Short-time Behavior Research Articles

Acoustic emission characteristics of creep fracture evolution in double-fracture fine sandstone under uniaxial compression

Acoustic emission (AE) experiments were carried out to study short-time creep behavior under uniaxial compression for cuboid-shaped fine sandstone specimens with two pre-existing cracks (the sample scale was 70 mm × 70 mm × 140 mm and the lengths of cracks were 2a = 20 mm and 2a = 30 mm) using the RLJW-2000 servo-controlled triaxial rheological test system of rock and PCI-II AE system. Based on the experimental results of creep curves, the mechanical properties of specimens with different pre-existing crack lengths were analyzed. The experimental results showed that compared to the intact specimen as the standard, the specimens with pre-existing crack lengths of 20 and 30 mm exhibited a decrease in the creep damage strength from 160.4 to 100 and 55 MPa, respectively. The pre-existing crack length significantly affected the strength of the specimen. Based on the experimental results of the spatial evolution of AE events, the three-dimensional evolution law of micro-cracks in the specimens was studied. The experimental results showed that the AE events could well reflect the crack evolution process of specimens with cracks. The occurrence of AE event points was mainly due to the initiation and propagation of micro-cracks within the sample; and with the increase of stress levels, the distribution of event points exhibited different characteristics. In the first two stages of creep loading, AE events were produced to a lesser extent and were evenly distributed inside the sample. With the increase of the creep stress level, the AE activity increased, in particular at the four ends of pre-existing cracks, and evolved from the ends of the fissures, thereby enabling the prediction of the penetration position of the micro-cracks within the fractured rock mass. Moreover, AE location results also directly reflected the interior stress field propagation process. AE location results directly reflected the spatial position, direction, and spatial curved face of crack propagation in the rock sample with pre-existing cracks, which are extremely significant to study the mechanism of rock failure.

Effect of Memory and Active Forces on Transition Path Time Distributions.

An analytical expression is derived for the transition path time distribution for a one-dimensional particle crossing of a parabolic barrier. Two cases are analyzed: (i) a non-Markovian process described by a generalized Langevin equation with a power-law memory kernel and (ii) a Markovian process with a noise violating the fluctuation-dissipation theorem, modeling the stochastic dynamics generated by active forces. In case i, we show that the anomalous dynamics strongly affect the short time behavior of the distributions, but this happens only for very rare events not influencing the overall statistics. At long times the decay is always exponential, in disagreement with a recent study suggesting a stretched exponential decay. In case ii, the active forces do not substantially modify the short time behavior of the distribution but do lead to an overall decrease of the average transition path time. These findings offer some novel insights, useful for the analysis of experiments of transition path times in (bio)molecular systems.

A two-layer model for buoyant displacement flows in a channel with wall slip

We study theoretically buoyant displacement flows of two generalized Newtonian fluids in a two-dimensional (2-D) channel with wall slip. We assume that a pseudo-interface separates two miscible (immiscible) fluids at the limit of negligible molecular diffusion (negligible surface tension). A heavy fluid displaces a light fluid at near-horizontal channel inclinations, implying that a stratified flow assumption is relevant. We develop a classical lubrication approximation model as a semi-analytical framework that includes a number of dimensionless parameters, such as a buoyancy number, the viscosity ratio, the non-Newtonian properties and the upper and lower wall slip coefficients. For specified interface heights and slopes, the reduced model can furnish the flux and velocity functions in displacing and displaced phases. We numerically solve the interface kinematic condition for four different wall slip cases: no slip (Case I), slip at the lower wall (Case II), slip at the upper wall (Case III) and slip at both walls (Case IV). The solutions for these cases deliver the interface propagation in time, for which leading and trailing displacement front heights, shapes and speeds and several key displacement features, such as front characteristic spreading lengths and short time behaviours, can be directly predicted by simplified analyses. The results reveal in detail how the presence of a channel wall slip may significantly affect the overall displacement flow and the interface evolution characteristics, for both Newtonian and non-Newtonian fluids. Regarding the latter, our analysis quantifies in particular the appearance and removal of static residual wall layers of the displaced phase, versus the wall slip cases.

