Power-law Kernel Research Articles

Quantifying Endogeneity of Cryptocurrency Markets

We construct a ‘reflexivity’ index to measure the activity generated endogenously within a market for cryptocurrencies. For this purpose, we fit a univariate self-exciting Hawkes process with two classes of parametric kernels to high-frequency trading data. A parsimonious model of both endogenous and exogenous dynamics enables a direct comparison with exchanges for traditional asset classes, in terms of identified branching ratios. We also formulate a ‘Hawkes disorder problem,’ as generalization of the established Poisson disorder problem, and provide a simulation-based approach to determining an optimal observation horizon. Our analysis suggests that Bitcoin mid-price dynamics feature long-memory properties, well explained by the power-law kernel, at a level of criticality similar to fiat-currency markets.

Memory effects in measure transport equations

Transport equations with a nonlocal velocity field have been introduced as a continuum model for interacting particle systems arising in physics, chemistry and biology. Fractional time derivatives, given by convolution integrals of the time-derivative with power-law kernels, are typical for memory effects in complex systems. In this paper we consider a nonlinear transport equation with a fractional time-derivative. We provide a well-posedness theory for weak measure solutions of the problem and an integral formula which generalizes the classical push-forward representation formula to this setting.

Magnetohydrodynamic Free Convection Flows with Thermal Memory over a Moving Vertical Plate in Porous Medium

The unsteady hydro-magnetic free convection flow with heat transfer of a linearly viscous, incompressible, electrically conducting fluid near a moving vertical plate with the constant heat is investigated. The flow domain is the porous half-space and a magnetic field of a variable direction is applied. The Caputo time-fractional derivative is employed in order to introduce a thermal flux constitutive equation with a weakly memory. The exact solutions for the fractional governing differential equations for fluid temperature, Nusselt number, velocity field, and skin friction are obtained by using the Laplace transform method. The numerical calculations are carried out and the results are presented in graphical illustrations. The influence of the memory parameter (the fractional order of the time-derivative) on the temperature and velocity fields is analyzed and a comparison between the fluid with the thermal memory and the ordinary fluid is made. It was observed that due to evolution in the time of the Caputo power-law kernel, the memory effects are stronger for the small values of the time t. Moreover, it is found that the fluid flow is accelerated / retarded by varying the inclination angle of the magnetic field direction.

Time fractional derivative model with Mittag-Leffler function kernel for describing anomalous diffusion: Analytical solution in bounded-domain and model comparison

Non-Fickian or anomalous diffusion had been well documented in material transport through heterogeneous systems at all scales, whose dynamics can be quantified by the time fractional derivative equations (fDEs). While analytical or numerical solutions have been developed for the standard time fDE in bounded domains, the standard time fDE suffers from the singularity issue due to its power-law function kernel. This study aimed at deriving the analytical solutions for the time fDE models with a modified kernel in bounded domains. The Mittag-Leffler function was selected as the alternate kernel to improve the standard power-law function in defining the time fractional derivative, which was known to be able to overcome the singularity issue of the standard fractional derivative. Results showed that the method of variable separation can be applied to derive the analytical solution for various time fDEs with absorbing and/or reflecting boundary conditions. Finally, numerical examples with detailed comparison for fDEs with different kernels showed that the models and solutions obtained by this study can capture anomalous diffusion in bounded domains.

The Small-Mass Limit and White-Noise Limit of an Infinite Dimensional Generalized Langevin Equation

We study asymptotic properties of the Generalized Langevin Equation (GLE) in the presence of a wide class of external potential wells with a power-law decay memory kernel. When the memory can be expressed as a sum of exponentials, a class of Markovian systems in infinite-dimensional spaces is used to represent the GLE. The solutions are shown to converge in probability in the small-mass limit and the white-noise limit to appropriate systems under minimal assumptions, of which no Lipschitz condition is required on the potentials. With further assumptions about space regularity and potentials, we obtain $L^1$ convergence in the white-noise limit.

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.

Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws

Variable-order differential operators can be employed as a powerful tool to modeling nonlinear fractional differential equations and chaotical systems. In this paper, we propose a new generalize numerical schemes for simulating variable-order fractional differential operators with power-law, exponential-law and Mittag-Leffler kernel. The numerical schemes are based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. These schemes were applied to simulate the chaotic financial system and memcapacitor-based circuit chaotic oscillator. Numerical examples are presented to show the applicability and efficiency of this novel method.

