Langevin-type Equation Research Articles

Introduction to dynamical large deviations of Markov processes

These notes give a summary of techniques used in large deviation theory to study the fluctuations of time-additive quantities, called dynamical observables, defined in the context of Langevin-type equations, which model equilibrium and nonequilibrium processes driven by external forces and noise sources. These fluctuations are described by large deviation functions, obtained by solving a dominant eigenvalue problem similar to the problem of finding the ground state energy of quantum systems. This analogy is used to explain the differences that exist between the fluctuations of equilibrium and nonequilibrium processes. An example involving the Ornstein–Uhlenbeck process is worked out in detail to illustrate these methods. Exercises, at the end of the notes, also complement the theory.

Collective phase reduction of globally coupled noisy dynamical elements.

We formulate a theory for the collective phase reduction of globally coupled noisy dynamical elements exhibiting macroscopic rhythms. We first transform the Langevin-type equation that represents a group of globally coupled noisy dynamical elements into the corresponding nonlinear Fokker-Planck equation and then develop the phase reduction method for limit-cycle solutions to the nonlinear Fokker-Planck equation. The theory enables us to describe the collective dynamics of a group of globally coupled noisy dynamical elements by a single degree of freedom called the collective phase. As long as the group collectively exhibits macroscopic rhythms, the theory is applicable even when the coupling and noise are strong; it is also independent of the assumption that each element of the group is a self-sustained oscillator. We also provide a simple and accurate numerical algorithm for the collective phase description method and numerically illustrate the theory using a group of globally coupled noisy FitzHugh-Nagumo elements.

Variational estimation of the drift for stochastic differential equations from the empirical density

We present a method for the nonparametric estimation of the drift function of certain types of stochastic differential equations from the empirical density. It is based on a variational formulation of the Fokker–Planck equation. The minimization of an empirical estimate of the variational functional using kernel based regularization can be performed in closed form. We demonstrate the performance of the method on second order, Langevin-type equations and show how the method can be generalized to other noise models.

Fluctuation scaling in the visual cortex at threshold.

Fluctuation scaling relates trial-to-trial variability to the average response by a power function in many physical processes. Here we address whether fluctuation scaling holds in sensory psychophysics and its functional role in visual processing. We report experimental evidence of fluctuation scaling in human color vision and form perception at threshold. Subjects detected thresholds in a psychophysical masking experiment that is considered a standard reference for studying suppression between neurons in the visual cortex. For all subjects, the analysis of threshold variability that results from the masking task indicates that fluctuation scaling is a global property that modulates detection thresholds with a scaling exponent that departs from 2, β=2.48±0.07. We also examine a generalized version of fluctuation scaling between the sample kurtosis K and the sample skewness S of threshold distributions. We find that K and S are related and follow a unique quadratic form K=(1.19±0.04)S^{2}+(2.68±0.06) that departs from the expected 4/3 power function regime. A random multiplicative process with weak additive noise is proposed based on a Langevin-type equation. The multiplicative process provides a unifying description of fluctuation scaling and the quadratic S-K relation and is related to on-off intermittency in sensory perception. Our findings provide an insight into how the human visual system interacts with the external environment. The theoretical methods open perspectives for investigating fluctuation scaling and intermittency effects in a wide variety of natural, economic, and cognitive phenomena.

Reconstruction of the modified discrete Langevin equation from persistent time series.

The discrete Langevin-type equation, which can describe persistent processes, was introduced. The procedure of reconstruction of the equation from time series was proposed and tested on synthetic data, with short and long-tail distributions, generated by different Langevin equations. Corrections due to the finite sampling rates were derived. For an exemplary meteorological time series, an appropriate Langevin equation, which constitutes a stochastic macroscopic model of the phenomenon, was reconstructed.

Fluctuation-dissipation and energy properties of a finite bath.

This paper expands a recent proposal by the authors to rederive the Langevin equation for a test particle in a finite-size thermal bath using a perturbation approach that yields a cascade of Langevin-type equations. Such an approach produces a different viewpoint for the fluctuation-dissipation duality by expressing them on similar scales. General properties of energy sharing between the test particle and the bath are outlined, investigating the resonant and nonresonant conditions.

Efficient estimators for likelihood ratio sensitivity indices of complex stochastic dynamics.

