The linear noise approximation for reaction-diffusion systems on networks

Stochastic reaction-diffusion models can be analytically studied on complex networks using the linear noise approximation. This is illustrated through the use of a specific stochastic model, which displays travelling waves in its deterministic limit. The role of stochastic fluctuations is investigated and shown to drive the emergence of stochastic waves beyond the region of the instability predicted from the deterministic theory. Simulations are performed to test the theoretical results and are analyzed via a generalized Fourier transform algorithm. This transform is defined using the eigenvectors of the discrete Laplacian defined on the network. A peak in the numerical power spectrum of the fluctuations is observed in quantitative agreement with the theoretical predictions.     

A high accuracy stochastic estimation of a nonlinear deterministic model

In this Letter, an approach to estimating a nonlinear deterministic model is presented. We introduce a stochastic model with extremely small variances so that the deterministic and stochastic models are essentially indistinguishable from each other. This point is explained in the Letter. The estimation is then carried out using stochastic optimization based on Markov chain Monte Carlo (MCMC) methods.     

An approximate method for solving radiation transport problems in temporally and spatially stochastic media

A new approach has been developed to deal with stochastic transport problems in three-dimensional media. It is assumed that the medium consists of randomly distributed lumps of material embedded in a background matrix and in each lump the properties may vary randomly with time. The coefficients for scattering and absorption are represented mathematically by members of a random characteristic set function, which depend on space and time. Different physical situations can be described by different forms and combinations of these set functions. In order to effect a solution of the resulting stochastic transport equation, which may be for photons or neutrons, we make the, a priori, assumption that the functional form for the solution of the transport equation, i.e. the stochastic flux, can be represented by the same mathematical form as the scattering and absorption coefficients (or cross sections), i.e. we introduce a stochastic ansatz. This procedure leads to a set of deterministic equations from which the mean and variance of the flux in space and time can be obtained. For the case of a two-phase medium, either two or four coupled integro-differential equations are obtained for the deterministic functions that arise (depending on the problem) and expressions are given for the mean and variance of the angular flux. There is a close relationship between these equations and those from the Levermore-Pomraning (LP) theory, but the new equations offer an opportunity to deal with more general forms of stochastic processes and combine simultaneously time and space fluctuations. The stochastic characteristics of the medium are defined by the correlation functions which appear in the equations and, by making plausible assumptions about the functional form of these autocorrelation functions, different physical situations can be simulated, according to the structure of the medium. The main contribution of the present work is to include space and time fluctuations simultaneously as a pseudo-dichotomic Markov process.     

The validity of nonlinear Langevin equations

In the presence of internal noise the variables describing a system are intrinsically stochastic. If they constitute a Markov process the expansion enables one to extract a deterministic macroscopic equation and to compute the fluctuations in successive approximations. In the lowest or linear noise approximation the fluctuations can be represented by a Langevin equation, provided it is handled appropriately. Higher orders cannot be described by any white noise Langevin equation. The question whether the equation has to be interpreted according to Itô or Stratonovich concerns these higher orders, for which the equation is not valid anyway.     

Kinetic Theory of Jet Dynamics in the Stochastic Barotropic and 2D Navier-Stokes Equations

We discuss the dynamics of zonal (or unidirectional) jets for barotropic flows forced by Gaussian stochastic fields with white in time correlation functions. This problem contains the stochastic dynamics of 2D Navier-Stokes equation as a special case. We consider the limit of weak forces and dissipation, when there is a time scale separation between the inertial time scale (fast) and the spin-up or spin-down time (large) needed to reach an average energy balance. In this limit, we show that an adiabatic reduction (or stochastic averaging) of the dynamics can be performed. We then obtain a kinetic equation that describes the slow evolution of zonal jets over a very long time scale, where the effect of non-zonal turbulence has been integrated out. The main theoretical difficulty, achieved in this work, is to analyze the stationary distribution of a Lyapunov equation that describes quasi-Gaussian fluctuations around each zonal jet, in the inertial limit. This is necessary to prove that there is no ultraviolet divergence at leading order, in such a way that the asymptotic expansion is self-consistent. We obtain at leading order a Fokker–Planck equation, associated to a stochastic kinetic equation, that describes the slow jet dynamics. Its deterministic part is related to well known phenomenological theories (for instance Stochastic Structural Stability Theory) and to quasi-linear approximations, whereas the stochastic part allows to go beyond the computation of the most probable zonal jet. We argue that the effect of the stochastic part may be of huge importance when, as for instance in the proximity of phase transitions, more than one attractor of the dynamics is present.     