Discrimination of Mild Alzheimer’s Disease Patients using Cluster Analysis of Information Transmission in EEG

A description method of information transmission was presented and applied to EEGs recorded from groups of mild Alzheimer’s disease (AD) patients and normal controls. To monitor the general interdependence of EEG channels, time-delayed mutual information was computed using the Fraser- Swinney algorithm. To manage the stochastic nature of mutual information and global connectivity of the brain, the modified Karhunen-Loeve decomposition (information transmission map: ITM), which was conducted as a quantification method of the resulting stochastic mutual information, was used. The integration of mutual information with Omega complexity elevated the discrimination power for mild AD. This provides a good procedure, capable of capturing the fine structure in the global information transmission between brain parts, compared with mutual information calculation alone. Compared with normal aged controls, mild AD patients showed significantly different short-time clustered behavior in the information transmission from the EEG. They also show a peculiar pattern of information flows from the successive snapshot monitoring. The observations suggest a discrimination of the mild AD patients from the normal aged controls.

Quantum Zeno and anti-Zeno dynamics in a spin environment

We analyze the quantum Zeno and anti-Zeno effect in a spin–spin model, where a two-level system is embedded in a sea of spins with nonuniform couplings. By employing a generalized polaron transformation as well as a perturbative approach, the analytical expression of the effective decay rate of the spin–spin model is obtained. Compared with the well-known spin-boson model, we show that the two models share the same short-time dynamical behavior if and only if the environmental temperature is zero. We also find that the effective decay rate in the spin environment is larger than that of the harmonic environment for the sub-Ohmic spectrum case. Moreover, it is demonstrated that the crossover between quantum Zeno effect and anti-Zeno effect is highly sensitive to the details of the bath spectral function regardless of the spin environment or the harmonic environment.

Short-time behavior for game-theoretic p-caloric functions

We consider the solution u of ut−ΔpGu=0 in a (not necessarily bounded) domain Ω, such that u=0 in Ω at time t=0 and u=1 on the boundary of Ω at all times. Here, ΔpG is the game-theoretic or normalizedp-laplacian. We derive new precise asymptotic formulas for t→0, that generalize that of S.R.S. Varadhan [39] for large deviations and that of the second author and S. Sakaguchi [26] for the heat content of a ball touching the boundary. We also determine the behavior for t→0 of the q-mean of u on such a ball. Applications to time-invariant level surfaces of u are then obtained.

Macroscopic and microscopic anomalous diffusion in comb model with fractional dual-phase-lag model

A novel constitutive equation which considers the macroscopic and microscopic relaxation characteristics and the memory and nonlocal characteristics is proposed to describe the anomalous diffusion in comb model. Formulated governing equation with the fractional derivative of order 1 + α corresponds to a diffusion-wave one and solutions are obtained analytically with the Laplace and Fourier transforms. As the solutions show, the existence of macroscopic relaxation parameter makes the expression of mean square displacement contain an integral form and the specific value for the microscopic relaxation parameter and macroscopic one changes the coefficient of fractional integral. The particle distribution and mean square displacement of Fick's model and the dual-phase-lag model are same at the short and long time behaviors and the special case of equal macroscopic and microscopic relaxation parameters. The particle distributions and mean square displacement with the effects of different parameters are presented graphically. Results show that the wave characteristic becomes stronger for a larger α, a larger τq or a smaller τP. For mean square displacement, the magnitude is larger at the short time behavior and smaller at the long time behavior for a smaller α. Besides, for a smaller τq or a larger τP, the magnitude is larger.