We demonstrate that centered likelihood ratio estimators for the sensitivity indices of complex stochastic dynamics are highly efficient with low, constant in time variance and consequently they are suitable for sensitivity analysis in long-time and steady-state regimes. These estimators rely on a new covariance formulation of the likelihood ratio that includes as a submatrix a Fisher information matrix for stochastic dynamics and can also be used for fast screening of insensitive parameters and parameter combinations. The proposed methods are applicable to broad classes of stochastic dynamics such as chemical reaction networks, Langevin-type equations and stochastic models in finance, including systems with a high dimensional parameter space and/or disparate decorrelation times between different observables. Furthermore, they are simple to implement as a standard observable in any existing simulation algorithm without additional modifications.

Steady and transient states in low-energy swarms: Stability and first-passage times.

We investigate a class of agent-based models of self-propelled particles (SPP) that interact according to a Morse potential in the presence of friction, a class which was able to reproduce many of the intriguing patterns of collective motion observed in nature. Specifically, we compare two closely related SPP models in the literature that differ by their prescription of particle drag and self-propulsion. Writing both models in terms of nondimensional parameters allows us to show that the dynamics in the highly viscous regime is independent of the precise forms of drag and propulsion. In contrast to what is indicated in the literature both models yield the same low-energy self-organized states: the coherent flock and the rigid rotation states which are highly ordered in both the coordinate and the velocity spaces and a velocity-disordered droplet state where particles are confined to rings which pass through the lattice points of the underlying Lagrange configuration. In contrast to the first two states which are stable, the third state is found to be a long-lived transient. In the regime studied, relaxing to one of the ordered steady states is inevitable, but how and when the transition occurs and what is the probability of ending in one state rather than the other are functions of the model parameters. Two types of transitions are characterized and first passage times are computed. Eventually, the evolution of the order parameter is explored in the framework of a Langevin-type equation, and the possible metastability of the random droplet state is discussed.

Nonlocal problems for Langevin-type differential equations with two fractional-order derivatives

In this paper, we intend to study nonlocal problems for Langevin-type differential equations with two fractional derivatives of orders $\alpha,\beta\in(1,2)$ . By using Laplace transform methods, formula of solutions involving Mittag-Leffler functions $A_{\alpha,\beta}(w)$ , $\alpha,\beta\in(1,2)$ , $w\in R$ , and nonlocal terms of such equations are presented by studying the corresponding linear Langevin-type equations with two fractional derivatives. Meanwhile, existence results of solutions are established by utilizing boundedness, continuity, monotonicity, nonnegative of Mittag-Leffler function $A_{\alpha,\beta}(w)$ , $\alpha,\beta\in(1,2)$ , $w\in R$ , and fixed point methods. Finally, two examples are presented to illustrate our theoretical results.

The crude oil price bubbling and universal scaling dynamics of price volatility

The main goal of this paper is to reveal the effect of crude oil price bubbling on the price volatility dynamics. For this purpose, the time series of volatility at different horizons are mapped into a model of kinetic roughening of interface growing in a stochastic environment. In this way, we found that the volatility dynamics obeys the Family-Viscek dynamic scaling ansatz. Although during the period from January 2, 1986 to July 25, 2014 the volatility remains a slightly anti-persistent, the dynamic exponent is found to be quite different during different regimes of price evolution. Accordingly, we define the intrinsic time of price volatility and metric of volatility horizons. This allows us to construct the Langevin-type equation governing the volatility dynamics during bubble and non-bubble periods. The data analysis indicates that the bubbling does not affect the intrinsic time of volatility, but strongly affect the metric of volatility horizons. In this regard, numerical data suggest the existence of two universal metrics characterizing the volatility dynamics during the bubble and non-bubble regimes of crude oil price evolution, respectively. The results of this work help us to get a further insight into the dynamics of crude oil price volatility.

Dynamic scaling behaviors of linear fractal Langevin-type equation driven by nonconserved and conserved noise

In order to study the effects of the microscopic details of fractal substrates on the scaling behavior of the growth model, a generalized linear fractal Langevin-type equation, ∂h/∂t=(−1)m+1ν∇mzrwh (zrw is the dynamic exponent of random walk on substrates), driven by nonconserved and conserved noise is proposed and investigated theoretically employing scaling analysis. Corresponding dynamic scaling exponents are obtained.

Fluctuation current in a superconducting ring caused by a thin solenoid

The fluctuating current due to the Cooper pair is calculated for a superconducting ring threaded by a thin magnetic solenoid that is arranged by time-varying flux. The formulation is based on the Fokker-Planck equation derived by the Langevin-type Landau-Ginzburg equation. The statistical average of the fluctuation current is calculated. As a special case, for the equilibrium state, the current shows a critical behavior near the transition point. In the presence of the time-varying flux, the current is calculated to be a highly nonlinear function of the electric field induced by the time variation of the flux, which yields a sort of the Ohmic law in the limit of the weak field.

Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation.

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.

Gaussian asymptotics for a non-linear Langevin type equation driven by an $\alpha$-stable Lévy noise

Consider the dynamics of a particle whose speed satisfies a one-dimensional stochastic differential equation driven by a small symmetric $\alpha$-stable Lévy process in a potential of the form a power function of exponent $\beta+1$. Two cases are studied: the noise could be path continuous, namely a standard Brownian motion, if $\alpha=2$, or pure jump Lévy process, if $\alpha\in(0,2)$. The main goal is to study a scaling limit of the position process with this speed, and one proves that the limit is Brownian in either case. This result is a generalization in some sense of the quadratic potential case studied recently by Hintze and Pavlyukevich.<br />

Determination of reaction flux from concentration fluctuations near a Hopf bifurcation.

Small open chemical systems, typically associated with far-from-equilibrium, nonlinear stochastic dynamics, offer the appropriate framework to elucidate biological phenomena at the cellular scale. Stochastic differential equations of Langevin-type are employed to establish the relation between the departure from equilibrium and the time cross-correlation functions of concentration fluctuations for chemical species susceptible to oscillate. Except in the immediate vicinity of the Hopf bifurcation, the results are in agreement with simulations of the chemical master equation but always differ from the prediction obtained for linear deterministic dynamics. In general, the magnitude of the asymmetry of time correlation functions definitely depends on the reaction flux circulating in an open system but also on the details of the nonlinearities of deterministic dynamics.

Temporal cross-correlation asymmetry and departure from equilibrium in a bistable chemical system.

This paper aims at determining sustained reaction fluxes in a nonlinear chemical system driven in a nonequilibrium steady state. The method relies on the computation of cross-correlation functions for the internal fluctuations of chemical species concentrations. By employing Langevin-type equations, we derive approximate analytical formulas for the cross-correlation functions associated with nonlinear dynamics. Kinetic Monte Carlo simulations of the chemical master equation are performed in order to check the validity of the Langevin equations for a bistable chemical system. The two approaches are found in excellent agreement, except for critical parameter values where the bifurcation between monostability and bistability occurs. From the theoretical point of view, the results imply that the behavior of cross-correlation functions cannot be exploited to measure sustained reaction fluxes in a specific nonlinear system without the prior knowledge of the associated chemical mechanism and the rate constants.

Langevin description of gauged scalar fields in a thermal bath

We study the dynamics of the oscillating gauged scalar field in a thermal bath. A Langevin type equation of motion of the scalar field, which contains both dissipation and fluctuation terms, is derived by using the real-time finite temperature effective action approach. The existence of the quantum fluctuation-dissipation relation between the non-local dissipation term and the Gaussian stochastic noise terms is verified. We find the noise variables are anti-correlated at equal-time. The dissipation rate for the each mode is also studied, which turns out to depend on the wavenumber.

Construction of a Langevin model from time series with a periodical correlation function: Application to wind speed data

A Langevin-type equation for stochastic processes with a periodical correlation function is introduced. A procedure of reconstruction of the equation from time series is proposed and verified on simulated data. The method is applied to geophysical time series–hourly time series of wind speed measured in northern Italy–constructing the macroscopic model of the phenomenon.

Three-dimensional domino model in geodynamo

The Lagrangian formalism is applied to consider temporal evolution of the ensemble of interacting magnetohydrodynamical cyclones governed by Langevin-type equations in a rotating medium. This problem is relevant for fast-rotating convective objects such as the cores of planets and a number of stars, where the Rossby numbers are far below unity and the geostrophic balance of the forces takes place. The paper presents the results of modeling for both the two-dimensional (2D) case when the cyclones can rotate relative to the rotation axis of the whole system in the vertical plane, and for the case of spatial rotation by two angles. It is shown that variations in the heat flux on the outer boundary of the spherical shell modulate the frequency of the reversals of the mean dipole magnetic field, which agrees with the three-dimensional (3D) modeling of the planetary dynamo. Applications of the model for giant planets are discussed, and an explanation for some episodes in the history of the geomagnetic field in the past is suggested.

Fluctuation limits of the single point catalytic super-stable processes

We prove a fluctuating limit theorem of a sequence of super-stable processes over ℝ with a single point catalyst. The weak convergence of the processes on the space of Schwartz distributions is established. The limiting process is an Ornstein-Uhlenbeck type process solving a Langevin type equation driven by a one-dimensional stable process.