions enable the user to use relatively few mesh points to accurately represent the regions of discontinuity. In addition, the spectral convergence and a linear relationship between the cost of computer memory/computation and particle numbers results in a very efficient method. In the present paper, the accuracy of the method is numerically investigated by example computations using one dielectric particle, one isolated conductor particle, one conductor particle connected to an external source with imposed voltage, and two conductor/dielectric particles with strong interactions. The efficiency of the method is demonstrated with one million particles, which suggests that the method can be used for many important engineering applications of broad interest.     

由含时边界条件的两种有限深量子势阱构造的哈密顿算符和它们的复BERRY相位

在量子基础理论的框架下分别从一维含时边界条件的半壁无限高量子势阱和三维含时边界条件的有限深球方势阱构造了两种描述带电粒子在电磁场作用下的新哈密顿算符.在绝热近似下，计算了这两种新量子体系的复Berry相位.     

Particle creation,classicality and related issues in quantum field theory: I. Formalism and toy models

The quantum theory of a harmonic oscillator with a time dependent frequency arises in several important physical problems, especially in the study of quantum field theory in an external background. While the mathematics of this system is straightforward, several conceptual issues arise in such a study. We present a general formalism to address some of the conceptual issues like the emergence of classicality, definition of particle content, back reaction etc. In particular, we parameterize the wave function in terms of a complex number (which we call excitation parameter) and express all physically relevant quantities in terms it. Many of the notions—like those of particle number density, effective Lagrangian etc., which are usually defined using asymptotic in–out states—are generalized as time-dependent concepts and we show that these generalized definitions lead to useful and reasonable results. Having developed the general formalism we apply it to several examples. Exact analytic expressions are found for a particular toy model and approximate analytic solutions are obtained in the extreme cases of adiabatic and highly non-adiabatic evolution. We then work out the exact results numerically for a variety of models and compare them with the analytic results and approximations. The formalism is useful in addressing the question of emergence of classicality of the quantum state, its relation to particle production and to clarify several conceptual issues related to this. In Paper II which is a sequel to this, the formalism will be applied to analyze the corresponding issues in the context of quantum field theory in background cosmological models and electric fields.     

On the continuity in energy of the free-bound one-photon cross-sections

For a charged particle in a central field of force we discuss the continuity in energy of the cross-sections for the related processes of ground state photoeffect and radiative capture. This continuity follows from general arguments, while for a Coulomb potential, where exact analytic expressions are available, the continuity can also be traced explicitly through the calculation. Our analysis allows us to identify an error in papers on radiative muon capture in the ground state which claim the existence of a discontinuity in the cross-section, occurring in hydrogen for muon energies in the neighbourhood of 2.8 keV. As a discontinuity in radiative capture would imply a discontinuity in atomic photoeffect, we note that in the latter case there are extensive experimental results in good qualitative agreement with the usual continuous result (Stobbe formula) for the cross-section. Received: 13 February 1998 / Accepted: 14 May 1998     

Gravitational Waves and Discontinuous Motions

The mathematical theory of gravitational wave-fronts is revisited with the help of orthogonal decomposition techniques. Thus many important well known results about gravitational waves in empty space are readily obtained in the local geometry framework and in general coordinates. Then different kinds of motion in the presence of a wave are investigated. Our study shows that a discontinuity effect, in traversing the wave-front, arises in the motion. The importance of this effect increases with the complexity of the considered particle. No essential discontinuity is present in simple motions such as that of a point particle and, differently from what usually believed, that of a dust of particles governed by the geodesic deviation equation.     

Higher-dimensional homogeneous cosmological models as autonomous dynamical systems

The field equations for homogeneous models in an arbitrary number of dimensions form a Hamiltonian system with constraint forces. Due to the monotonic behavior of the determinant of the induced metric, the evolution of the system can be interpreted as the motion of a particle in an explicitly time-dependent potential. Considering vacuum models, we show that this explicit time-dependence can be eliminated. Using the scaling properties of the Ricci tensor we obtain an autonomous system, for which we can also find a Liapunov function in terms of the n-dimensional Ricci curvature.