A Non-Linear Stress-Strain Relation Endowed With Fractional Derivative Elements

In this paper a non-linear stress-strain relation based on an integral formulation with a power-law kernel is proposed. This constitutive law is able to reproduce both the viscoelastic behavior and the inelastic irreversible phenomenon. It is shown how the proposed stress-strain law is capable to fit experimental data obtained from tensile tests on two kind of metal alloys. Such best-fitting procedure have shown the accuracy of the proposed model and its results are compared to other ones obtained with the aid of classical non-linear constitutive law.

Convective flows of generalized time-nonlocal nanofluids through a vertical rectangular channel

Time-nonlocal generalized model of the natural convection heat transfer and nanofluid flows through a rectangular vertical channel with wall conditions of the Robin type are studied. The generalized mathematical model with time-nonlocality is developed by considering the fractional constitutive equations for the shear stress and thermal flux defined with the time-fractional Caputo derivative. The Caputo power-law non-local kernel provides the damping to the velocity and temperature gradient; therefore, transport processes are influenced by the histories at all past and present times. Analytical solutions for dimensionless velocity and temperature fields are obtained by using the Laplace transform coupled with the finite sine-cosine Fourier transform which is suitable to problems with boundary conditions of the Robin type. Particularizing the fractional thermal and velocity parameters, solutions for three simplified models are obtained (classical linear momentum equation with damped thermal flux; fractional shear stress constitutive equation with classical Fourier’s law for thermal flux; classical shear stress and thermal flux constitutive equations). It is found that the thermal histories strongly influence the thermal transport for small values of time t. Also, the thermal transport can be enhanced if the thermal fractional parameter decreases or by increasing the nanoparticles’ volume fraction. The velocity field is influenced on the one hand by the temperature of the fluid and on the other by the damping of the velocity gradient introduced by the fractional derivative. Also, the transport motions of the channel walls influence the motion of the fluid layers located near them.

Dynamic disorder characteristics of single-molecule pulling experiments on two-dimensional free-energy landscape

The dynamic characteristic of macromolecule is mainly subject to the fluctuation of rate constant and this phenomenon is usually considered as dynamic disorder (DD). In order to detect the DD nature in the bio-molecule system more accurately, here we propose a theoretical framework based on the two-dimensional (2D) free-energy landscape including the pulling coordinate and other slow conformational variables. The generalized Langevin equation (GLE) with fractional Gaussian noise (fGn) and the power-law memory kernel are used on this landscape for the research. The transition rate, which depends on both intrinsic barrier height and noise strength ratio, has been analyzed under the condition of external force. The particular discrepancies were investigated between the kinetics of the transitions with and without DD. We find that the discrepancies relied on the barrier height and the noise strength ratio. Taken together, our study illustrates the importance of the DD characteristics, which should be taken into account during the research into the single-molecule pulling experiments.

Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena

To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamental differences between the power law, exponential decay, Mittag-Leffler law and their possible applications in nature. We demonstrate the failure of the semi-group principle in modeling real-world problems. We use natural phenomena to illustrate the importance of non-commutative and non-associative operators under which the Caputo-Fabrizio and Atangana-Baleanu fractional operators fall. We present statistical properties of generator for each fractional derivative, including Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu ones. The Atangana-Baleanu and Caputo-Fabrizio fractional derivatives show crossover properties for the mean-square displacement, while the Riemann-Liouville is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the Caputo Fabrizio kernel has a steady state between the transition. Only the Atangana-Baleanu kernel is a crossover for the waiting time distribution from stretched exponential to power law. A new criterion was suggested, namely the Atangana-Gomez fractional bracket, that helps describe the energy needed by a fractional derivative to characterize a 2-pletic manifold. Based on these properties, we classified fractional derivatives in three categories: weak, mild and strong fractional differential and integral operators. We presented some applications of fractional differential operators to describe real-world problems and we proved, with numerical simulations, that the Riemann-Liouville power-law derivative provides a description of real-world problems with much additional information, that can be seen as noise or error due to specific memory properties of its power-law kernel. The Caputo-Fabrizio derivative is less noisy while the Atangana-Baleanu fractional derivative provides an excellent description, due to its Mittag-Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state. The study suggests that the properties of associativity and commutativity or the semi-group principle are just irrelevant in fractional calculus. Properties of classical derivatives were established for the ordinary calculus with no memory effect and it is a failure of mathematical investigation to attempt to describe more complex natural phenomena using the same notions.

