Short-time Expansion Research Articles

Where Are We Going Now? The Current and Future Distributions of the Monk Parakeet (Myiopsitta monachus) and Eurasian Collared Dove (Streptopelia decaocto) in a Megalopolis

The monk parakeet (Myiopsitta monachus) and the Eurasian collared dove (Streptopelia decaocto) are two of the most prevalent invasive species globally due to their high dispersal ability. Since these birds were first recorded (1999 for the monk parakeet and 2013 for Eurasian collared dove) in the Mexico City Metropolitan Area (MCMA), both species have spread rapidly throughout the area. However, the impacts of global climate changes on the distribution patterns of these species remain poorly studied across the MCMA. Therefore, based on an ecological niche modeling approach, we assessed the expansion and potential invasion of both species in this megalopolis using current and future climate projections (year 2050). Our results estimated that the current suitable areas are 5564 km2 for the monk parakeet and 5489 km2 for the Eurasian collared dove, covering ~70% of the study area, suggesting a rapidly invading species, as expected. We observed a slight decrease (up to 24%) in both species in future climate scenarios, but our models estimated that the sizes of the suitable areas would remain stable. We found that the range expansion of these species in the megalopolis may be largely attributed to their propensity for jump dispersion and short-time niche expansion ability. Our findings allow for a better understanding of the factors contributing to the range expansion of the monk parakeet and the Eurasian collared dove in Mexico and can better inform the monitoring guidelines for and assessments of these invasive species.

Short-time expansion of one-dimensional Fokker-Planck equationswith heterogeneous diffusion.

We formulate a short-time expansion for one-dimensional Fokker-Planck equationswith spatially dependent diffusion coefficients, derived from stochastic processes with Gaussian white noise, for general values of the discretization parameter 0≤α≤1 of the stochastic integral. The kernel of the Fokker-Planck equation(the propagator) can be expressed as a product of a singular and a regular term. While the singular term can be given in closed form, the regular term can be computed from a Taylor expansion whose coefficients obey simple ordinary differential equations. We illustrate the application of our approach with examples taken from statistical physics and biophysics. Furthermore, we show how our formalism allows us to define a class of stochastic equationswhich can be treated exactly. The convergence of the expansion cannot be guaranteed independently from the discretization parameter α.

Semiclassical analysis of a nonlocal boundary value problem related to magnitude

AbstractWe study a Dirichlet boundary problem related to the fractional Laplacian in a manifold. Its variational formulation arises in the study of magnitude, an invariant of compact metric spaces given by the reciprocal of the ground state energy. Using recent techniques developed for pseudodifferential boundary problems we discuss the structure of the solution operator and resulting properties of the magnitude. In a semiclassical limit we obtain an asymptotic expansion of the magnitude in terms of curvature invariants of the manifold and the boundary, similar to the invariants arising in short-time expansions for heat kernels.

The jump leverage risk premium

Jumps in asset prices are ubiquitous, yet the apparent high price of jump risk observed empirically is commonly viewed as puzzling. We develop new model-free short-time risk-neutral variance expansions, allowing us to clearly delineate the importance of jumps in generating both price and variance risks. We find that simultaneous jumps in the price and the stochastic volatility and/or jump intensity of the market commands a sizeable risk premium. The existence of “jump leverage” risk premium may be rationalized in the context of equilibrium-based models by jumps in the conditional moments of the underlying fundamentals and/or changes in investors' risk aversion.

Estimation of non-uniqueness and short-time asymptotic expansions for Navier–Stokes flows

There is considerable evidence that solutions to the three-dimensional Navier–Stokes equations in the natural energy space are not unique. Assuming this is the case, it becomes important to quantify how non-uniqueness evolves. In this paper we provide an algebraic estimate for how rapidly two possibly non-unique solutions can separate over a compact spatial region in which the initial data has sub-critical regularity. Outside of this compact region, the data is only assumed to be in the scaling critical weak Lebesgue space and can be large. To establish this upper bound on the separation rate, we develop a new spatially local, short-time asymptotic expansion which is of independent interest.

