Random Forcing Term Research Articles

Rare-Event Simulation for the Stochastic Korteweg--de Vries Equation

An asymptotic analysis of the tail probabilities for the dynamics of a soliton wave $U(x,t)$ under a stochastic time-dependent force is developed. The dynamics of the soliton wave $U(x,t)$ is described by the Korteweg--de Vries (KdV) equation with homogeneous Dirichlet boundary conditions under a stochastic time-dependent force, which is modeled as a time-dependent Gaussian noise with amplitude $\epsilon$. The tail probability considered is $w(b):=P(\sup_{t\in[0,T]}U(x,t)>b)$, as $b\to\infty$, for some constant $T>0$ and a fixed $x$, which can be interpreted as tail probability of the amplitude of a water wave on the shallow surface of a fluid or long internal wave in a density-stratified ocean. Our goal is to characterize the asymptotic behaviors of $w(b)$ and evaluate the tail probability of the event that the soliton wave exceeds a certain threshold value under a random force term. Such rare-event calculation of $w(b)$ is especially useful for fast estimation of the risk of the potential damage that co...

Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term

We study the mixed formulation of the stochastic Hodge-Laplace problem dened on a $n$-dimensional domain $D (n ≥ 1)$, with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three dimensional case. We derive and analyze the moment equations, that is the deterministic equations solved by the $m-th$ moment $(m ≥ 1$) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order of convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element spaces.

Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation

We consider the non-homogeneous generalised Burgers equation $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} -\nu \frac{\partial^2 u}{\partial x^2} = \eta, t \geq 0, x \in S^1.$$ Here f is strongly convex and satisfies a growth condition, ν is small and positive, while η is a random forcing term, smooth in space and white in time. For any solution u of this equation we consider the quasi-stationary regime, corresponding to $${t \geq T_1}$$ , where T 1 depends only on f and on the distribution of η. We obtain sharp upper and lower bounds for Sobolev norms of u averaged in time and in ensemble. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence. All our bounds do not depend on the initial condition or on t for $${t \geq T_1}$$ , and hold uniformly in ν. Estimates similar to some of our results have been obtained by Aurell, Frisch, Lutsko and Vergassola on a physical level of rigour; we use an argument from their article.

On a stochastic Leray-[formula omitted] model of Euler equations

We deal with the 3D inviscid Leray-α model. The well posedness for this problem is not known; by adding a random perturbation we prove that there exists a unique (in law) global solution. The random forcing term formally preserves conservation of energy. The result holds for the initial velocity of finite energy and the solution has finite energy a.s. These results continue to hold in the 2D case.

Dependence of polynomial chaos on random types of forces of KdV equations

In this study, one-dimensional stochastic Korteweg–de Vries equation with uncertainty in its forcing term is considered. Extending the Wiener chaos expansion, a numerical algorithm based on orthonormal polynomials from the Askey scheme is derived. Then dependence of polynomial chaos on the distribution type of the random forcing term is inspected. It is numerically shown that when Hermite (Laguerre or Jacobi) polynomial chaos is chosen as a basis in the Gaussian (Gamma or Beta, respectively) random space for uncertainty, the solution to the KdV equation converges exponentially. If a proper polynomial chaos is not used, however, the solution converges with slower rate.

Fluctuating Navier-Stokes equations for inelastic hard spheres or disks

Starting from the fluctuating Boltzmann equation for smooth inelastic hard spheres or disks, closed equations for the fluctuating hydrodynamic fields to Navier-Stokes order are derived. This requires deriving constitutive relations for both the fluctuating fluxes and the correlations of the random forces. The former are identified as having the same form as the macroscopic average fluxes and involving the same transport coefficients. On the other hand, the random force terms exhibit two peculiarities as compared with their elastic limit for molecular systems. First, they are not white but have some finite relaxation time. Second, their amplitude is not determined by the macroscopic transport coefficients but involves new coefficients.