Uniqueness of Mass-Conserving Self-similar Solutions to Smoluchowski’s Coagulation Equation with Inverse Power Law Kernels

Uniqueness of mass-conserving self-similar solutions to Smoluchowski's coagulation equation is shown when the coagulation kernel $K$ is given by $K(x,x\_*)=2(x x\_*)^{-\alpha}$, $(x,x\_*)\in (0,\infty)^2$, for some $\alpha>0$.

Exact and Nonlocal Solutions for Vibration of Axial Lattice with Direct and Indirect Neighboring Interactions

This paper investigates the effect of long-range interaction forces in the longitudinal vibration of an axial linear elastic lattice. The eigenfrequencies of this axial lattice with direct and indirect linear elastic interactions are expressed in closed-form solutions for fixed-fixed and fixed-free boundary conditions. The paper starts with the two-neighbor interaction problem, and then generalizes the result to the more general N-neighbor interaction problem. Starting from a continualization of the higher-order difference equations, a nonlocal elastic continuum is constructed to approximate the behavior of the axial lattice with generalized short and long-range interactions. It is shown, while preserving the definite positiveness of the generalized lattice strain energy, that the associated nonlocal continuum is equivalent to a stress gradient type of Eringen's nonlocal model, which is able to capture the main scale phenomena of the generalized lattice. The length scale factor of the nonlocal model is calibrated to the size of the influence domain for the long-range interaction. The paper ends with a discussion about the effect of the discrete interaction kernel on the sensitivity of the eigenfrequencies with respect to scale effects. Exact analytical natural frequencies are calculated for discrete power law or exponential-based kernels. The stress gradient nonlocal theory offers an efficient engineering continuum framework, which accurately fits the response of generalized lattice with N-neighbor interaction.

Analysis of projectile motion: A comparative study using fractional operators with power law, exponential decay and Mittag-Leffler kernel

In this paper, the two-dimensional projectile motion was studied; for this study two cases were considered, for the first one, we considered that there is no air resistance and, for the second case, we considered a resisting medium k . The study was carried out by using fractional calculus. The solution to this study was obtained by using fractional operators with power law, exponential decay and Mittag-Leffler kernel in the range of $ \gamma \in (0,1]$ . These operators were considered in the Liouville-Caputo sense to use physical initial conditions with a known physical interpretation. The range and the maximum height of the projectile were obtained using these derivatives. With the aim of exploring the validity of the obtained results, we compared our results with experimental data given in the literature. A multi-objective particle swarm optimization approach was used for generating Pareto-optimal solutions for the parameters k and $ \gamma$ for different fixed values of velocity v0 and angle $ \theta$ . The results showed some relevant qualitative differences between the use of power law, exponential decay and Mittag-Leffler law.

Memory effects for a stochastic fractional oscillator in a magnetic field

The problem of random motion of harmonically trapped charged particles in a constant external magnetic field is studied. A generalized three-dimensional Langevin equation with a power-law memory kernel is used to model the interaction of Brownian particles with the complex structure of viscoelastic media (e.g., dusty plasmas). The influence of a fluctuating environment is modeled by an additive fractional Gaussian noise. In the long-time limit the exact expressions of the first-order and second-order moments of the fluctuating position for the Brownian particle subjected to an external periodic force in the plane perpendicular to the magnetic field have been calculated. Also, the particle's angular momentum is found. It is shown that an interplay of external periodic forcing, memory, and colored noise can generate a variety of cooperation effects, such as memory-induced sign reversals of the angular momentum, multiresonance versus Larmor frequency, and memory-induced particle confinement in the absence of an external trapping field. Particularly in the case without external trapping, if the memory exponent is lower than a critical value, we find a resonancelike behavior of the anisotropy in the particle position distribution versus the driving frequency, implying that it can be efficiently excited by an oscillating electric field. Similarities and differences between the behaviors of the models with internal and external noises are also discussed.

Memory-induced resonancelike suppression of spike generation in a resonate-and-fire neuron model.