Optimization of a global seventh-order dissipative compact finite-difference scheme by a genetic algorithm

A global seventh-order dissipative compact finite-difference scheme is optimized in terms of time stability. The dissipative parameters appearing in the boundary closures are assumed to be different, resulting in an optimization problem with several parameters determined by applying a generic algorithm. The optimized schemes are analyzed carefully from the aspects of the eigenvalue distribution, the e-pseudospectra, the short time behavior, and the Fourier analysis. Numerical experiments for the Euler equations are used to show the effectiveness of the final recommended scheme.

Transition path time distributions for Lévy flights

This paper presents a study of transition path time distributions for Lévy noise-induced barrier crossing. Transition paths are short segments of the reactive trajectories and span the barrier region of the potential without spilling into the reactant/product wells. The time taken to traverse this segment is referred to as the transition path time. Since the transition path is devoid of excursions in the minimum, the corresponding time will give the exclusive barrier crossing time, unlike . This work explores the distribution of transition path times for superdiffusive barrier crossing, analytically. This is made possible by approximating the barrier by an inverted parabola. Using this approximation, the distributions are evaluated in both over- and under-damped limits of friction. The short-time behaviour of the distributions, provide analytical evidence for single-step transition events—a feature in Lévy-barrier crossing as observed in prior simulation studies. The average transition path time is calculated as a function of the Lévy index (α), and the optimal value of α leading to minimum average transition path time is discussed, in both the limits of friction. Langevin dynamics simulations corroborating with the analytical results are also presented.

Transient Charge and Energy Flow in the Wide-Band Limit.

The wide-band limit is a commonly used approximation to analyze transport through nanoscale devices. In this work we investigate its applicability to the study of charge and heat transport through molecular break junctions exposed to voltage biases and temperature gradients. We find by comparative simulations that while the wide-band-limit approximation faithfully describes the long-time charge and heat transport, it fails to characterize the short-time behavior of the junction. In particular, we show that the charge current flowing through the device shows a discontinuity when a temperature gradient is applied, while the energy flow is discontinuous when a voltage bias is switched on and even diverges when the junction is exposed to both a temperature gradient and a voltage bias. We provide an explanation for this pathological behavior and propose two possible solutions to this problem.

Enhancing Metastability by Dissipation and Driving in an Asymmetric Bistable Quantum System.

The stabilizing effect of quantum fluctuations on the escape process and the relaxation dynamics from a quantum metastable state are investigated. Specifically, the quantum dynamics of a multilevel bistable system coupled to a bosonic Ohmic thermal bath in strong dissipation regime is analyzed. The study is performed by a non-perturbative method based on the real-time path integral approach of the Feynman-Vernon influence functional. We consider a strongly asymmetric double well potential with and without a monochromatic external driving, and with an out-of-equilibrium initial condition. In the absence of driving we observe a nonmonotonic behavior of the escape time from the metastable region, as a function both of the system-bath coupling coefficient and the temperature. This indicates a stabilizing effect of the quantum fluctuations. In the presence of driving our findings indicate that, as the coupling coefficient increases, the escape time, initially controlled by the external driving, shows resonant peaks and dips, becoming frequency-independent for higher values. Moreover, the escape time from the metastable state displays a nonmonotonic behavior as a function of the temperature, the frequency of the driving, and the thermal-bath coupling, which indicates the presence of a quantum noise enhanced stability phenomenon. Finally, we investigate the role of different spectral densities, both in sub-Ohmic and super-Ohmic dissipation regime and for different cutoff frequencies, on the relaxation dynamics from the quantum metastable state. The results obtained indicate that, in the crossover dynamical regime characterized by damped intrawell oscillations and incoherent tunneling, the spectral properties of the thermal bath influence non-trivially the short time behavior and the time scales of the relaxation dynamics from the metastable state.

