Langevin-type Equation Research Articles

A class of processes defined in the white noise space through generalized fractional operators

The generalization of fractional Brownian motion in the white and grey noise spaces has been recently carried over, following the Mandelbrot–Van Ness representation, through Riemann–Liouville type fractional operators. Our aim is to extend this construction by means of more general fractional derivatives and integrals, which we define as Fourier-multiplier operators and then specialize by means of Bernstein functions. More precisely, we introduce a general class of kernel-driven processes which encompasses, as special cases, a number of models in the literature, including fractional Brownian motion, tempered fractional Brownian motion, Ornstein–Uhlenbeck process. The greater generality of our model, with respect to the previous ones, allows a higher flexibility and a wider applicability. We derive here some properties of this class of processes (such as continuity, occupation density, variance asymptotics and persistence) according to the conditions satisfied by the Fourier symbol of the operator or the Bernstein function chosen. On the other hand, these processes are proved to display short- or long-range dependence, if obtained by means of a derivative or an integral type operator, respectively, regardless of the kernel used in their definition. Finally, this kind of construction allows us to define the corresponding noise and to solve a Langevin-type integral equation.

Fractional differential equations of Bagley-Torvik and Langevin type

Fractional differential equations of Bagley-Torvik and Langevin type

Theoretical modeling of the above-barrier fusion process for nuclei with [formula omitted] based on the experimental nuclear charge densities

Most of theoretical models of nucleus-nucleus collision are based on the nucleus-nucleus potential. Semi-microscopical double-folding model with the M3Y-Paris NN-forces represents one of the advanced methods for evaluating this potential. Nucleon densities (NDs), being crucial ingredient of this model, must reproduce the experimental nuclear charge densities (NCDs). In theoretical approaches for modeling the fusion process, it is not always so. In the present work, we aim to reach satisfactory description of both the charge nucleon density and the above-barrier fusion cross sections of even-even nuclei with Z=N. We construct an approximation for the ND which is abbreviated as FEV-density (Fermi + Exponential tail with Variable diffuseness). The proton and neutron densities are considered to be identical and having a coordinate-dependent diffuseness. Parameters of the nucleon density are varied to obtain the best fit of the experimental NCD. Then the resulting FEV nucleon densities are used to calculate the nucleus-nucleus potential applying the double-folding model with the density-dependent M3Y-Paris NN-forces. This potential is employed for evaluating the above-barrier fusion cross sections for ten reactions: 32S+12C, 24Mg, 40Ca; 28Si+12C, 16O, 24Mg, 28Si; and 24Mg+12C, 16O, 24Mg where the experimental data are available. The cross sections are calculated using either the barrier penetration model or the trajectory model with surface friction. To find the transmission coefficients for the trajectory model, the Langevin-type equations are solved numerically. For all considered reactions, our trajectory model typically reproduces the above-barrier experimental cross sections within 10–15 % with typical χ2<2. The adjustable parameter of the model, the optimal friction strength KRm, was found to be between 20 and 40 zs·GeV−1 which is in reasonable agreement with two systematics found earlier. We also make predictions for the fusion excitation functions for reactions 32S+16O, 32S+28Si, 24Si+40Ca where we did not manage finding experimental data.

Stochastic melonic kinetics with random initial conditions

The probability laws associated with random tensors or tensor field theories are traditionally equilibrium distributions. In this paper, we consider a stochastic point of view, and approach the quantization by a Langevin type equation. We especially address the low-temperature behavior of the phase ordering kinetics of a stochastic complex tensor field Ti1⋯id(x,t) of size N and rank d in dimension D. The method we propose use the self averaging property of the tensorial invariants in the large N limit. In this regime, the dynamics is governed by the melonic sector, whose behavior is studied in the quenched limit, where the contractions involving d−1 indices self-average around a diagonal matrix proportional to the identity. The following work especially focuses on the cyclic (i.e. non-branching) melonic sector, and we study the way that the system returns to the equilibrium regime regarding the temperature and the shape of the potential. In particular, we provide a general formula for the transition temperature between these regimes. The manuscript is accompanied by numerical simulations to support the theoretical analysis, and essentially aims to open towards this new field of investigation.

