Analytical Approximation Scheme Research Articles

Boosted fluctuation responses in power grids with active voltage dynamics

Secure electric energy supply and thus stable operation of power grids fundamentally relies on their capability to cope with fluctuations. Here, we study how active voltage dynamics impacts the collective response dynamics of networked power grids. We find that the systems driven by ongoing fluctuating inputs exhibit a bulk, a resonance, and a localized grid frequency response regime, as for static voltages. However, active voltage dynamics generically weakens the degree of localization in the grid, thereby intensifying and spatially extending the high-frequency responses. An analytic approximation scheme that takes into account shortest signal propagation paths among the voltage, phase angle and frequency variables result in an asymptotic lowest-order expansion that helps understanding the boosted high-frequency responses. These results moreover offer a generic tool to systematically investigate fluctuation response patterns in power grid models with and without active voltage dynamics.

An analytic approach to quasinormal modes for coupled linear systems

Quasinormal modes describe the ringdown of compact objects deformed by small perturbations. In generic theories of gravity that extend General Relativity, the linearized dynamics of these perturbations is described by a system of coupled linear differential equations of second order. We first show, under general assumptions, that such a system can be brought to a Schrödinger-like form. We then devise an analytic approximation scheme to compute the spectrum of quasinormal modes. We validate our approach using a toy model with a controllable mixing parameter ε and showing that the analytic approximation for the fundamental mode agrees with the numerical computation when the approximation is justified. The accuracy of the analytic approximation is at the (sub-) percent level for the real part and at the level of a few percent for the imaginary part, even when ε is of order one. Our approximation scheme can be seen as an extension of the approach of Schutz and Will [1] to the case of coupled systems of equations, although our approach is not phrased in terms of a WKB analysis, and offers a new viewpoint even in the case of a single equation.

Charging and energy fluctuations of a driven quantum battery

We consider a quantum battery modelled as a set of N independent two-level quantum systems driven by a time dependent classical source. Different figures of merit, such as stored energy, time of charging and energy quantum fluctuations during the charging process, are characterized in a wide range of parameters, by means of numerical approach and suitable analytical approximation scheme. Particular emphasis is put on the role of different initial conditions, describing the preparation state of the quantum battery, as well as on the sensitivity to the functional form of the external time-dependent drive. It is shown that an optimal charging protocol, characterized by fast charging time and the absence of charging fluctuations, can be achieved starting from the ground state of each two-level system, while other pure preparation states are less efficient. Moreover, we argue that a periodic train of peaked rectangular pulses can lead to fast charging. This study aims at providing a useful theoretical background in view of future experimental solid-state implementations.

Quantum-Granularity Effect in the Formation of Supermixed Solitons in Ring Lattices

We investigate a notable class of states peculiar to a bosonic binary mixture featuring repulsive intraspecies and attractive interspecies couplings. We evidence that, for small values of the hopping amplitudes, one can access particular regimes marked by the fact that the interwell boson transfer occurs in a jerky fashion. This property is shown to be responsible for the emergence of a staircase-like structure in the phase diagram of a mixture confined in a ring trimer and to resemble the mechanism of the superfluid-Mott insulator transition strongly. Under certain conditions, in fact, we show that it is possible to interpret the interspecies attraction as an effective chemical potential and the supermixed soliton as an effective particle reservoir. Our investigation is developed both within a fully quantum approach based on the analysis of several quantum indicators and by means of a simple analytical approximation scheme capable of capturing the essential features of this ultraquantum effect.

Universal approximations for flavor models

We develop a systematic analytical approximation scheme for the singular value decompositions of arbitrary complex three dimensional matrices Y with non-degenerate singular values. We derive exact expressions for the errors of this approximation and show that they are bounded from above by very simple ratios of the form (yi/yj )2n where yi< yj are singular values of Y and n is the order of the approximation. The applications we have in mind are the analytical and numerical treatments of arbitrary theories of flavor. We also compute upper bounds for the errors of the Cabbibo Kobayashi Maskawa (CKM) matrix that only depend on the ratios of the masses and the physical CKM angles.

