External Stochastic Force Research Articles

Distributions and extreme value analysis of critical response rate and split-time metric in nonlinear oscillators with stochastic excitation

A single-degree-of-freedom oscillator with a nonlinear restoring force and stochastic external forcing is studied with the goal of understanding the distributions of certain response rates. Motivated by applications to ship motions, the response rates concern the minimum response rate leading to capsizing (the critical response rate) and a measurement comparing an observed response rate to the critical response rate (the split-time metric), whose distributions are investigated both analytically and numerically. Three nonlinear restoring forces are considered: piecewise linear (experiencing linearly softening stiffness above a “knuckle” point), doubly piecewise linear (experiencing piecewise linearly softening stiffness above a “knuckle” point), and the cubic softening restoring force of the Duffing oscillator. In the first two cases, an efficient numerical simulation of the critical response rate and split-time metric is proposed from a derived distribution; in the latter case, the density of the critical response rate is approximated assuming white noise excitation. A key interest is in the nature of the right tail of the split-time metric, specifically as it relates to its extrapolation through extreme value analysis in estimating the probability of capsizing. The distribution is found to have a “light” tail, which motivates the use of exponential rather than the generalized Pareto distribution for exceedances above threshold in extreme value analysis. Finally, threshold selection through a prediction error criterion for the exponential distribution is examined, and the Weibull distribution tail is suggested as a useful means for a more refined examination of the distribution tail.

Large deviation principles for a 2D liquid crystal model with jump noise

We consider a stochastic 2D liquid crystal model with a multiplicative noise of Lévy type, which models the dynamic of nematic liquid crystals under the influence of stochastic external forces of jump type. We derive a large deviation principle for the model. The proof is based on the weak convergence method introduced in [Budhiraja A, Dupuis P, Maroulas V. Variational representations for continuous time processes. Ann Inst Henri Poincar Probab Stat. 2011;47(3):725–747].

Steady-state statistics, emergent patterns and intermittent energy transfer in a ring of oscillators

Networks of coupled nonlinear oscillators model a broad class of physical, chemical and biological systems. Understanding emergent patterns in such networks is an ongoing effort with profound implications for different fields. In this work, we analytically and numerically study a symmetric ring of N coupled self-oscillators of van der Pol type under external stochastic forcing. The system is proposed as a model of the thermo- and aeroacoustic interactions of sound fields in rigid enclosures with compact source regions in a can-annular combustor. The oscillators are connected via linear resistive coupling with nonlinear saturation. After transforming the system to amplitude-phase coordinates, deterministic and stochastic averaging is performed to eliminate the fast oscillating terms. By projecting the potential of the slow-flow dynamics onto the phase-locked quasi-limit cycle solutions, we obtain a compact, low-order description of the (de-)synchronization transition for an arbitrary number of oscillators. The stationary probability density function of the state variables is derived from the Fokker–Planck equation, studied for varying parameter values and compared to time series simulations. We leverage our analysis to offer explanations for the intermittent energy transfer between Bloch waves observed in acoustic pressure spectrograms observed of real-world gas turbines.

Global existence of [formula omitted] solutions to the Zakharov equations with additive noises

We show the global existence of L2 solutions for the Cauchy problem of the Zakharov equations with additive noises in one spatial dimension. For the proof, we use the argument by Colliander, Holmer and Tzirakis, which was applied to the deterministic Zakharov equations. Our proof is an extension of their proof to the problem with stochastic external forces.

Weak-force sensing in optomechanical systems with Kalman filtering

We investigate the weak-force sensing in an optomechanical resonator by Kalman filter. By discretizing the continuous-time optomechanical system, the state of the resulting system is estimated by the unbiased minimum variance Kalman filter. Subsequently, the external stochastic force is estimated, provided that all noises in the system are white and Gaussian. Furthermore, the accuracy of force estimation, described by the mean squared error, is derived theoretically. The proposed algorithm is finally illustrated by comparing the theoretical accuracy with the numerical accuracy in an explicit example.

Approximation of a stochastic two-phase flow model by a splitting-up method

<p style='text-indent:20px;'>In this paper, we consider a stochastic Allen-Cahn Navier-Stokes system in a bounded domain of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^d, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ d = 2,3 $\end{document}</tex-math></inline-formula>. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on splitting-up method as well as some compactness method.