The behavior of a stochastic resonate-and-fire neuron model based on a reduction of a fractional noise-driven generalized Langevin equation (GLE) with a power-law memory kernel is considered. The effect of temporally correlated random activity of synaptic inputs, which arise from other neurons forming local and distant networks, is modeled as an additive fractional Gaussian noise in the GLE. Using a first-passage-time formulation, in certain system parameter domains exact expressions for the output interspike interval (ISI) density and for the survival probability (the probability that a spike is not generated) are derived and their dependence on input parameters, especially on the memory exponent, is analyzed. In the case of external white noise, it is shown that at intermediate values of the memory exponent the survival probability is significantly enhanced in comparison with the cases of strong and weak memory, which causes a resonancelike suppression of the probability of spike generation as a function of the memory exponent. Moreover, an examination of the dependence of multimodality in the ISI distribution on input parameters shows that there exists a critical memory exponent α_{c}≈0.402, which marks a dynamical transition in the behavior of the system. That phenomenon is illustrated by a phase diagram describing the emergence of three qualitatively different structures of the ISI distribution. Similarities and differences between the behavior of the model at internal and external noises are also discussed.

Generalized time-dependent Schrödinger equation in two dimensions under constraints

We investigate a generalized two-dimensional time-dependent Schrödinger equation on a comb with a memory kernel. A Dirac delta term is introduced in the Schrödinger equation so that the quantum motion along the x-direction is constrained at y = 0. The wave function is analyzed by using Green’s function approach for several forms of the memory kernel, which are of particular interest. Closed form solutions for the cases of Dirac delta and power-law memory kernels in terms of Fox H-function, as well as for a distributed order memory kernel, are obtained. Further, a nonlocal term is also introduced and investigated analytically. It is shown that the solution for such a case can be represented in terms of infinite series in Fox H-functions. Green’s functions for each of the considered cases are analyzed and plotted for the most representative ones. Anomalous diffusion signatures are evident from the presence of the power-law tails. The normalized Green’s functions obtained in this work are of broader interest, as they are an important ingredient for further calculations and analyses of some interesting effects in the transport properties in low-dimensional heterogeneous media.

New numerical approach for fractional differential equations

In the present case, we propose the correct version of the fractional Adams-Bashforth methods which take into account the nonlinearity of the kernels including the power law for the Riemann-Liouville type, the exponential decay law for the Caputo-Fabrizio case and the Mittag-Leffler law for the Atangana-Baleanu scenario.The Adams-Bashforth method for fractional differentiation suggested and are commonly use in the literature nowadays is not mathematically correct and the method was derived without taking into account the nonlinearity of the power law kernel. Unlike the proposed version found in the literature, our approximation, in all the cases, we are able to recover the standard case whenever the fractional powerα= 1. Numerical results are finally given to justify the effectiveness of the proposed schemes.

Exceptional times for percolation under exclusion dynamics

We analyse in this paper a conservative analogue of the celebrated model of dynamical percolation introduced by H\aggstr\om, Peres and Steif in [HPS97]. It is simply defined as follows: start with an initial percolation configuration $\omega(t=0)$. Let this configuration evolve in time according to a simple exclusion process with symmetric kernel $K(x,y)$. We start with a general investigation (following [HPS97]) of this dynamical process $t \mapsto \omega_K(t)$ which we call $K$-exclusion dynamical percolation. We then proceed with a detailed analysis of the planar case at the critical point (both for the triangular grid and the square lattice $Z^2$) where we consider the power-law kernels $K^\alpha$ \[ K^{\alpha}(x,y) \propto \frac 1 {\|x-y\|_2^{2+\alpha}} \, . \] We prove that if $\alpha > 0$ is chosen small enough, there exist exceptional times $t$ for which an infinite cluster appears in $\omega_{K^{\alpha}}(t)$. (On the triangular grid, we prove that it holds for all $\alpha < \alpha_0 = \frac {217}{816}$.) The existence of such exceptional times for standard i.i.d. dynamical percolation (where sites evolve according to independent Poisson point processes) goes back to the work by Schramm-Steif in [SS10]. In order to handle such a $K$-exclusion dynamics, we push further the spectral analysis of exclusion noise sensitivity which had been initiated in [BGS13]. (The latter paper can be viewed as a conservative analogue of the seminal paper by Benjamini-Kalai-Schramm [BKS99] on i.i.d. noise sensitivity.) The case of a nearest-neighbour simple exclusion process, corresponding to the limiting case $\alpha = +\infty$, is left widely open.

Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.