Short-time Behavior Research Articles

Dynamical stability of the weakly nonharmonic propeller-shaped planar Brownian rotator.

Dynamical stability is a prerequisite for control and functioning of desired nanomachines. We utilize the Caldeira-Leggett master equation to investigate dynamical stability of molecular cogwheels modeled as a rigid, propeller-shaped planar rotator. To match certain expected realistic physical situations, we consider a weakly nonharmonic external potential for the rotator. Two methods for investigating stability are used. First, we employ a quantum-mechanical counterpart of the so-called "first passage time" method. Second, we investigate time dependence of the standard deviation of the rotator for both the angle and angular momentum quantum observables. A perturbationlike procedure is introduced and implemented to provide the closed set of differential equations for the moments. Extensive analysis is performed for different combinations of the values of system parameters. The two methods are, in a sense, mutually complementary. Appropriate for the short time behavior, the first passage time exhibits a numerically relevant dependence only on the damping factor as well as on the rotator size. However, the standard deviations for both the angle and angular momentum observables exhibit strong dependence on the parameter values for both short and long time intervals. Contrary to our expectations, the time decrease of the standard deviations is found for certain parameter regimes. In addition, for certain parameter regimes nonmonotonic dependence on the rotator size is observed for the standard deviations and for the damping of the oscillation amplitude. Hence, nonfulfillment of the classical expectation that the size of the rotator can be reduced to the inertia of the rotator. In effect, the task of designing the desired protocols for the proper control of the molecular rotations becomes an optimization problem that requires further technical elaboration.

Short-time behavior of continuous-time quantum walks on graphs

Dynamical evolution of systems with sparse Hamiltonians can always be recognized as continuous time quantum walks (CTQWs) on graphs. In this paper, we analyze the short time asymptotics of CTQWs. In recent studies, it was shown that for the classical diffusion process the short time asymptotics of the transition probabilities follows power laws whose exponents are given by the usual combinatorial distances of the nodes. Inspired by this result, we perform a similar analysis for CTQWs both in closed and open systems, including time-dependent couplings. For time-reversal symmetric coherent quantum evolutions, the short time asymptotics of the transition probabilities is completely determined by the topology of the underlying graph analogously to the classical case, but with a doubled power-law exponent. Moreover, this result is robust against the introduction of on-site potential terms. However, we show that time-reversal symmetry breaking terms and non-coherent effects can significantly alter the short time asymptotics. The analytical formulas are checked against numerics, and excellent agreement is found. Furthermore, we discuss in detail the relevance of our results for quantum evolutions on particular network topologies.

Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains.

How long does a diffusing molecule spend in a close vicinity of a confining boundary or a catalytic surface? This quantity is determined by the boundary local time, which plays thus a crucial role in the description of various surface-mediated phenomena, such as heterogeneous catalysis, permeation through semipermeable membranes, or surface relaxation in nuclear magnetic resonance. In this paper, we obtain the probability distribution of the boundary local time in terms of the spectral properties of the Dirichlet-to-Neumann operator. We investigate the short-time and long-time asymptotic behaviors of this random variable for both bounded and unbounded domains. This analysis provides complementary insights onto the dynamics of diffusing molecules near partially reactive boundaries.

Inferring the smoothness of the autocorrelation function from that of the initial state

We point out that by the ‘smoothness means fast decay’ principle in Fourier analysis, it is possible to infer the smoothness (or nonsmoothness) of the autocorrelation function from a mere glimpse of the initial state. Specifically, for a generic system with smooth eigenstates, the smoother an initial state is, the faster its decomposition coefficients with respect to the eigenstates of the system decay, and in turn the smoother the autocorrelation function is. The idea is illustrated with three increasingly smooth functions in an infinite square well. By using the Mellin transform, we also find that the nonsmoothness or singularity of the initial state affects the short-time behavior of its autocorrelation function. In particular, a sufficiently nonsmooth or singular initial state could decay in a nonquadratic power law, with the exponent continuously tunable. We now understand the periodic cusps of the autocorrelation function in the quench dynamics of a Bloch state, which was observed previously (Zhang and Yang 2016 Europhys. Lett. 114 60001).

Vibronic Coupling and Excitation Transfer in Hydrogen-Bonded Molecular Dimers: A Quantum Dynamical Analysis.