Long Time Scale Dynamics of Molecules With Internal Rigid Fragments

Due to the large number of degrees of freedom which are present in protein systems it is still a major challenge to monitor the dynamics of molecules out to the long time scales on which functionally important conformational changes occur. We have developed a rigid body Newtonian dynamics method, in which local high frequency motion of the molecule are naturally frozen out by decomposing the molecule into linked rigid bodies, thus decreasing the number of degrees of freedom monitored in the system. The factor of ca. 10 increase in the time step which comes from the elimination of high frequency motion in rigid body Newtonian dynamics can be further extended by elimination of explicit water molecules using a Langevin dynamics prescription in which friction and random force terms are added to the potential function to mimic the effect of the solvent. As a first step in this direction we have implemented a Langevin prescription for normal modes vibrational analysis of molecules with internal rigid fragments which is developed by Durand et al. [Biopolymers, 34, 759 (1994)]. Two simple illustrative examples representing signal propagation in the membrane-bound gramicidin-A dimer with two different initial conditions are presented as numerical applications for i) rigid body Newtonian dynamics (anharmonic PES) and ii) Langevin dynamics with internal rigid fragments in the harmonic field, respectively.

Acoustic Observation of the Time Dependence of the Roughness of Sandy Seafloors

A statistical model for the time evolution of seafloor roughness due to biological activity is applied to photographic and acoustic data. In this model, the function describing small scale seafloor topography obeys a time-evolution equation with a random forcing term that creates roughness and a diffusion term that degrades roughness. When compared to acoustic data from the 1999 and 2004 Sediment Acoustics Experiments (SAX99 and SAX04), the model yields diffusivities in the range from 3.5 times 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-11</sup> to 2.5 times 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-10</sup> m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> (from 10 to 80 cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> yr <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ), with the larger values occurring at sites where bottom-feeding fish were active. While the experimental results lend support to the model, a more focused experimental and simulation effort is required to test several assumptions intrinsic to the model.

On Simulating Randomly Driven Dynamic Systems

Much of modern estimation and control theory is based on dynamic models with random forcing terms. As long as these models are linear, the behavior of the system is well understood; in fact, the state space system can be solved exactly. In this paper we address the problem of simulating nonlinear systems such that the average properties of the simulation match those of the true system being modeled. We present previous results on Runge-Kutta methods for linear, stochastic state space systems and show how even these need to be treated carefully. We then discuss nonlinear stochastic models and how the two main types, Ito and Stratonovich, relate to the physical systems being considered. We present a Runge-Kutta type algorithm for simulating nonlinear stochastic systems and demonstrate the validity of the approach on a simple laboratory experiment.

Noise influence on pole solutions of the Sivashinsky equation for planar and outward propagating flames

The dynamics of planar and outward propagating cylindrical flames has been studied in terms of exact solutions of the Sivashinsky equation with a random force term. The force term models the computational roundoff errors or a variety of perturbations of physical origins. In contrast to noiseless conditions, the number of poles in the system does not conserve and new poles appear due to the external forcing. It was found that modification of the pole solutions taking into account the appearance of new poles captures the features typical for the hydrodynamically unstable flames, which cannot be detected by the pole solutions with a fixed number of poles. Investigations based on the pole solutions make it possible to exclude the uncontrolled numerical noise that is always present in direct computations of the Sivashinsky equation, and to examine the interplay between noises and hydrodynamic instability. The study clearly demonstrates that the presence of noises is a necessary condition for flame acceleration.

Invariant Measures for a Stochastic Kuramoto–Sivashinsky Equation

For the one-dimensional Kuramoto–Sivashinsky equation with random forcing term, existence and uniqueness of solutions is proved. Then, the Markovian semigroup is well defined; its properties are analyzed in order to provide sufficient conditions for existence and uniqueness of invariant measures for this stochastic equation. Finally, regularity results are presented.