An Asymptotic Expansion for Forward–Backward SDEs: A Malliavin Calculus Approach

This paper proposes a new analytical approximation scheme for the representation of the forward–backward stochastic differential equations (FBSDEs) of Ma and Zhang (Ann Appl Probab, 2002). In particular, we obtain an error estimate for the scheme applying Malliavin calculus method for the forward SDEs combined with the Picard iteration scheme for the BSDEs. We also show numerical examples for pricing option with counterparty risk under local and stochastic volatility models, where the credit value adjustment is taken into account.

Point spread functions and deconvolution of ultrasonic images.

This article investigates the restoration of ultrasonic pulse-echo C-scan images by means of deconvolution with a point spread function (PSF). The deconvolution concept from linear system theory (LST) is linked to the wave equation formulation of the imaging process, and an analytic formula for the PSF of planar transducers is derived. For this analytic expression, different numerical and analytic approximation schemes for evaluating the PSF are presented. By comparing simulated images with measured C-scan images, we demonstrate that the assumptions of LST in combination with our formula for the PSF are a good model for the pulse-echo imaging process. To reconstruct the object from a C-scan image, we compare different deconvolution schemes: the Wiener filter, the ForWaRD algorithm, and the Richardson-Lucy algorithm. The best results are obtained with the Richardson-Lucy algorithm with total variation regularization. For distances greater or equal twice the near field distance, our experiments show that the numerically computed PSF can be replaced with a simple closed analytic term based on a far field approximation.

Analytical Approximation Schemes for Mean Nucleation Rate in Turbulent Flows

Nucleation rate is a very sensitive function of the temperature and vapor mole fraction. Analytical approximation schemes for the mean nucleation rate in turbulent flows are derived using Laplace’s approximation method. The schemes only require the derivative of the nucleation rate function and the probability density function (pdf) of the vapor mole fraction and/or temperature at the point of maximum nucleation rate. Based on the relation between the mole fraction and temperature, i.e., linearly correlated or not, different approximation schemes are developed. Numerical examples are constructed to investigate the accuracy of these approximation. In the examples, the pdfs of mole fraction and/or temperature in various turbulent flows are assumed to come from the beta distribution with five distinct forms. The mean nucleation rate of dibutyl phthalate (DBP) aerosol in these turbulent flows are calculated from the approximation schemes, and compared with exact numerical integration. The relative errors are less than 1% for cases when nucleation rate diminishes at the bounds of temperature fluctuations, and no more than 50% for all studied examples. Furthermore, the approximation schemes are not sensitive to the precise form of the pdfs. Hence, these developed approximation schemes can be used to estimate the mean nucleation rate in a broad range of turbulent flows conveniently. Copyright 2014 American Association for Aerosol Research

Resonance structures in coupled two-componentϕ4model

We present a numerical study of the process of the kink-antikink collisions in the coupled one-dimensional two-component $\phi^4$ model. Our results reveal two different soliton solutions which represent double kink configuration and kink-non-topological soliton (lump) bound state. Collision of these solitons leads to very reach resonance structure which is related to reversible energy exchange between the kinks, non-topological solitons and the internal vibrational modes. Various channels of the collisions are discussed, it is shown there is a new type of self-similar fractal structure which appears in the collisions of the relativistic kinks, there the width of the resonance windows increases with the increase of the impact velocity. An analytical approximation scheme is discussed in the limit of the perturbative coupling between the sectors. Considering the spectrum of linear fluctuations around the solitons we found that the double kink configuration is unstable if the coupling constant between the sectors is negative.

An Asymptotic Expansion for Forward-Backward SDEs: A Malliavin Calculus Approach

This paper proposes a new analytical approximation scheme for the representation of the forward- backward stochastic differential equations (FBSDEs) of Ma and Zhang (2002). In particular, we obtain an error estimate for the scheme applying Malliavin calculus method for the forward SDEs combined with the Picard iteration scheme for the BSDEs. We also show numerical examples for pricing option with counterparty risk under local and stochastic volatility models, where the credit value adjustment (CVA) is taken into account.