Diabatic quantum and classical annealing of the Sherrington-Kirkpatrick model

Quantum annealing is a contender to solve combinatorial optimization problems based on quantum dynamics. While significant efforts have been undertaken to investigate the quality of the solutions and the required run times, much less attention has been paid to understanding the dynamics of quantum annealing and the process leading to the solution during the sweep itself. In this comprehensive study, we investigate various aspects of the quantum annealing dynamics using different approaches. We perform quantum annealing, simulated quantum annealing, and classical annealing on several hundred instances of the Sherrington-Kirkpatrick model with intermediate system sizes up to 22 spins using numerical simulations. We observe qualitative differences between the quantum and classical methods, in particular at intermediate times, where a peak in the fidelity, also known as diabatic bump, appears for hard instances. Furthermore, we investigate the two-point correlation functions, which feature differences at intermediate times as well. At short times, however, the methods are similar again, which can be explained by relating the short-time expansion of quantum annealing to a high-temperature expansion, thus allowing one in principle to find the classical solution already at short times, albeit at prohibitive sampling cost.

A modified tissue expander method for ear reconstruction in patients with excessively insufficient postauricular skin.

For microtia patients with excessively insufficient postauricular skin, it is difficult to obtain a satisfied outcome with existing strategies. In this study, we developed a modified tissue expander method for auricular reconstruction. The modified tissue expander method divided into 4 stages. In the first stage, a 30ml or 50ml kidney-shaped tissue expander was implanted in the mastoid region. A short time expansion (average 33.5days) was conducted subsequently. In the second stage, the expander was removed and a modified cartilage framework without tragus was inserted through the same incision. A crescent-shaped cartilage pad was inserted into the incision of cartilage-harvest site at the same time. In the third stage, the reconstructed ear was elevated. Lobule rotation and remanent modification were performed in the fourth stage. The patients were followed up between half a year and 10years. The outcomes of the reconstructed ears were scored with evaluation criteria. From January 2010 to December 2019, a total of 45 microtia patients with excessively insufficient postauricular skin were performed the modified tissue expander method. Fourty-two patients showed satisfied outcomes. Complications such as hyperpigmentation in the skin graft area (3, 6.7%), scar hyperplasia (3, 6.7%) and folliculitis (1, 2.2%) were found. There were no complications related to the tissue expander. The modified tissue expander method is an effective and safe technique for auricular reconstruction in patients having excessively insufficient postauricular skin, with satisfying medium-term results.

Mixedness timescale in non-Hermitian quantum systems

We discuss the short-time perturbative expansion of the linear entropy for finite-dimensional quantum systems whose dynamics can be effectively described by a non-Hermitian Hamiltonian. We derive a timescale for the degree of mixedness for an input state undergoing non-Hermitian dynamics and specialize these results in the case of a driven-dissipative two-level system. Next, we derive a timescale for the growth of mixedness for bipartite quantum systems that depends on the effective non-Hermitian Hamiltonian. In the Hermitian limit, this result recovers the perturbative expansion for coherence loss in Hermitian systems, while it provides an entanglement timescale for initial pure and uncorrelated states. To illustrate these findings, we consider the many-body transverse-field XY Hamiltonian coupled to an imaginary all-to-all Ising model. We find that the non-Hermitian Hamiltonian enhances the short-time dynamics of the linear entropy for the considered input states. Overall, each timescale depends on minimal ingredients such as the probe state and the non-Hermitian Hamiltonian of the system, and its evaluation requires low computational cost. Our results find applications to non-Hermitian quantum sensing, quantum thermodynamics of non-Hermitian systems, and $\mathcal{PT}$-symmetric quantum field theory.

A Short Time Expansion Measurement Method for the Detection of Aging Effect of Lithium Ion Cells Using a High Resolution Laser Interfometric Setup

A Short Time Expansion Measurement Method for the Detection of Aging Effect of Lithium Ion Cells Using a High Resolution Laser Interfometric Setup

Reversible bond kinetics from single-molecule force spectroscopy experiments close to equilibrium

Analysis of bond rupture data from single-molecule force spectroscopy experiments commonly relies on the strong assumption that the bond dissociation process is irreversible. However, with increased spatiotemporal resolution of instruments it is now possible to observe multiple unbinding-rebinding (or unfolding-refolding) events in a single pulling experiment. Here we augment the theory of force-induced unbinding by explicitly taking into account rebinding kinetics and provide approximate analytic solutions of the resulting rate equations. Furthermore, we use a short-time expansion of the exact kinetics to construct numerically efficient maximum likelihood estimators for the parameters of the force-dependent unbinding and rebinding rates, which pair well with and complement established methods, such as the analysis of rate maps. The estimators are independent of the applied force protocol and can therefore be used to globally analyze multiple force-extension traces for both force-ramp and force-clamp experiments. We provide an open-source implementation of the theory, evaluated for Bell-like rates, which we apply to synthetic data generated by a Gillespie stochastic simulation algorithm for time-dependent rates.

