Memory Kernel Research Articles

Stability of a viscoelastic Timoshenko system with non-monotonic kernel

In this paper, the stability of a linear Timoshenko beam system involved with infinite memory is considered. Different from the previous results on where the monotony of kernel is always fulfilled, the memory kernel under consideration is assumed to be non-monotonic. The well-posedness of the system is obtained by means of resolvent family theory and the exponential stability is proved under certain conditions. Numerical simulations are also presented to verify the main results.

Well-posedness and stability results for the Korteweg–de Vries–Burgers and Kuramoto–Sivashinsky equations with infinite memory: A history approach

The main concern of the present paper is to study the well-posedness and stability problem of two different dispersive systems subject to the effect of a distributed infinite memory term. The two systems are respectively governed by the one-dimensional Korteweg–de Vries–Burgers and Kuramoto–Sivashinsky equations in a bounded domain [0,1]. In order to deal with the presence of the memory term, we adopt the history approach. First, we show that both problems are well-posed in appropriate functional spaces by means of the Fixed-Point Theorem provided that the initial condition is sufficiently small. Then, the energy method enables us to provide a decay estimate of the systems’ energy according to the assumptions satisfied by the physical parameters and the memory kernel.

Simulating the dynamics of electronic observables via reduced-dimensionality generalized quantum master equations.

We describe a general-purpose framework for formulating the dynamics of any subset of electronic reduced density matrix elements in terms of a formally exact generalized quantum master equation (GQME). Within this framework, the effect of coupling to the nuclear degrees of freedom, as well as to any projected-out electronic reduced density matrix elements, is captured by a memory kernel and an inhomogeneous term, whose dimensionalities are dictated by the number of electronic reduced density matrix elements included in the subset of interest. We show that the memory kernel and inhomogeneous term within such GQMEs can be calculated from projection-free inputs of the same dimensionality, which can be cast in terms of the corresponding subsets of overall system two-time correlation functions. The applicability and feasibility of such reduced-dimensionality GQMEs is demonstrated on the two-state spin-boson benchmark model. To this end, we compare and contrast the following four types of GQMEs: (1) a full density matrix GQME, (2) a single-population scalar GQME, (3) a populations-only GQME, and (4) a subset GQME for any combination of populations and coherences. Using a method based on the mapping Hamiltonian approach and linearized semiclassical approximation to calculate the projection-free inputs, we find that while single-population GQMEs and subset GQMEs containing only one population are less accurate, they can still produce reasonable results and that the accuracy of the results obtained via the populations-only GQME and a subset GQME containing both populations is comparable to that obtained via the full density matrix GQMEs.

Effective Mori-Zwanzig equation for the reduced-order modeling of stochastic systems

<p style='text-indent:20px;'>Built upon the hypoelliptic analysis of the effective Mori-Zwanzig (EMZ) equation for observables of stochastic dynamical systems, we show that the obtained semigroup estimates for the EMZ equation can be used to derive prior estimates of the observable statistics for systems in the equilibrium and nonequilibrium state. In addition, we introduce both first-principle and data-driven methods to approximate the EMZ memory kernel and prove the convergence of the data-driven parametrization schemes using the regularity estimate of the memory kernel. The analysis results are validated numerically via the Monte-Carlo simulation of the Langevin dynamics for a Fermi-Pasta-Ulam chain model. With the same example, we also show the effectiveness of the proposed memory kernel approximation methods.</p>

Long-time memory effects in a localizable central spin problem

We study the properties of the Nakajima–Zwanzig memory kernel for a qubit immersed in a many-body localized (i.e. disordered and interacting) bath. We argue that the memory kernel decays as a power law in both the localized and ergodic regimes, and show how this can be leveraged to extract t → ∞ populations for the qubit from finite time (Jt ⩽ 102) data in the thermalizing phase. This allows us to quantify how the long-time values of the populations approach the expected thermalized state as the bath approaches the thermodynamic limit. This approach should provide a good complement to state-of-the-art numerical methods, for which the long-time dynamics with large baths are impossible to simulate in this phase. Additionally, our numerics on finite baths reveal the possibility for unbounded exponential growth in the memory kernel, a phenomenon rooted in the appearance of exceptional points in the projected Liouvillian governing the reduced dynamics. In small systems amenable to exact numerics, we find that these pathologies may have some correlation with delocalization.