ANALYTICAL APPROXIMATION FOR NON-LINEAR FBSDEs WITH PERTURBATION SCHEME

In this work, we have presented a simple analytical approximation scheme for generic non-linear FBSDEs. By treating the interested system as the linear decoupled FBSDE perturbed with non-linear generator and feedback terms, we have shown that it is possible to carry out a recursive approximation to an arbitrarily higher order, where the required calculations in each order are equivalent to those for standard European contingent claims. We have also applied the perturbative method to the PDE framework following the so-called Four Step Scheme. The method is found to render the original non-linear PDE into a series of standard parabolic linear PDEs. Due to the equivalence of the two approaches, it is also possible to derive approximate analytic solution for the non-linear PDE by applying the asymptotic expansion to the corresponding probabilistic model. Two simple examples are provided to demonstrate how the perturbation works and show its accuracy relative to known numerical techniques. The method presented in this paper may be useful for various important problems which have eluded analytical treatment so far.

Energy Versus Angular Momentum in Black Hole Binaries

Using accurate numerical-relativity simulations of (nonspinning) black-hole binaries with mass ratios 1:1, 2:1, and 3:1, we compute the gauge-invariant relation between the (reduced) binding energy E and the (reduced) angular momentum j of the system. We show that the relation E(j) is an accurate diagnostic of the dynamics of a black-hole binary in a highly relativistic regime. By comparing the numerical-relativity E(NR)(j) curve with the predictions of several analytic approximation schemes, we find that, while the canonically defined, nonresummed post-Newtonian-expanded E(PN)(j) relation exhibits large and growing deviations from E(NR)(j), the prediction of the effective one body formalism, based purely on known analytical results (without any calibration to numerical relativity), agrees strikingly well with the numerical-relativity results.

Dimers, Effective Interactions, and Pauli Blocking Effects in a Bilayer of Cold Fermionic Polar Molecules

We consider a bilayer setup with two parallel planes of cold fermionic polar molecules when the dipole moments are oriented perpendicular to the planes. The binding energy of two-body states with one polar molecule in each layer is determined and compared to various analytic approximation schemes in both coordinate- and momentum-space. The effective interaction of two bound dimers is obtained by integrating out the internal dimer bound state wave function and its robustness under analytical approximations is studied. Furthermore, we consider the effect of the background of other fermions on the dimer state through Pauli blocking, and discuss implications for the zero-temperature many-body phase diagram of this experimentally realizable system.

Analytical Approximation for Non-Linear FBSDEs with Perturbation Scheme

In this work, we have presented a simple analytical approximation scheme for generic non-linear FBSDEs. By treating the interested system as the linear decoupled FBSDE perturbed with non-linear generator and feedback terms, we have shown that it is possible to carry out a recursive approximation to an arbitrarily higher order, where the required calculations in each order are equivalent to those for standard European contingent claims. We have also applied the perturbative method to the PDE framework following the so-called Four Step Scheme. The method is found to render the original non-linear PDE into a series of standard parabolic linear PDEs. Due to the equivalence of the two approaches, it is also possible to derive approximate analytic solution for the non-linear PDE by applying the asymptotic expansion to the corresponding probabilistic model. Two simple examples are provided to demonstrate how the perturbation works and show its accuracy relative to known numerical techniques. The method presented in this paper may be useful for various important problems which have eluded analytical treatment so far.

Periastron Advance in Black-Hole Binaries

The general relativistic (Mercury-type) periastron advance is calculated here for the first time with exquisite precision in full general relativity. We use accurate numerical relativity simulations of spinless black-hole binaries with mass ratios 1/8≤m(1)/m(2)≤1 and compare with the predictions of several analytic approximation schemes. We find the effective-one-body model to be remarkably accurate and, surprisingly, so also the predictions of self-force theory [replacing m(1)/m(2)→m(1)m(2)/(m(1)+m(2))(2)]. Our results can inform a universal analytic model of the two-body dynamics, crucial for ongoing and future gravitational-wave searches.