An asymptotic Green's function method for time-dependent Schrödinger equations with application to Kohn-Sham equations

We present an effective asymptotic Green's function method for solving the time-dependent Schrödinger equation with scalar and vector potentials, and apply it to the Kohn-Sham equations for electronic structure calculation within the time-dependent density functional theory, where the perfectly matched layer approach is incorporated such that the computation can be performed in a bounded domain of physical interest. The method, which extends the approach in Leung et al. (2014) [40], combines the Huygens' principle or Feynman's path integral for propagating the wavefunction and the semi-classical approximations for approximating the retarded Green's function. Once the analytic approximations for the phase and amplitudes of the asymptotic retarded Green's function are obtained through short-time Taylor series expansions, a short-time propagator for the wavefunction is derived, and the resulting integral can be evaluated by fast Fourier transform after appropriate lowrank approximations. Numerical experiments are presented for demonstration.

Short-Time Expansion of Characteristic Functions in a Rough Volatility Setting With Applications

Short-Time Expansion of Characteristic Functions in a Rough Volatility Setting With Applications

Dimensional regularization for the particle transition amplitude in curved space

We compute the perturbative short-time expansion for the transition amplitude of a particle in curved space time, by employing dimensional regularization (DR) to treat the divergences which occur in some Feynman diagrams. The present work generalizes known results where DR was applied to the computation of one-loop effective actions, which in the worldline approach are linked to particle path integrals on the circle, i.e. with periodic boundary conditions. The main motivation of the present work comes from revived interest in particle transition amplitudes in curved space-times, due to their use in the recently proposed worldline quantum field theory (in curved space-time).

An asymptotic Green's function method for the wave equation

We present an effective asymptotic Green's function method for propagating the waves through the linear scalar wave equation. The wave is first split into its forward-propagating and backward-propagating parts. Following that, the method, which combines the Huygens' principle and the geometrical optics approximations, is designed to propagate the forward-propagating and backward-propagating waves, where an integral with Green's functions that is based on the Huygens' principle is used to propagate the waves, and the Green's functions are approximated by the geometrical optics approximations. Upon obtaining analytic approximations for the phase and amplitude in the geometrical optics approximations through short-time Taylor series expansions, a short-time propagator for the waves is derived and the resulting integral can be evaluated efficiently by fast Fourier transform after appropriate lowrank approximations. The short-time propagator can be applied repeatedly to propagate the waves for a long time. In order to restrict the computation onto a bounded domain of interest, the perfectly matched layer technique with complex coordinate stretching transformation is incorporated. The method is first-order accurate, and has complexity O(tϵNlog⁡N) per time step with N the number of spatial points and tϵ the low rank for a prescribed accuracy requirement ϵ>0. Numerical experiments are presented to demonstrate the proposed method.

Dissipative spin dynamics in hot quantum paramagnets

We use the functional renormalization approach for quantum spin systems developed by Krieg and Kopietz [Phys. Rev. B $\mathbf{99}$, 060403(R) (2019)] to calculate the spin-spin correlation function $G (\boldsymbol{k}, \omega )$ of quantum Heisenberg magnets at infinite temperature. For small wavevectors $\boldsymbol{k} $ and frequencies $\omega$ we find that $G ( \boldsymbol{k}, \omega )$ assumes in dimensions $d > 2$ the diffusive form predicted by hydrodynamics. In three dimensions our result for the spin-diffusion coefficient ${\cal{D}}$ is somewhat smaller than previous theoretical predictions based on the extrapolation of the short-time expansion, but is still about $30 \%$ larger than the measured high-temperature value of ${\cal{D}}$ in the Heisenberg ferromagnet Rb$_2$CuBr$_4\cdot$2H$_2$O. In reduced dimensions $d \leq 2$ we find superdiffusion characterized by a frequency-dependent complex spin-diffusion coefficient ${\cal{D}} ( \omega )$ which diverges logarithmically in $d=2$, and as a power-law ${\cal{D}} ( \omega ) \propto \omega^{-1/3}$ in $d=1$. Our result in one dimension implies scaling with dynamical exponent $z =3/2$, in agreement with recent calculations for integrable spin chains. Our approach is not restricted to the hydrodynamic regime and allows us to calculate the dynamic structure factor $S ( \boldsymbol{k} , \omega )$ for all wavevectors. We show how the short-wavelength behavior of $S ( \boldsymbol{k}, \omega )$ at high temperatures reflects the relative sign and strength of competing exchange interactions.