A phonon heat bath approach for the atomistic and multiscale simulation of solids

AbstractWe present a novel approach to numerical modelling of the crystalline solid as a heat bath. The approach allows bringing together a small and a large crystalline domain, and model accurately the resultant interface, using harmonic assumptions for the larger domain, which is excluded from the explicit model and viewed only as a hypothetic heat bath. Such an interface is non‐reflective for the elastic waves, as well as providing thermostatting conditions for the small domain. The small domain can be modelled with a standard molecular dynamics approach, and its interior may accommodate arbitrary non‐linearities. The formulation involves a normal decomposition for the random thermal motion term R(t) in the generalized Langevin equation for solid–solid interfaces. Heat bath temperature serves as a parameter for the distribution of the normal mode amplitudes found from the Gibbs canonical distribution for the phonon gas. Spectral properties of the normal modes (polarization vectors and normal frequencies) are derived from the interatomic potential. Approach results in a physically motivated random force term R(t) derived consistently to represent the correlated thermal motion of lattice atoms. We describe the method in detail, and demonstrate applications to one‐ and two‐dimensional lattice structures. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Covariance kernel representations of multidimensional second-order stochastic processes

The dynamics of stationary stochastic processes in space is not exactly analogous to that of stationary stochastic processes in the time domain. This is due to the unilateral nature of the time series that is only influenced by past values as opposed to the dependence in all directions of the spatial process. In this work, we unfold the connection that exits between the covariance kernel of a multi-dimensional second-order autoregressive random process and its underlying discrete random dynamical system. Starting from a discrete random dynamical system, we show that the random process satisfying that system is governed by the modified Helmholtz equation in the continuous limit. We establish the dependence of the correlation constant on the grid size of the discretization. We also show that the random forcing term in the continuous case turns out to be a white noise process. A number of covariance functions are worked out for simple and more complex geometrical domains with various boundary conditions in multi-dimensions. We use both the discrete and the continuous systems in our computations.

A finite-temperature dynamic coupled atomistic/discrete dislocation method

A method for simultaneously thermostatting an atomistic region and absorbing energetic pulses impinging on the atomistic/continuum interface from the atomistic region is developed to operate within the framework of the coupled atomistic/discrete dislocation method. The approach inserts an additional Langevin damping term and a random force term into the equations of motion for atoms in a ‘stadium’ boundary region near the atom/continuum interface, with the damping coefficient ramped linearly over the width of the region, as suggested by Holian and Ravelo. The remaining interior atom dynamics are computed using a standard MD algorithm with no artificial damping or thermostatting. The continuum region deformations are computed using static FEM updated stochastically over time scales comparable to the Debye frequency of the atoms using time-averaged displacements at the atom/continuum interface, thereby providing an evolution of the continuum region that tracks the atomistic deformation. The method is evaluated by studying the ability of the coupled system: (i) to equilibrate the inner atomistic region at a desired temperature under conditions of no external or internal loading, (ii) to produce the proper canonical thermal fluctuations and (iii) to absorb deformation pulses initiated in the interior region and incident upon the atomistic/continuum boundary. With an optimal maximum damping coefficient of approximately 1/2 of the Debye frequency, temperature stability is attained at values very close to the target temperature. The temperature variance agrees well with the canonical expectation for various temperatures. For the same damping parameters and at low temperature, high-energy deformation pulses propagate unimpeded up to the stadium boundary region and then are completely damped out upon approach to the atomistic/continuum interface with no measurable reflections. At higher temperatures, thermal fluctuations in the total energy make analyses difficult, but damping of high-energy deformation pulses is achieved within the limits of the thermal noise in the system while observation of the time-dependent displacements shows no observable reflections.

Statistics of intense turbulent vorticity events.

We investigate statistical properties of vorticity fluctuations in fully developed turbulence, which are known to exhibit a strong intermittent behavior. Taking as the starting point the Navier-Stokes equations with a random force term correlated at large scales, we obtain in the high Reynolds number regime a closed analytical expression for the probability distribution function of an arbitrary component of the vorticity field. The central idea underlying the analysis consists in the restriction of the velocity configurational phase-space to a particular sector where the rate of strain and the rotation tensors can be locally regarded as slow and fast degrees of freedom, respectively. This prescription is implemented along the Martin-Siggia-Rose functional framework, whereby instantons and perturbations around them are taken into account within a steepest-descent approach.