ON TEMPORAL VARIATIONS IN THE GRANULOMETRIC CHARACTERISTICS OF BINARY MIXED MICROPOWDERS TREATED IN A PLANETARY-TYPE BALL MILL

Integral and differential particle size distribution functions have been experimentally determined for two-component mixed micropowders, which contain stoichiometric amounts of polycrystalline aluminum, nickel, and titanium, as well as amorphous boron, and have been mechanically treated in a planetary-type ball mill under different time conditions of the process. The influence of the duration of treatment of the above mixtures on the mathematical parameters of the found distribution functions has been analyzed. It has been shown that, in all considered cases, these functions can be represented in a lognormal form. The most informative statistical characteristics (moments) of these functions have been determined. The dependences of these characteristics on the duration of the mechanical treatment of the mixtures have been revealed. Within the framework of an approximation based on the use of generalized dynamic-stochastic Langevin-type equations, a kinetic model has been proposed for the process of mechanical treatment of metallic and non-metallic mixed micropowders in planetary-type mills. The model makes it possible to describe, as a first approximation, the temporal evolution of the fractional composition of the mixtures during their machining, as well as to record temporal variations of the main statistical characteristics of the corresponding distribution functions in this process.

Learning effective stochastic differential equations from microscopic simulations: Linking stochastic numerics to deep learning.

We identify effective stochastic differential equations (SDEs) for coarse observables of fine-grained particle- or agent-based simulations; these SDEs then provide useful coarse surrogate models of the fine scale dynamics. We approximate the drift and diffusivity functions in these effective SDEs through neural networks, which can be thought of as effective stochastic ResNets. The loss function is inspired by, and embodies, the structure of established stochastic numerical integrators (here, Euler-Maruyama and Milstein); our approximations can thus benefit from backward error analysis of these underlying numerical schemes. They also lend themselves naturally to "physics-informed" gray-box identification when approximate coarse models, such as mean field equations, are available. Existing numerical integration schemes for Langevin-type equations and for stochastic partial differential equations can also be used for training; we demonstrate this on a stochastically forced oscillator and the stochastic wave equation. Our approach does not require long trajectories, works on scattered snapshot data, and is designed to naturally handle different time steps per snapshot. We consider both the case where the coarse collective observables are known in advance, as well as the case where they must be found in a data-driven manner.

Theoretical and Numerical Study of Self-Organizing Processes in a Closed System Classical Oscillator and Random Environment

A self-organizing joint system classical oscillator–random environment is considered within the framework of a complex probabilistic process that satisfies a Langevin-type stochastic differential equation. Various types of randomness generated by the environment are considered. In the limit of statistical equilibrium (SEq), second-order partial differential equations (PDE) are derived that describe the distribution of classical environmental fields. The mathematical expectation of the oscillator trajectory is constructed in the form of a functional-integral representation, which, in the SEq limit, is compactified into a two-dimensional integral representation with an integrand: the solution of the second-order complex PDE. It is proved that the complex PDE in the general case is reduced to two independent PDEs of the second order with spatially deviating arguments. The geometric and topological features of the two-dimensional subspace on which these equations arise are studied in detail. An algorithm for parallel modeling of the problem has been developed.

On a boundary value problem for a class of fractional Langevin type differential equations in a Banach space

In this paper, we consider a boundary value problem for differential equations of Langevin type with the Caputo fractional derivative in a Banach space. It is assumed that the nonlinear part of the equation is a Caratheodory type map. Equations of this type generalize equations of motion in various kinds of media, for example, viscoelastic media or in media where a drag force is expressed using a fractional derivative. We will use the theory of fractional mathematical analysis, the properties of the Mittag-Leffler function, as well as the theory of measures of non-compactness and condensing operators to solve the problem. The initial problem is reduced to the problem of the existence of fixed points of the corresponding resolving integral operator in the space of continuous functions. We will use Sadovskii type fixed point theorem to prove the existence of fixed points of the resolving operator. We will show that the resolving integral operator is condensing with respect to the vector measure of non-compactness in the space of continuous functions and transforms a closed ball in this space into itself.