Stochastic resonance in periodic potential system with memory damping function

The stochastic dynamical system with memory effects describes a non-Markovian process that can happen in some complex systems or disordered media, such as viscoelastic media and living cell. Its velocity yields the memory effects because of the nonlocality in time, giving rise to a generalized Langevin equation for describing the dynamics of the system. In particular, the friction term in generalized Langevin equation is given by the time-dependent memory kernel. Besides, the research of stochastic resonance in periodic potential models emerges as an important subject because such systems have potential applications in diverse areas of natural sciences. However, the analysis of the influence of memory on stochastic resonance has not been reported so far in periodic potential model. In this paper, the phenomenon of stochastic resonance is investigated in the periodic potential system with friction memory kernel driven by an external periodic signal and internal noise. The generalized Langevin equation is converted into the three-dimensional Markovian Langevin equations. Analytical expression for the spectral amplification, together with the amplitude of the response, is derived in the periodic potential with an arbitrary number of simultaneously stable steady states, which can be applied to the general multi-stable dynamical model. The obtained results indicate that the curve of spectral amplification versus temperature exhibits a pronounced peak. Obviously, this typical phenomenon is a signature of stochastic resonance. The stochastic resonance effect is enhanced with the increase of the memory time or the number of stable steady states. For a certain range of the particle motion, the existence of an optimal number of stable steady states for which the output of the system can be maximized is established. Moreover, the phenomenon of stochastic resonance is studied according to the stochastic energetics. The average input energy per period is calculated over all the trajectories for quantifying stochastic resonance. It is found that the stochastic resonance effect is first weakened and then enhanced with increasing memory time. Specifically, under appropriate temperature conditions, there is an optimal memory time, which can maximize the work done by the external periodic force on the system.

Stabilization of an Overhead Crane System With Infinite Memory in the Boundary Control

This paper is concerned with the boundary stabilization problem of an overhead crane with infinite memory. We show, under some assumptions of the memory kernel, that the problem is well-posed in the sense of semigroups theory. Furthermore, the uniform exponential stability of the solutions to the system is established by means of the frequency-domain method.

Indirect stabilization of coupled abstract evolution equations with memory: Different propagation speeds

We consider the Cauchy problem for coupled second‐order evolution equations in a Hilbert space with indirect memory damping, which arises from the theory of viscoelasticity. Here, the coefficients of the main operators in the system are non‐equal (the case of different speeds of propagation). Under much weaker assumptions on memory kernels, we obtain a polynomial decay rate, whereas it is known that exponential stability does not hold. This work essentially generalizes and improves the related ones in the literature. As an application of our abstract results, we also establish for the Timoshenko system the so far best decay rate under such weak assumptions on the memory kernel in the case of different speeds of propagation.

General decay results for viscoelastic systems with memory and time‐varying delay

In this paper, we consider a wave equation with memory term and time‐varying delayed velocity term. By some subtle analysis techniques, assuming memory kernel function with a minimal and general condition, we establish a general uniform energy decay result for the system. The condition here substantially weakens the conditions in earlier literature. From the result, we can give the explicit energy decay rates directly if the memory kernel is expressed explicitly. The result is new even for the case that memory kernel decays polynomially.