Analytical and numerical study of trapped strongly correlated bosons in two- and three-dimensional lattices

We study the ground-state properties of trapped inhomogeneous systems of hardcore bosons in two- and three-dimensional lattices. We obtain results both numerically, using quantum Monte Carlo techniques, and via several analytical approximation schemes, such as the Gutzwiller mean-field approach, a cluster mean-field method, and a spin-wave analysis which takes quantum fluctuations into account. We first study the homogeneous case, for which simple analytical expressions are obtained for all observables of interest, and compare the results with the numerical ones. We obtain the equation of state of the system along with other thermodynamic properties such as the free energy, kinetic energy, superfluid density, condensate density, and compressibility. In the presence of a trap, there is in general a spatial coexistence of superfluid and insulating domains. We show that the spin-wave-based method reproduces the quantum Monte Carlo results for global as well as for local quantities with a high degree of accuracy. We also discuss the validity of the local density approximation. Our analysis can be used to describe bosons in optical lattices where the onsite interaction $U$ is much larger than the hopping amplitude $t$.

Energy spectra for quantum wires and two-dimensional electron gases in magnetic fields with Rashba and Dresselhaus spin-orbit interactions

We introduce an analytical approximation scheme to diagonalize parabolically confined two dimensional electron systems with both the Rashba and Dresselhaus spin-orbit interactions. The starting point of our perturbative expansion is a zeroth-order Hamiltonian for an electron confined in a quantum wire with an effective spin-orbit induced magnetic field along the wire, obtained by properly rotating the usual spin-orbit Hamiltonian. We find that the spin-orbit-related transverse coupling terms can be recast into two parts W and V, which couple crossing and non-crossing adjacent transverse modes, respectively. Interestingly, the zeroth-order Hamiltonian together with W can be solved exactly, as it maps onto the Jaynes-Cummings model of quantum optics. We treat the V coupling by performing a Schrieffer-Wolff transformation. This allows us to obtain an effective Hamiltonian to third order in the coupling strength k_Rl of V, which can be straightforwardly diagonalized via an additional unitary transformation. We also apply our approach to other types of effective parabolic confinement, e.g., 2D electrons in a perpendicular magnetic field. To demonstrate the usefulness of our approximate eigensolutions, we obtain analytical expressions for the n^th Landau-level g_n-factors in the presence of both Rashba and Dresselhaus couplings. For small values of the bulk g-factors, we find that spin-orbit effects cancel out entirely for particular values of the spin-orbit couplings. By solving simple transcendental equations we also obtain the band minima of a Rashba-coupled quantum wire as a function of an external magnetic field. These can be used to describe Shubnikov-de Haas oscillations. This procedure makes it easier to extract the strength of the spin-orbit interaction in these systems via proper fitting of the data.

First-passage time distribution for a random walker on a random forcing energylandscape

We present an analytical approximation scheme for the disordered average first-passagetime distribution on a finite interval of a random walker on a random forcing energylandscape. The approximation scheme captures the behavior of the distribution over alltimescales in the problem. The results are carefully checked against numericalsimulations.

Recursive Schrödinger equation approach to faster converging path integrals

By recursively solving the underlying Schrödinger equation, we set up an efficient systematic approach for deriving analytic expressions for discretized effective actions. With this, we obtain discrete short-time propagators for both one and many particles in arbitrary dimension to orders that have not been accessible before. They can be used to substantially speed up numerical Monte Carlo calculations of path integrals, as well as for setting up an alternative analytical approximation scheme for energy spectra, density of states, and other statistical properties of quantum systems.

Evolution in random fitness landscapes: the infinite sites model

We consider the evolution of an asexually reproducing population in an uncorrelatedrandom fitness landscape in the limit of infinite genome size, which implies that eachmutation generates a new fitness value drawn from a probability distributiong(w). This is the finite population version of Kingman’s house of cards model (Kingman1978 J. Appl. Probab. 15 1). In contrast to Kingman’s work, the focus here is on unbounded distributionsg(w) which lead to an indefinite growth of the population fitness. The model is solvedanalytically in the limit of infinite population size and simulated numerically for finiteN. When the genome-wide mutation probabilityU is small, the long-time behavior of the model reduces to a point process of fixationevents, which is referred to as a diluted record process (DRP). The DRP is similarto the standard record process except that a new record candidate (a numberthat exceeds all previous entries in the sequence) is accepted only with a certainprobability that depends on the values of the current record and the candidate.We develop a systematic analytic approximation scheme for the DRP. At finiteU the fitness frequency distribution of the population decomposes into astationary part due to mutations and a traveling wave component due toselection, which is shown to imply a reduction of the mean fitness by a factor of1−U compared to the limit.