Finite Iterative Forecasting Model Based on Fractional Generalized Pareto Motion

In this paper, an efficient prediction model based on the fractional generalized Pareto motion (fGPm) with Long-Range Dependent (LRD) and infinite variance characteristics is proposed. Firstly, we discuss the meaning of each parameter of the generalized Pareto distribution (GPD), and the LRD characteristics of the generalized Pareto motion are analyzed by taking into account the heavy-tailed characteristics of its distribution. Then, the mathematical relationship H=1⁄α between the self-similar parameter H and the tail parameter α is obtained. Also, the generalized Pareto increment distribution is obtained using statistical methods, which offers the subsequent derivation of the iterative forecasting model based on the increment form. Secondly, the tail parameter α is introduced to generalize the integral expression of the fractional Brownian motion, and the integral expression of fGPm is obtained. Then, by discretizing the integral expression of fGPm, the statistical characteristics of infinite variance is shown. In addition, in order to study the LRD prediction characteristic of fGPm, LRD and self-similarity analysis are performed on fGPm, and the LRD prediction conditions H&gt;1⁄α is obtained. Compared to the fractional Brownian motion describing LRD by a self-similar parameter H, fGPm introduces the tail parameter α, which increases the flexibility of the LRD description. However, the two parameters are not independent, because of the LRD condition H&gt;1⁄α. An iterative prediction model is obtained from the Langevin-type stochastic differential equation driven by fGPm. The prediction model inherits the LRD condition H&gt;1⁄α of fGPm and the time series, simulated by the Monte Carlo method, shows the superiority of the prediction model to predict data with high jumps. Finally, this paper uses power load data in two different situations (weekdays and weekends), used to verify the validity and general applicability of the forecasting model, which is compared with the fractional Brown prediction model, highlighting the “high jump data prediction advantage” of the fGPm prediction model.

Quantum theory of electronic friction

Electronic friction is an important energy loss channel for atoms and molecules scattering off, reacting, or simply vibrating at metallic surfaces. It is usually well described by mixed classical-quantum approaches where the nuclei evolve classically according to Langevin-type equations of motion, and Born-Oppenheimer forces and friction kernels are obtained from first-principles electronic structure calculations. However, classical dynamics falls short when light atoms are involved, which is also the situation where electronic friction becomes the dominant dissipation channel and its role in the dynamics can be unambiguously assessed. Furthermore, the interplay between electronic friction and nuclear quantum effects in molecular processes at surfaces is largely unknown; in fact, it is not even clear how to include electronic friction in a quantum setting. Here we fill this gap by developing a fully quantum theory of electronic friction at $T=0\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. The electronic bath is considered to be entirely general and can be made of interacting electrons, potentially in a strongly correlated state. The derived friction kernel agrees with a recently obtained mixed quantum-classical result [Dou, Miao, and Subotnik, Phys. Rev. Lett. 119, 046001 (2017)], except for a pseudomagnetic contribution in the latter that is removed here. The ensuing equation of motion for the nuclear wave function is a nonlinear Schr\"odinger equation with a frictional vector potential that depends on the past wave function behavior. The equation becomes local-in-time in the typical situation where the electrons respond rapidly on the slow timescale of the nuclear dynamics (Markov limit) and generalizes previously known Schr\"odinger-Langevin equations to coordinate-dependent, tensorial friction kernels.