A brief pedagogic rederivation is given of basic exciton coupling theory, taking the nuclear coordinates to be fixed. This is then extended to take variations of these coordinates into account by adopting suitable multimode coupling models and extracting the transfer of excitation energy from the populations of the locally excited states. The dynamical problem thus defined is solved numerically in a fully quantal manner. Two doubly hydrogen bonded dimers of (hetero)aromatic systems are selected as representative cases whose electronic excitation spectra have been analyzed previously based on ab initio data, and good agreement with experiment has been found. The numerical calculations of the electronic populations reveal a complex time dependence of the excitation transfer that is far from being oscillatory or exponential. For localized excitation, the short-time behavior can be understood in terms of the quenched excitonic energy splitting, while for delocalized excitation a complex time dependence with rapidly changing features results. Some of these can be interpreted in terms of the vibronic structure of the excitation spectra. The importance of the quenched excitonic splitting for the short-time behavior of the excitation energy transfer is emphasized.

Hamiltonian short-time critical dynamics of the three-dimensional XY model.

The short-time relaxation critical behavior of the XY model on a simple-cubic lattice is investigated within the scope of deterministic Hamiltonian dynamics. The Hamiltonian includes a first-neighbor interaction between planar vectors and a rotational kinetic term from which the motion equations are derived. The dynamical evolution from a fully ordered initial state is followed by employing a symplectic algorithm based on a high-order Trotter-Suzuki decomposition of the time-evolution operator. A finite-time scaling analysis is performed to provide accurate estimates of the critical energy density, the order-parameter relaxation exponent, and the dynamical critical exponent. The estimated critical exponents are consistent with prior theoretical and experimental values reported for the superfluid ^{4}He, extreme type-II superconducting, and Bose-Einstein condensation transitions.

Identifying short- and long-time modes of the mean-square displacement: An improved nonlinear fitting approach

This paper is concerned with fitting the mean-square displacement (MSD) function, and extract reliable and accurate values for the diffusion coefficient D. In this work, we present a new optimal and robust nonlinear regression model capable of fitting the MSD function with different regimes corresponding to different time scales. The algorithm presented here achieves two major goals; a more accurate estimation of D as well as extracting information about the short time behavior. The algorithm fits the MSD to a continuous piece-wise function and predicts all the coefficients in the model including the breakpoints. The novelty of this approach lies in its ability to find the breakpoints which separate different modes of motion. We tested our algorithm using numerical experiments, and our fits described the data remarkably well. In addition, we applied our algorithm to extract D for water based on Molecular Dynamics (MD) simulations. The results of our fits are in good agreement with the experimentally reported values.

Revisiting the heat kernel on isotropic and nonisotropic Heisenberg groups*

The aim of this paper is threefold. First, we obtain the precise bounds for the heat kernel on isotropic Heisenberg groups by using well-known results in the three-dimensional case. Second, we study the asymptotic estimates at infinity for the heat kernel on nonisotropic Heisenberg groups. As a consequence, we give uniform upper and lower estimates of the heat kernel, and complete its short-time behavior obtained by Beals–Gaveau–Greiner. Third, we prove that the uniform asymptotic behaviour at infinity (so the small-time asymptotic behaviour) of the heat kernel for Grushin operators, obtained by the first author, are still valid in two and three dimensions.

Evolution of interfaces for the nonlinear parabolic p-Laplacian-type reaction-diffusion equations. II. Fast diffusion vs. absorption

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic p-Laplacian-type reaction-diffusion equation of non-Newtonian elastic filtration$$u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ 1 \lt p \lt 2, \beta \gt 0.$$If the interface is finite, it may expand, shrink or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters p, β, sign b, and asymptotics of the initial function near its support. In some range of parameters, strong domination of the diffusion causes infinite speed of propagation and interfaces are absent. In all cases with finite interfaces, we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The methods of the proof are based on nonlinear scaling laws and a barrier technique using special comparison theorems in irregular domains with characteristic boundary curves. A full description of small-time behaviour of the interfaces and local solutions near the interfaces for slow diffusion case when p>2 is presented in a recent paper by Abdulla and Jeli [(2017) Europ. J. Appl. Math.28(5), 827–853].

Classical stochastic systems with fast-switching environments: Reduced master equations, their interpretation, and limits of validity.