Comb model for the anomalous diffusion with dual-phase-lag constitutive relation

As a development of the Fick's model, the dual-phase-lag constitutive relationship with macroscopic and microscopic relaxation characteristics is introduced to describe the anomalous diffusion in comb model. The Dirac delta function in the formulated governing equation represents the special spatial structure of comb model that the horizontal current only exists on the x axis. Solutions are obtained by analytical method with Laplace transform and Fourier transform. The dependence of concentration field and mean square displacement on different parameters are presented and discussed. Results show that the macroscopic and microscopic relaxation parameters have opposite effects on the particle distribution and mean square displacement. Furthermore, four significant results with constant 1/2 are concluded, namely the product of the particle number and the mean square displacement on the x axis equals to 1/2, the exponent of mean square displacement is 1/2 at the special case τq = τP, an asymptotic form of mean square displacement (MSD∼t1/2 as t→0, ∞) is obtained as well at the short time behavior and the long time behavior.

Bulk dynamics of Brownian hard disks: Dynamical density functional theory versus experiments on two-dimensional colloidal hard spheres.

Using dynamical density functional theory (DDFT), we theoretically study Brownian self-diffusion and structural relaxation of hard disks and compare to experimental results on quasi two-dimensional colloidal hard spheres. To this end, we calculate the self-van Hove correlation function and distinct van Hove correlation function by extending a recently proposed DDFT-approach for three-dimensional systems to two dimensions. We find that the theoretical results for both self-part and distinct part of the van Hove function are in very good quantitative agreement with the experiments up to relatively high fluid packing fractions of roughly 0.60. However, at even higher densities, deviations between the experiment and the theoretical approach become clearly visible. Upon increasing packing fraction, in experiments, the short-time self-diffusive behavior is strongly affected by hydrodynamic effects and leads to a significant decrease in the respective mean-squared displacement. By contrast, and in accordance with previous simulation studies, the present DDFT, which neglects hydrodynamic effects, shows no dependence on the particle density for this quantity.

Black–Scholes in a CEV random environment

Classical (Ito diffusions) stochastic volatility models are not able to capture the steepness of small-maturity implied volatility smiles. Jumps, in particular exponential Levy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see Tankov in Pricing and hedging in exponential Levy models: review of recent results. Paris-Princeton Lecture Notes in Mathematical Finance, Springer, Berlin, 2010 for an overview), and more recently rough volatility models (Alos et al. in On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch 11(4):571–589, 2007, Fukasawa in Asymptotic analysis for stochastic volatility: martingale expansion. Finance Stoch 15:635–654, 2011). We suggest here a different route, randomising the Black–Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential Levy models and fractional stochastic volatility models.

Time-dependent variational principle in matrix-product state manifolds: Pitfalls and potential

We study the applicability of the time-dependent variational principle in matrix product state manifolds for the long time description of quantum interacting systems. By studying integrable and nonintegrable systems for which the long time dynamics are known we demonstrate that convergence of long time observables is subtle and needs to be examined carefully. Remarkably, for the disordered nonintegrable system we consider the long time dynamics are in good agreement with the rigorously obtained short time behavior and with previous obtained numerically exact results, suggesting that at least in this case the apparent convergence of this approach is reliable. Our study indicates that while great care must be exercised in establishing the convergence of the method, it may still be asymptotically accurate for a class of disordered nonintegrable quantum systems.

Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities

We study the short and large time behaviour of solutions of nonlocal heat equations of the form $\partial_tu+\mathcal{L} u = 0$. Here $\mathcal{L}$ is an integral operator given by a symmetric nonnegative kernel of Lévy type, that includes bounded and unbounded transition probability densities. We characterize when a regularizing effect occurs for small times and obtain $L^q$-$L^p$ decay estimates, $1≤ q < p < ∞$ when the time is large. These properties turn out to depend only on the behaviour of the kernel at the origin or at infinity, respectively, without need of any information at the other end. An equivalence between the decay and a restricted Nash inequality is shown. Finally we deal with the decay of nonlinear nonlocal equations of porous medium type $\partial_tu+\mathcal{L}Φ(u) = 0$.