Bounding the quality of stochastic oscillations in population models

We analyze general two-species stochastic models, of the kind generally used for the study of population dynamics. Although usually defined a priori, the deterministic version of these models can be obtained as the infinite volume limit of many stochastic models (which are necessarily defined by more parameters than the deterministic one). It is known that damped oscillations in a deterministic model usually correspond to oscillatory-like fluctuations in their deterministic counterparts. The quality of these “oscillations" depends on details of each stochastic model. We show, however, that the parameters of the deterministic system are generally enough to obtain very good bounds for the quality of “oscillations" in any of its stochastic counterparts. These bounds are shown to depend on only one dimensionless parameter.     

用随机方法解河渠非恒定流

随机水力学是水力学研究的一个新发展。本文应用确定性模型与随机模型相结合的方法,对明渠非恒定渐变流的计算进行了研究。采用Saint Venant方程组为系统模型,但对系统的输入如明渠的糙率、过水断面、水力半径等,则采用适当的随机模型,产生的输出(流量、水位随时间和位置的变化)是随机性的、本文以柘溪水电站下游河道作为实例,用计算机进行了计算,并与实测结果作了印证。     

Responses of a Hodgkin-Huxley neuron to various types of spike-train inputs

Numerical investigations have been made of responses of a Hodgkin-Huxley (HH) neuron to spike-train inputs whose interspike interval (ISI) is modulated by deterministic, semi-deterministic (chaotic), and stochastic signals. As deterministic one, we adopt inputs with the time-independent ISI and with time-dependent ISI modulated by sinusoidal signal. The R?ssler and Lorentz models are adopted for chaotic modulations of ISI. Stochastic ISI inputs with the gamma distribution are employed. It is shown that distribution of output ISI data depends not only on the mean of ISIs of spike-train inputs but also on their fluctuations. The distinction of responses to the three kinds of inputs can be made by return maps of input and output ISIs, but not by their histograms. The relation between the variations of input and output ISIs is shown to be different from that of the integrate and fire (IF) model because of the refractory period in the HH neuron.     

Nonstationarity induced by long-time noise correlations in the langevin equation

We solve the generalized Langevin equation driven by a stochastic force with a power-law autocorrelation function. A stationary Markov process has been applied as a model of the noise. However, the resulting velocity variance does not stabilize but diminishes with time. It is shown that algebraic distributions can induce such effects. Results are compared to those obtained with a deterministic random force. Consequences for the diffusion process are also discussed.     

Dynamics of oscillators with periodic dichotomous noise

The dynamics of bistable oscillators driven by periodic dichotomous noise is described. The stochastic differential equation governing the flow implies smooth trajectories between noise switching events. The dynamics of the two-branched map induced by this flow is a Markov process. Harmonic and quartic models of the bistable potential are studied in the overdamped limit. In the linear (harmonic) case the dynamics can be reduced to a stochastic one-dimensional map with two branches. The moments decay exponentially in this case, although the invariant measure may be multifractal. For strong damping, relaxation induces a cascade leading to a Cantor set and anomalous decay of the density in this case is modeled by a Markov chain. For the physically more realistic case of a quartic potential many additional features arise since the contraction factor is distance dependent. By tuning the barrier-height parameter in the quartic potential, noise-induced transition rates with the characteristics of intermittency are found.     

The construction of an Ito model for geoelectrical signals

The Ito stochastic differential equation governs the one-dimensional diffusive Markov process. Geoelectrical signals measured in seismic areas can be considered as the result of competitive and collective interactions among system elements. The Ito equation may constitute a good macroscopic model of such a phenomenon in which microscopic interactions are adequately averaged. The present study shows how to construct an Ito model for a geoelectrical time series measured in a seismic area of southern Italy. Our results reveal that the Ito model describes the whole time series quite well, but it performs better when one considers fragments of the data set with lower variability range (absent or rare large fluctuations). Our findings show that generally detrended geoelectrical time series can be considered as approximations of Markov diffusion processes.     

Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems

We consider a Bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field, endowed with a hierarchical Gaussian process prior. Computational challenges in this construction arise from the need for repeated evaluations of the forward model (e.g., in the context of Markov chain Monte Carlo) and are compounded by high dimensionality of the posterior. We address these challenges by introducing truncated Karhunen–Loève expansions, based on the prior distribution, to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior. We seek a solution of this problem using Galerkin projection on a polynomial chaos basis, and use the solution to construct a reduced-dimensionality surrogate posterior density that is inexpensive to evaluate. We demonstrate the formulation on a transient diffusion equation with prescribed source terms, inferring the spatially-varying diffusivity of the medium from limited and noisy data.     

Stochastic differential equation derivation: Comparison of the Markov method versus the additive method

There are several methods of transforming an ordinary differential equation into a stochastic differential equation (SDE). The two most common are adding noise to a system parameter or variable and transforming to a SDE or deriving the SDE by assuming an underlying Markov process. Using simple one- and two-dimensional systems we investigate the differences in dynamics and bifurcations between SDE derived by each method from simple deterministic population models.     