Collective resonant behaviors in two coupled fluctuating-mass oscillators with tempered Mittag-Leffler memory kernel

In this study, we propose the coupled fluctuating-mass oscillators with a tempered Mittag-Leffler (M-L) memory kernel, and investigate the collective resonant behaviors. Using the stochastic average method, we acquire the analytical expression of steady-state output amplitude. Based on numerical results, we further study the dependence on various system parameters in detail, and find that coupling strength, memory exponent and memory time exert different effects on the non-monotonic behaviors. We also observe that stochastic resonance (SR) in the wide sense occurs in the coupled system. These results can provide more extensive support for manipulating the collective behaviors in the potential applications.

Data-driven learning of Mori–Zwanzig operators for isotropic turbulence

Developing reduced-order models for turbulent flows, which contain dynamics over a wide range of scales, is an extremely challenging problem. In statistical mechanics, the Mori–Zwanzig (MZ) formalism provides a mathematically exact procedure for constructing reduced-order representations of high-dimensional dynamical systems, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. Turbulence models based on MZ formalism have been scarce due to the limited knowledge of the MZ operators, which originates from the difficulty in deriving MZ kernels for complex nonlinear dynamical systems. In this work, we apply a recently developed data-driven learning algorithm, which is based on Koopman's description of dynamical systems and Mori's linear projection operator, on a set of fully resolved isotropic turbulence datasets to extract the Mori–Zwanzig operators. With data augmentation using known turbulence symmetries, the extracted Markov term, memory kernel, and orthogonal dynamics are statistically converged and the generalized fluctuation–dissipation relation can be verified. The properties of the memory kernel and orthogonal dynamics, and their dependence on the choices of observables are investigated to address the modeling assumptions that are commonly used in MZ-based models. A series of numerical experiments are then constructed using the extracted kernels to evaluate the memory effects on prediction. The results show that the prediction errors are strongly affected by the choice of observables and can be further reduced by including the past history of the observables in the memory kernel.

Editorial: Overview and Some New Directions

The fractional derivative holds historical dependence or memory effects. But it also brings error accumulation of the numerical solutions as well as the theoretical analysis since many properties from the integer order case cannot hold. The fractional difference is discrete counterpart of the fractional derivative. The memory kernel is defined by applying the discrete functions on time scale and avoids the errors from the numerical discretization. It is particularly suitable for fractional modeling with computer implementations. Many applications of fractional dynamics to neural networks, signal processing, time series and big data become possible now. This issue of work presents the latest progress in the neighborhood related to fractional calculus.

PhotoNs-GPU: A GPU accelerated cosmological simulation code

We present a GPU-accelerated cosmological simulation code, PhotoNs-GPU, based on an algorithm of Particle Mesh Fast Multipole Method (PM-FMM), and focus on the GPU utilization and optimization. A proper interpolated method for truncated gravity is introduced to speed up the special functions in kernels. We verify the GPU code in mixed precision and different levels of theinterpolated method on GPU. A run with single precision is roughly two times faster than double precision for current practical cosmological simulations. But it could induce an unbiased small noise in power spectrum. Compared with the CPU version of PhotoNs and Gadget-2, the efficiency of the new code is significantly improved. Activated all the optimizations on the memory access, kernel functions and concurrency management, the peak performance of our test runs achieves 48% of the theoretical speed and the average performance approaches to ∼35% on GPU.

Time fractional advection-dispersion model to study transportation of particles with time-memory for unsteady nonequilibrium suspension in open-channel turbulent flows

In this study, the effects of time-memory on the mixing and nonequilibrium transportation of particles in an unsteady turbulent flow are investigated. The memory effect of particles is captured through a time-fractional advection-dispersion equation rather than a traditional advection-dispersion equation. The time-fractional derivative is considered in Caputo sense which includes a power-law memory kernel that captures the power-law jumps of particles. The time-fractional model is solved using the Chebyshev collocation method. To make the solution procedure more robust three different kinds of Chebyshev polynomials are considered. The time-fractional derivative is approximated using the finite difference method at small time intervals and numerical solutions are obtained in terms of Chebyshev polynomials. The model solutions are compared with existing experimental data of traditional conditions and satisfactory results are obtained. Apart from this, the effects of time-memory are analyzed for bottom concentration and transient concentration distribution of particles. The results show that for uniform initial conditions, bottom concentration increases with time as the order of fractional derivative decreases. In the case of transient concentration, the value of concentration initially decreases when T < 1 and thereafter increases throughout the flow depth. The effects of time-memory are also analyzed under steady flow conditions. Results show that under steady conditions, transient concentration is more sensitive for linear, parabolic, and parabolic-constant models of sediment diffusivity rather than the constant model.