Lyapunov-type inequalities for fractional Langevin-type equations involving Caputo-Hadamard fractional derivative

In this study, some new Lyapunov-type inequalities are presented for Caputo-Hadamard fractional Langevin-type equations of the forms Da+βHC(HCDa+α+p(t))x(t)+q(t)x(t)=0,0<a<t<b,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\begin{aligned} &{}_{H}^{C}D_{a + }^{\\beta } \\bigl({}_{H}^{C}D_{a + }^{\\alpha }+ p(t)\\bigr)x(t) + q(t)x(t) = 0,\\quad 0 < a < t < b, \\end{aligned} $$\\end{document} and Da+ηHCϕp[(HCDa+γ+u(t))x(t)]+v(t)ϕp(x(t))=0,0<a<t<b,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\begin{aligned} &{}_{H}^{C}D_{a + }^{\\eta }{ \\phi _{p}}\\bigl[\\bigl({}_{H}^{C}D_{a + }^{\\gamma }+ u(t)\\bigr)x(t)\\bigr] + v(t){\\phi _{p}}\\bigl(x(t)\\bigr) = 0,\\quad 0 < a < t < b, \\end{aligned} $$\\end{document} subject to mixed boundary conditions, respectively, where p(t), q(t), u(t), v(t) are real-valued functions and 0 < beta < 1 < alpha < 2, 1 < gamma , eta < 2, {phi _{p}}(s) = |s{|^{p - 2}}s, p > 1. The boundary value problems of fractional Langevin-type equations were firstly converted into the equivalent integral equations with corresponding kernel functions, and then the Lyapunov-type inequalities were derived by the analytical method. Noteworthy, the Langevin-type equations are multi-term differential equations, creating significant challenges and difficulties in investigating the problems. Consequently, this study provides new results that can enrich the existing literature on the topic.

Application of projection operator method to coarse-grained dynamics with transient potential.

We show that the coarse-grained dynamics model with the time-dependent and fluctuating potential (transient potential) can be derived from the microscopic Hamiltonian dynamics. The concept of the transient potential was first introduced rather phenomenologically, and its relation to the underlying microscopic dynamics has not been clarified yet. This is in contrast to the generalized Langevin equation, the relation of which to the microscopic dynamics is well-established. In this work, we show that the dynamic equationswith the transient potential can be derived for the coupled oscillator model, without any approximations. It is known that the dynamics of the coupled oscillator model can be exactly described by the generalized Langevin-type equations. This fact implies that the dynamic equationswith the transient potential can be utilized as a coarse-grained dynamics model in a similar way to the generalized Langevin equation. Then we show that the dynamic equationsfor the transient potential can also be formally derived for the microscopic Hamiltonian dynamics, without any approximations. We use the projection operator method for the coarse-grained variables and transient potential. The dynamic equationsfor the coarse-grained positions and momenta are similar to those in the Hamiltonian dynamics, but the interaction potential is replaced by the transient potential. The dynamic equationfor the transient potential is the generalized Langevin equationwith the memory effect. Our result justifies the use of the transient potential to describe the coarse-grained dynamics. We propose several approximations to obtain the simplified dynamics model. We show that, under several approximations, the dynamic equationfor the transient potential reduces to the relatively simple Markovian dynamic equationfor the potential parameters. We also show that with several additional approximations, the approximate dynamics model further reduces to the Markovian Langevin-type equationswith the transient potential.

A Langevin-Type q-Variant System of Nonlinear Fractional Integro-Difference Equations with Nonlocal Boundary Conditions

We introduce a new class of boundary value problems consisting of a q-variant system of Langevin-type nonlinear coupled fractional integro-difference equations and nonlocal multipoint boundary conditions. We make use of standard fixed-point theorems to derive the existence and uniqueness results for the given problem. Illustrative examples for the obtained results are also presented.

Fractional Langevin Type Equations for White Noise Distributions

Abstract In this paper, by applying the intertwining properties, we introduce the fractional powers of the number operator perturbed by generalized Gross Laplacians (infinite dimensional Laplacians), which are special types of the infinitesimal generators of generalized Mehler semigroups. By applying the intertwining properties and semigroup approach, we study the Langevin type equations associated with the infinite dimensional Laplacians and with white noise distributions as forcing terms. Then we investigate the unique solution of the fractional Langevin type equations associated with the Riemann-Liouville and Caputo time fractional derivatives, and the fractional power of the infinite dimensional Laplacians, for which we apply the intertwining properties again. For our purpose, we discuss the fractional integrals and fractional derivatives of white noise distribution valued functions.