Long time tails in stationary random media II: Applications

In a previous paper we developed a mode-coupling theory to describe the long time properties of diffusion in stationary, statistically homogeneous, random media. Here the general theory is applied to deterministic and stochastic Lorentz models and several hopping models. The mode-coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse-grained diffusion coefficient and a coarse-grained free volume. For one-dimensional models these amplitudes can be evaluated, and the mode-coupling theory is shown to agree with exact solutions obtained for these models. For higher-dimensional Lorentz models the formal theory yields expressions which are difficult to evaluate. For these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers.Work supported by grant No. CHE 77-16308 from the National Science Foundation and by a Nato Travel Grant.     

随机Ginzburg-Landau方程的数值模拟

对随机Ginzburg-Landau方程进行数值研究,构造一个非线性差分格式和一个线性化差分格式.通过对确定性和随机Ginzburg-Landau方程的计算,表明所构造的格式具有较高的精度和较快的计算效率.对随机Ginzburg-Landau方程就噪声振幅的不同取值进行了数值模拟,并对由此引发的各种行为进行了描述.     

Proposed stochastic parameterisations of subgrid turbulence in large eddy simulations of turbulent channel flow

Stochastic and deterministic subgrid parameterisations are developed for the large eddy simulation (LES) of a turbulent channel flow with friction-velocity-based Reynolds number of Re_τ = 950 and centreline-based Reynolds number of Re₀ = 20,580. The subgrid model coefficients (eddy viscosities) are determined from the statistics of truncated reference direct numerical simulations (DNSs). The stochastic subgrid model consists of a mean-field shift, a drain eddy viscosity acting on the resolved field and a stochastic backscatter force of variance proportional to the backscatter eddy viscosity. The deterministic variant consists of a net eddy viscosity acting on the resolved field, which represents the net effect of the drain and backscatter. LES adopting the stochastic and deterministic models is shown to reproduce the time-averaged kinetic energy spectra of the DNS within the resolved scales.     

A causal multifractal stochastic equation and its statistical properties

Multiplicative cascades have been introduced in turbulence to generate random or deterministic fields having intermittent values and long-range power-law correlations. Generally this is done using discrete construction rules leading to discrete cascades. Here a causal log-normal stochastic process is introduced; its multifractal properties are demonstrated together with other properties such as the composition rule for scale dependence and stochastic differential equations for time and scale evolutions. This multifractal stochastic process is continuous in scale ratio and in time. It has a simple generating equation and can be used to generate sequentially time series of any length.Received: 15 April 2003, Published online: 23 July 2003PACS: 02.50.Ey Stochastic processes - 05.45.Df Fractals - 47.27.Eq Turbulence simulation and modeling     

Stochastic Processes and Mean Field Systems Defined by Nonlinear Markov Chains: An Illustration for a Model of Evolutionary Population Dynamics

In physics, there is a growing interest in studying stochastic processes described by evolution equations such as nonlinear master equations and nonlinear Fokker–Planck equations that define the so-called nonlinear Markov processes and are nonlinear with respect to probability densities. In this context, however, relatively little is known about nonlinear Markov processes defined by nonlinear Markov chains. In the present work, we demonstrate explicitly how the nonlinear Markov chain approach can be carried out by addressing a model for evolutionary population dynamics. In line with the nonlinear Markov chain approach, we derive a measure that tells us how attractive it is for a biological entity to evolve towards a particular biological type. Likewise, a measure for the noise level of the evolutionary process is obtained. Both measures are found to be implicitly time dependent. Finally, a simulation scheme for the many-body system corresponding to the Markov chain model is discussed.     

Spatially adaptive stochastic numerical methods for intrinsic fluctuations in reaction–diffusion systems

Stochastic partial differential equations are introduced for the continuum concentration fields of reaction–diffusion systems. The stochastic partial differential equations account for fluctuations arising from the finite number of molecules which diffusively migrate and react. Spatially adaptive stochastic numerical methods are developed for approximation of the stochastic partial differential equations. The methods allow for adaptive meshes with multiple levels of resolution, Neumann and Dirichlet boundary conditions, and domains having geometries with curved boundaries. A key issue addressed by the methods is the formulation of consistent discretizations for the stochastic driving fields at coarse-refined interfaces of the mesh and at boundaries. Methods are also introduced for the efficient generation of the required stochastic driving fields on such meshes. As a demonstration of the methods, investigations are made of the role of fluctuations in a biological model for microorganism direction sensing based on concentration gradients. Also investigated, a mechanism for spatial pattern formation induced by fluctuations. The discretization approaches introduced for SPDEs have the potential to be widely applicable in the development of numerical methods for the study of spatially extended stochastic systems.