Path-integral approach to the thermodynamics of bosons with memory: Density and correlation functions

Expanding upon previous work, using the path-integral formalism we derive expressions for the one-particle reduced density matrix and the two-point correlation function for a quadratic system of bosons that interact through a general class of memory kernels. The results are applied to study the density, condensate fraction and pair correlation function of trapped bosons harmonically coupled to external distinguishable masses.

Pólya Urn with Memory Kernel and Asymptotic Behaviors of Autocorrelation Function

P\'{o}lya urn is a stochastic process in which balls are randomly drawn from an urn of red and blue balls, and balls of the same color as the drawn balls are added. The probability of a ball of a certain color being drawn is equal to the percentage of balls of that color in the urn. We introduce arbitrary memory kernels to modify this probability. If the memory kernel decays exponentially, it is a stationary process and is mean-reverting. If the memory kernel decays by a power-law, a phase transition occurs and the asymptotic behavior of the autocorrelation function changes. An auxiliary field variable is introduced to transform the process Markovian and the field obeys a multivariate Ornstein-Uhlenbeck process. The exponents of the power law are estimated for the decay of the leading and subleading terms of the autocorrelation function. It is shown that the power law exponents changes discontinuously at the critical point.

Stable numerical schemes for time-fractional diffusion equation with generalized memory kernel

The paper aims to develop the stable numerical schemes for generalized time-fractional diffusion equations (GTFDEs) with smooth and non-smooth solutions on the non-uniform grid. In time, the generalized Caputo derivative is discretized by a difference scheme of order (2−α) on a non-uniform grid where 0<α<1. Choosing the non-uniform meshes in the case of the smooth and non-smooth solution is also essential, so we graded the mesh in both cases separately. Stability and convergence for smooth as well as non-smooth solutions are obtained in L2-norm and L∞-norm respectively. Several numerical results are presented to show how the grading of meshes is essential. Also, numerical results validate the efficiency and effectiveness of proposed schemes and show how a non-uniform grid produces better results.

Flow decomposition for heat equations with memory

We build up a decomposition for the flow generated by the heat equation with a real analytic memory kernel. It consists of three components: The first one is of parabolic nature; the second one gathers the hyperbolic component of the dynamics, with null velocity of propagation; the last one exhibits a finite smoothing effect. This decomposition reveals the hybrid parabolic-hyperbolic nature of the flow and clearly illustrates the significant impact of the memory term on the parabolic behavior of the system in the absence of memory terms.

Approximate controllability of a parabolic integro-differential equation with nonconstant memory kernel under a weak constraint condition

The approximate controllability of a parabolic integro-differential equation system has already been shown for the case where the memory kernel is a constant. However, when this condition on the memory kernel and considering instead only the weak constraint of boundedness, the problem becomes more complex and it is less easy to prove the approximate controllability of the system. In this study, based on the operator semigroup method, a detailed proof of the approximate controllability of the system is given for the case where the memory kernel is nonconstant and bounded. In addition, three numerical simulation studies are performed to verify the theoretical proof.

Engineering Classical Capacity of Generalized Pauli Channels with Admissible Memory Kernels.

In this paper, we analyze the classical capacity of the generalized Pauli channels generated via memory kernel master equations. For suitable engineering of the kernel parameters, evolution with non-local noise effects can produce dynamical maps with a higher capacity than a purely Markovian evolution. We provide instructive examples for qubit and qutrit evolution. Interestingly, similar behavior is not observed when analyzing time-local master equations.