Direct numerical simulation of the Brownian motion of particles by using fluctuating hydrodynamic equations

In this paper, we present a direct numerical simulation scheme for the Brownian motion of particles. In this approach, the thermal fluctuations are included in the fluid equations via random stress terms. Solving the fluctuating hydrodynamic equations coupled with the particle equations of motion result in the Brownian motion of the particles. There is no need to add a random force term in the particle equations. The particles acquire random motion through the hydrodynamic force acting on its surface from the surrounding fluctuating fluid. The random stress in the fluid equations are easy to calculate unlike the random terms in the conventional Brownian dynamics type approaches. We present a three-dimensional implementation along with validation.

On a complex beam–beam interaction model with random forcing

In recent years, many studies have been devoted to complex differential equations (CDE), which appear in very important applications in physics and engineering. This paper aims to investigate one such CDE, containing a random forcing term: (∗) z ̈ +ω o 2z+ε 2λg( z ̇ )+ε 2f(z, z ̄ )p(ω ot)=αn(t) , where z( t)= x( t)+i y( t) a complex function, i= −1 , n( t) is a broad-band process with zero mean and ε 2 and λ small real parameters. In particular, we use Eq. (∗) to model the interaction between two colliding beams in particle accelerators, setting g( z ̇ )= z ̇ , f(z, z ̄ )=z|z| 2 and p(ω ot)= cos ω ot and extend the work we had started in an earlier publication (Mahmoud, Physica A 216 (1995) 445). We apply the stochastic averaging method to derive a Fokker–Planck–Kolmogorov equation for this equation and obtain analytically the exact stationary probability density function and the first and second moments in the amplitude of the solutions. Numerical simulations are carried out to compare with the theoretical ones and excellent agreement is found.

Turbulence Statistics Inside and Over Forest: Influence on Footprint Prediction

Observations of wind statistics within and above a Scots pine forest are comparedwith those predicted from an analytical second-order closure model. The roughnesssublayer (RSL) effects, and the influence of stability on similarity functions, arestudied using observations. The commonly accepted forms of similarity functionsdescribe the influence of diabatic effects above the RSL well. According to earlierstudies they are expected also to apply within the RSL. As an exception, the averagewind speed normalised with friction velocity was found to be invariant with stabilityclose to the canopy top under unstable conditions. Lagrangian stochastic trajectorysimulations were used to evaluate the influence of canopy turbulence profiles onfootprint prediction. The main uncertainty was found to arise from parameterisationof the random forcing term in the Lagrangian velocity equation. The influence ofdiabatic conditions was studied, and it was found that thermal stability affectssignificantly the footprint function above the forest canopy, but significantuncertainty exists because of uncertainties in the formulation of stability functions.

Study of Stochastic Fluctuations in a Shell Dynamo

We present a one-dimensional αω-dynamo model to study the role of stochastic fluctuations in nonlinear kinematic dynamos. We introduce nonlinearities in the form of α-quenching and consider the effect of a meridional flow. Solutions are seen to settle down to a nonlinear periodic mode with a given amplitude. We show that the system displays relaxation oscillations that are in good agreement with the oscillations observed in the sunspot number time series. Irregularities are generated by adding a random forcing term to the α-coefficient to simulate the spatial and temporal complexity of the turbulent motions in the dynamo region. These fluctuations induce amplitude and cycle period variability, fluctuations in the dividing line between polarities around the equator, emergence of regions of reverse polarity, and north-south asymmetries. These results are in good agreement with observations and can be explained as the interaction of the deterministic solution with overtones excited by noise.

Lattice Boltzmann Numerical Simulation of a Circular Cylinder

The lattice Boltzmann equation (LBE) model based on the Boltzmann equation issuitable for the numerical simulation of various flow fields. The fluiddynamics equation can be recovered from the LBE model. However, compared tothe Navier-Stokes transport equation, the fluid dynamics equation derivedfrom the LBE model is somewhat different in the viscosity transport term,which contains not only the Navier-Stokes transport equation but also nonsteadypressure and momentum flux terms. The two nonsteady terms can produce thesame function as the random stirring force term introduced in the directnumerical or large-eddy vortex simulation of turbulence. Through computationof a circular cylinder, it is verified that the influence of the twononsteady terms on flow field stability cannot be ignored, which is helpfulfor the study of turbulence.