We study classical Markovian stochastic systems with discrete states, coupled to randomly switching external environments. For fast environmental processes we derive reduced dynamics for the system itself, focusing on corrections to the adiabatic limit of infinite timescale separation. We show that this can lead to master equations with bursting events. Negative transition rates can result in the reduced master equation, leading to unphysical short-time behavior. However, the reduced master equation can describe stationary states better than a leading-order adiabatic calculation, similar to what is known for Kramers-Moyal expansions in the context of the Pawula theorem [R. F. Pawula, Phys. Rev. 162, 186 (1967)PHRVAO0031-899X10.1103/PhysRev.162.186; H. Risken and H. Vollmer, Z. Phys. B 35, 313 (1979)ZPBBDJ0340-224X10.1007/BF01319854]. We provide an interpretation of the reduced dynamics in discrete time and a criterion for the occurrence of negative rates for systems with two environmental states.

A note on the implied volatility of floating strike Asian options

In this paper, we study the short-time behavior of the implied volatility for short-time floating strike Asian options. Our method is based on Malliavin calculus techniques and allows us to construct an approximation formula for the corresponding option prices. Numerical examples are given.

Low cycle fatigue modelling of a steam turbine rotor steel

Materials in steam turbine rotors are subjected to cyclic loads at high temperature, causing cracks to initiate and grow. To allow for more flexible operation, accurate fatigue models for life prediction must not be overly conservative. In this study, fully reversed low cycle fatigue tests were performed on a turbine rotor steel called FB2. The tests were done isothermally, within temperature range of room temperature to 600 °C, under strain control with 0.8-1.2 % total strain range. Some tests included hold time to calibrate the short-time creep behaviour of the material. Different fatigue life models were constructed. The life curve in terms of stress amplitude was found unusable at 600 °C, while the life curve in terms of total strain or inelastic strain amplitudes displayed inconsistent behaviour at 500 °C. To construct better life model, the inelastic strain amplitudes were separated into plastic and creep components by modelling the deformation behaviour of the material, including creep. Based on strain range partitioning approach, the fatigue life depends on different damage mechanisms at different strain ranges. This allowed the formulation of life curves based on plasticity or creep domination, which showed creep domination at 600 °C, while at 500 °C, creep only dominates for higher strain range.

Concentration for Coulomb gases on compact manifolds

We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequality in Kantorovich-Wasserstein distance inspired from the work of Chafa\"i, Hardy and Ma\"ida on the Euclidean space. Their proof involves large deviation techniques together with an energy-distance comparison and a regularization procedure based on the superharmonicity of the Green function. This last ingredient is not available on a manifold. We solve this problem by using the heat kernel and its short-time asymptotic behavior.

Effect of abrupt change of the wall temperature in the kinetic theory

We consider a semi-infinite expanse of a rarefied gas bounded by an infinite plane wall. The temperature of the wall is $ T_0 $, and the gas is initially in equilibrium with density $ \rho_0 $ and temperature $ T_0 $. The temperature of the wall is suddenly changed to $ T_w $ at time $ t = 0 $ and is kept at $ T_w $ afterward. We study the quantitative short time behavior of the gas in response to the abrupt change of the wall temperature on the basis of the linearized Boltzmann equation. Our approach is based on a straightforward calculation of the exact formulas derived by Duhamel's integral. Our method allows us to establish the pointwise estimates of the microscopic distribution and the macroscopic variables in short time. We show that the short-time solution consists of the free molecular flow and its perturbation, which exhibits logarithmic singularities along the characteristic line and on the boundary.

Influence of calcium-binding salts on heat stability and fouling of whey protein isolate dispersions

The effect of the calcium-binding salts (CBS), trisodium citrate (TSC), tripotassium citrate (TPC) and disodium hydrogen phosphate (DSHP) at concentrations of 1–45 mm on the heat stability and fouling of whey protein isolate (WPI) dispersions (3%, w/v, protein) was investigated. The WPI dispersions were assessed for heat stability in an oil bath at 95 °C for 30 min, viscosity changes during simulated high-temperature short-time (HTST) and fouling behaviour using a lab-scale fouling rig. Adding CBS at levels of 5–30 mm for TSC and TPC and 25–35 mm for DSHP improved thermal stability of WPI dispersions by decreasing the ionic calcium (Ca2+) concentration; however, lower or higher concentrations destabilised the systems on heating. Adding CBS improved heat transfer during thermal processing, and resulted in lower viscosity and fouling. This study demonstrates that adding CBS is an effective means of increasing WPI protein stability during HTST thermal processing.