Jamming densities of random sequential adsorption on d-dimensional cubic lattices.

The rate of convergence of the jamming densities to their asymptotic high-dimensional tree approximation is studied, for two types of random sequential adsorption (RSA) processes on a d-dimensional cubic lattice. The first RSA process has an exclusion shell around a particle of nearest neighbors in all d dimensions (N1 model). In the second process the exclusion shell consists of a d-dimensional hypercube with length k=2 around a particle (N2 model). For the N1 model the deviation of the jamming density ρ_{r}(d) from its asymptotic high d value ρ_{asy}(d)=ln(1+2d)/2d vanishes as [ln(1+2d)/2d]^{3.41}. In addition, it has been shown that the coefficients a_{n}(d) of the short-time expansion of the occupation density of this model (at least up to n=6) are given for all d by a finite correction sum of order (n-2) in 1/d to their asymptotic high d limit. The convergence rate of the jamming densities of the N2 model to their high d limits ρ_{asy}(d)=dln3/3^{d} is slow. For 2≤d≤4 the generalized Palasti approximation provides by far a better approximation. For higher d values the jamming densities converge monotonically to the above asymptotic limits, and their decay with d is clearly faster than the decay as (0.432332...)^{d} predicted by the generalized Palasti approximation.

Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model.

We study a two state “jumping diffusivity” model for a Brownian process alternating between two different diffusion constants, , with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, Gaussian behavior with an effective diffusion coefficient is recovered. We show that, for equilibrium initial conditions and when the limit of the diffusion coefficient is taken, the short time behavior leads to a cusp, namely a non-analytical behavior, in the distribution of the displacements for . Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state to a -function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non-analyticity in . We demonstrate how super-statistical framework is a zeroth order short time expansion of , in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory.

Active Brownian and inertial particles in disordered environments: Short-time expansion of the mean-square displacement.

We consider an active Brownian particle moving in a disordered two-dimensional energy or motility landscape. The averaged mean-square displacement (MSD) of the particle is calculated analytically within a systematic short-time expansion. As a result, for overdamped particles, both an external random force field and disorder in the self-propulsion speed induce ballistic behavior adding to the ballistic regime of an active particle with sharp self-propulsion speed. Spatial correlations in the force and motility landscape contribute only to the cubic and higher-order powers in time for the MSD. Finally, for inertial particles two superballistic regimes are found where the scaling exponent of the MSD with time is α=3 and α=4. We confirm our theoretical predictions by computer simulations. Moreover, they are verifiable in experiments on self-propelled colloids in random environments.

Asymptotic behavior and calibration of short-time option prices under the normal tempered stable model

In recent years, small-time asymptotics of option prices have received much attention. In this paper, we derive short-time first-order expansions for price and implied volatility of at-the-money call options under the exponential Normal Tempered Stable model in two methods. The first method is similar to the approach used by Figueroa-López and Forde for the CGMY model. The other is based on the explicit form of characteristic functions of the Normal Tempered Stable processes. Our numerical results show that the expansions herein remarkably approximate the prices obtained by Fourier transform, even for options with maturities as long as a few years. For this reason, our expansions can be used for parameter calibration and model testing in a simple and efficient way.

Minakshisundaram–Pleijel coefficients for non-compact higher rank symmetric spaces

We assign to a non-compact Riemannian symmetric space X a theta function and a zeta function. We compute all of the Minakshisundaram–Pleijel coefficients in a short-time asymptotic expansion of the theta function especially when X is of complex type, which we use to compute the one-loop effective potential—whose relevance for quantum field theory, for example, is briefly commented on. These coefficients, for X of general type, are also shown to be specifically related to residues and special values of the zeta function. The results presented extend previous ones in the literature where the rank of X is assumed to be one.