Characterization of Log-convex decay in non-selfadjoint dynamics

The short-time and global behaviour are studied for an autonomous linear evolution equation, which is defined by a generator inducing a uniformly bounded holomorphic semigroup in a Hilbert space. A general necessary and sufficient condition is introduced under which the norm of the solution is shown to be a log-convex and strictly decreasing function of time, and differentiable also at the initial time with a derivative controlled by the lower bound of the generator, which moreover is shown to be positively accretive. Injectivity of holomorphic semigroups is the main technical tool.

Fractional anomalous convection diffusion in comb structure with a non-Fick constitutive model

A novel constitutive model that contains the fractional, convection velocity, and macroscopic and microcosmic relaxation parameters is proposed to describe anomalous diffusion in the comb model. The formulated governing equation contains mixed derivatives of time and space, and the solutions are obtained by a numerical method in which the L1 scheme is used. The temporal evolution mechanism of the monotonically decreasing distribution of particles and the monotonically increasing distribution of the mean square displacement with the involvement of different parameter effects is presented and analyzed in detail. Power functions are applied to fit the distributions of the mean square displacement with five kinds of modified models and the power law indexes are estimated. It is noteworthy that subdiffusion occurs for the Fick model and superdiffusion occurs for short-time behavior, while subdiffusion occurs for long-time behavior with the Cattaneo and dual-phase-lag models, and superdiffusion occurs for the Cattaneo–Christov model and the newly proposed model.

Identification of Capacitance Distribution in Neuronal Membranes from a Fractional-Order Electrical Circuit and Whole-Cell Patch-Clamped Cells

Passive electrical membrane properties are key determinants of signaling processes in brain cells, influencing input-output responses as well as action potential firing. We propose a novel interpretation of patch-clamp recordings that brings out the fractional dynamic of the electrical properties of cell membranes and provides a better knowledge of their microscopic behavior. The passive electrical properties of the cell membrane were modeled using an electrical equivalent circuit (EEC) consisting of a constant phase element (CPE) in parallel with a resistor. The Mittag-Leffler function was used to describe the non-exponential behavior of the voltage transients that are attributable to the processes of charging and discharging the membrane capacitance. The procedure proposed, based on circuit theory and fractional calculus, was used to study the voltage transients obtained in response to low-amplitude hyperpolarizing current pulses applied to cultured mice dorsal root ganglion (DRG) neurons under whole-cell current-clamp configuration. To further validate the method, we also analyzed the voltage transients obtained from hippocampal pyramidal neurons and glial cells recorded in mice brain slices in vitro using the short-time behavior of the resulting membrane voltages.

Understanding evolution of vortex rings in viscous fluids

The evolution of vortex rings in isodensity and isoviscosity fluid has been studied analytically using a novel mathematical model. The model predicts the spatiotemporal variation in peak vorticity, circulation, vortex size and spacing based on instantaneous vortex parameters. This proposed model is quantitatively verified using experimental measurements. Experiments are conducted using high-speed particle image velocimetry (PIV) and laser induced fluorescence (LIF) techniques. Non-buoyant vortex rings are generated from a nozzle using a constant hydrostatic pressure tank. The vortex Reynolds number based on circulation $(\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708})$ is varied in the range 100–1500 to account for a large range of operating conditions. Experimental results show good agreement with theoretical predictions. However, it is observed that neither Saffman’s thin-core model nor the thick-core equations could correctly explain vortex evolution for all initial conditions. Therefore, a transitional theory is framed using force balance equations which seamlessly integrate short- and long-time asymptotic theories. It is found that the parameter $A=(a/\unicode[STIX]{x1D70E})^{2}$, where $a$ is the vortex half-spacing and $\unicode[STIX]{x1D70E}$ denotes the standard deviation of the Gaussian vorticity profile, governs the regime of vortex evolution. For higher values of $A$, evolution follows short-time behaviour, while for $A=O(1)$, long-time behaviour is prominent. Using this theory, many reported anomalous observations have been explained.