Mott, Floquet, and the response of periodically driven Anderson insulators

We consider periodically driven Anderson insulators. The short time behavior for weak, monochromatic, uniform electric fields is given by linear response theory and was famously derived by Mott. We go beyond this to consider both long times---which is the physics of Floquet late time states---and strong electric fields. This results in a `phase diagram' in the frequency-field strength plane, in which we identify four distinct regimes. These are: a linear response regime dominated by pre-existing Mott resonances, which exists provided Floquet saturation is not reached within a period; a non-linear perturbative regime, which exhibits multiphoton-absorption in response to the field; a near-adiabatic regime, which exhibits a primarily reactive response spread over the entire sample and is insensitive to pre-existing resonances; and finally an enhanced dissipative regime.

Distribution of waiting times between electron cotunneling events

In the resonant tunneling regime sequential processes dominate single electron transport through quantum dots or molecules that are weakly coupled to macroscopic electrodes. In the Coulomb blockade regime, however, cotunneling processes dominate. Cotunneling is an inherently quantum phenomenon and thus gives rise to interesting observations, such as an increase in the current shot noise. Since cotunneling processes are inherently fast compared to the sequential processes, it is of interest to examine the short time behaviour of systems where cotunneling plays a role, and whether these systems display nonrenewal statistics. We consider three questions in this paper. Given that an electron has tunneled from the source to the drain via a cotunneling or sequential process, what is the waiting time until another electron cotunnels from the source to the drain? What are the statistical properties of these waiting time intervals? How does cotunneling affect the statistical properties of a system with strong inelastic electron-electron interactions? In answering these questions, we extend the existing formalism for waiting time distributions in single electron transport to include cotunneling processes via an $n$-resolved Markovian master equation. We demonstrate that for a single resonant level the analytic waiting time distribution including cotunneling processes yields information on individual tunneling amplitudes. For both a SRL and an Anderson impurity deep in the Coulomb blockade there is a nonzero probability for two electrons to cotunnel to the drain with zero waiting time inbetween. Furthermore, we show that at high voltages cotunneling processes slightly modify the nonrenewal behaviour of an Anderson impurity with a strong inelastic electron-electron interaction.

Chaos Signatures in the Short and Long Time Behavior of the Out-of-Time Ordered Correlator.

Two properties are needed for a classical system to be chaotic: exponential stretching and mixing. Recently, out-of-time order correlators were proposed as a measure of chaos in a wide range of physical systems. While most of the attention has previously been devoted to the short time stretching aspect of chaos, characterized by the Lyapunov exponent, we show for quantum maps that the out-of-time correlator approaches its stationary value exponentially with a rate determined by the Ruelle-Pollicot resonances. This property constitutes clear evidence of the dual role of the underlying classical chaos dictating the behavior of the correlator at different timescales.

Front propagation dynamics: Qualitative differences revealed by very high intensity fluctuations

This paper presents a detailed analysis of the effect of a large external noise on the propagation of a front between two stable steady states. Noise affects one of the steady states or the thresholds. Significant differences with respect to the case of small noise intensity are carefully studied using a Langevin approach. The average effect of noise is modeled by means of an additional nonlinear term that enables an analysis of short-time and long-time behaviors. Nontrivial behaviors are numerically observed and analytically discussed: (i) Two symmetry breakings of the fluctuation-free stable states are observed and (ii) two fronts propagating in opposite directions are simultaneously observed and the propagation direction may be exchanged by changing the diffusion coefficient.

Diffusion in Mesoscopic Lattice Models of Amorphous Plasticity

We present results on tagged particle diffusion in a mesoscale lattice model for sheared amorphous material in athermal quasistatic conditions. We find a short time diffusive regime and a long time diffusive regime whose diffusion coefficients depend on system size in dramatically different ways. At short time, we find that the diffusion coefficient, D, scales roughly linearly with system length, D∼L^{1.05}. This short time behavior is consistent with particle-based simulations. The long-time diffusion coefficient scales like D∼L^{1.6}, close to previous studies which found D∼L^{1.5}. Furthermore, we show that the near-field details of the interaction kernel do not affect the short time behavior but qualitatively and dramatically affect the long time behavior, potentially causing a saturation of the mean-squared displacement at long times. Our finding of a D∼L^{1.05} short time scaling resolves a long standing puzzle about the disagreement between the diffusion coefficient measured in particle-based models and mesoscale lattice models of amorphous plasticity.