Discrete Langevin-type equation for p-order persistent time series and procedure of its reconstruction.

The stochastic discrete Langevin-type equation, which can describe p-order persistent processes, was introduced. The procedure of reconstruction of the equation from time series was proposed and tested on synthetic data. The approach was applied to hydrological data leading to the stochastic model of the phenomenon. The work is a substantial extension of our paper [Chaos 26, 053109 (2016)], in which the persistence of order 1 was taken into account.

Stochastic models with multiplicative noise for economic inequality and mobility

Abstract In this article, we discuss a dynamical stochastic model that represents the time evolution of income distribution of a population, where the dynamics develops from an interplay of multiple economic exchanges in the presence of multiplicative noise. The model remit stretches beyond the conventional framework of a Langevin-type kinetic equation in that our model dynamics is self-consistently constrained by dynamical conservation laws emerging from population and wealth conservation. This model is numerically solved and analysed to evaluate the inequality of income in correlation to other relevant dynamical parameters like the mobility M and the total income μ. Inequality is quantified by the Gini index G. In particular, correlations between any two of the mobility index M and/or the total income μ with the Gini index G are investigated and compared with the analogous quantities resulting from an additive noise model.

Gazing at Social Interactions Between Foraging and Decision Theory.

Finding the underlying principles of social attention in humans seems to be essential for the design of the interaction between natural and artificial agents. Here, we focus on the computational modeling of gaze dynamics as exhibited by humans when perceiving socially relevant multimodal information. The audio-visual landscape of social interactions is distilled into a number of multimodal patches that convey different social value, and we work under the general frame of foraging as a tradeoff between local patch exploitation and landscape exploration. We show that the spatio-temporal dynamics of gaze shifts can be parsimoniously described by Langevin-type stochastic differential equations triggering a decision equation over time. In particular, value-based patch choice and handling is reduced to a simple multi-alternative perceptual decision making that relies on a race-to-threshold between independent continuous-time perceptual evidence integrators, each integrator being associated with a patch.

On the numerical schemes for Langevin-type equations

In this paper, a numerical approach is proposed based on the variation-of-constants formula for the numerical discretization Langevin-type equations. Linear and non-linear cases are treated separately. The proofs of convergence have been provided for the linear case, and the numerical implementation has been executed for the non-linear case. The order one convergence for the numerical scheme has been shown both theoretically and numerically. The stability of the numerical scheme has been shown numerically and depicted graphically.

Testing the energy diffusion approximation for the escape of a Brownian particle from a potential pocket

For the first time, the energy diffusion approximation is confronted at the percent level with the exact numerical modeling of thermal decay of a metastable state. This model is useful in many branches of natural sciences: e.g. in biology, nuclear physics, chemistry, etc. The exact (within the statistical errors about 2%) quasistationary decay rates result from the Langevin equations for the coordinate and conjugated momentum. For the energy (or action) diffusion approach, a Langevin-type equation for the action is constructed, validated, and solved numerically. The comparison of these two approaches is performed for four potentials (two of which are anharmonic) in a wide range of two dimensionless scaling parameters: i) the governing parameter G reflecting how high is the barrier with respect to the temperature and ii) the damping parameter φ expressing the friction strength. It turns out that the energy diffusion approach produces the rate which comes into 50%-agreement with the exact one only at φ < 0.02. Thus, we quantify, for the first time by our knowledge, the condition φ ≪ 1 known in the literature.

Accurate Sampling with Noisy Forces from Approximate Computing

In scientific computing, the acceleration of atomistic computer simulations by means of custom hardware is finding ever-growing application. A major limitation, however, is that the high efficiency in terms of performance and low power consumption entails the massive usage of low precision computing units. Here, based on the approximate computing paradigm, we present an algorithmic method to compensate for numerical inaccuracies due to low accuracy arithmetic operations rigorously, yet still obtaining exact expectation values using a properly modified Langevin-type equation.