Heat dissipation rate in a nonequilibrium viscoelastic medium

A living non-Newtonian matter like the cell cortex and tissues are driven out-of-equilibrium at multiple spatial and temporal scales. The stochastic dynamics of a particle embedded in such a medium are non-Markovian, given by a generalized Langevin equation. Due to the non-Markovian nature of the dynamics, the heat dissipation and the entropy production rate cannot be computed using the standard methods for Markovian processes. In this work, to calculate heat dissipation, we use an effective Markov description of the non-Markovian dynamics, which includes the degrees-of-freedom of the medium. Specifically, we calculate entropy production and heat dissipation rate for a spherical colloid in a non-Newtonian medium whose rheology is given by a Maxwell viscoelastic element in parallel with a viscous fluid element, connected to different temperature baths. This problem is nonequilibrium for two reasons: the medium is nonequilibrium due to different effective temperatures of the bath, and the particle is driven out-of-equilibrium by an external stochastic force. When the medium is nonequilibrium, the effective non-Markov dynamics of the particle may lead to a negative value of heat dissipation and entropy production rate. The positivity is restored when the medium’s degree-of-freedom is considered. When the medium is at equilibrium, and the only nonequilibrium component is the external driving, the correct dissipation is obtained from the effective description of the particle.

Existence of a solution to the stochastic nonlocal Cahn–Hilliard Navier–Stokes model via a splitting-up method

We consider a stochastic diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids under the influence of stochastic external forces in a bounded domain of , d = 2, 3. The model consists of the stochastic Navier–Stokes equations coupled with a nonlocal Cahn–Hilliard equation. We prove the existence of a global weak martingale solution via a numerical scheme based on splitting-up method.

On the weak solutions to a stochastic two-phase flow model

ABSTRACT We study, in this article, a stochastic version of a coupled Allen–Cahn–Navier–Stokes model in a two- or three-dimensional bounded domain. The model consists of the Navier–Stokes equations for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter. These equations are motivated by the dynamic of binary fluids under the influence of stochastic external forces. We prove the existence of a probabilistic weak solutions. The proof relies on a Galerkin approximation as well as some compactness results. In the two-dimensional case, we prove the uniqueness of the weak solutions.

Splitting-up scheme for the stochastic Cahn–Hilliard Navier–Stokes model

In this paper, we consider a stochastic Cahn–Hilliard Navier–Stokes system in a bounded domain of [Formula: see text] [Formula: see text]. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on the splitting-up method as well as some compactness method.

Perturbation dynamics in Keplerian flow under external stochastic forcing

ABSTRACT We investigate the dynamics of linear perturbations in Keplerian flow under external stochastic force. To abstract from the details of flow structure and boundary conditions, we consider the problem in the shearing box approximation. An external force is assumed to have zero mean, even so, induced perturbations form a steady state, which provides angular momentum transfer to the periphery of the flow. The most effective scenario is based on the transient amplification of induced vortices with the following emission of a shearing sound wave, wherein the maximum of the flux linearly depends on Reynolds number. Thus such a mechanism is significant for astrophysical flows, for which enormous Reynolds numbers are typical. At the same time, addressing the problem analytically, we find that for incompressible fluid in the shearing box approximation stochastic forcing does not lead to average angular momentum transfer. Thus the compressibility of the fluid plays an important role here, and one cannot neglect it.

Weak solution of a stochastic 2D Ericksen–Leslie model driven by jump noise

We investigate a stochastic 2D Ericksen–Leslie model with a multiplicative noise of Lévy type. The system models the dynamic of nematic liquid crystals under the influence of stochastic external forces and stretching effects. We prove the existence of a weak martingale solution of the system. The proof of the existence is based on a classical Galerkin approximation as well as some compactness methods. We also prove the pathwise uniqueness of the weak solution when the stretching effect is neglected.

Global existence and analyticity of mild solutions for the stochastic Navier–Stokes–Coriolis equations in Besov spaces

Consider the stochastic Navier–Stokes–Coriolis equation in R3 driven by an additive white noise, we establish the unique existence and spatial analyticity of global mild solution even when initial data is essentially arbitrarily large and even when stochastic external force is also large provided that the speed of the rotation is fast enough. The proof is based on the Picard contraction principle and a priori estimates to the stochastic parabolic equation.

On the Weak Solutions to a Stochastic 2D Simplified Ericksen-Leslie Model

We study in this article a stochastic version of a 2D Ericksen-Leslie systems. The system model the dynamic of nematic liquid crystals under the influence of stochastic external forces and stretching effects. We prove the existence of a probabilistic weak solutions. The proof relies on a reformulation of the model proposed in Gong et al. (Nonlinearity 28(10), 3677–3694 2015) as well as a Galerkin approximation and some compactness results. We also prove the pathwise uniqueness of the weak solution when the stretching effect is neglected.

Some results on the penalised nematic liquid crystals driven by multiplicative noise: weak solution and maximum principle

In this paper, we prove several mathematical results related to a system of highly nonlinear stochastic partial differential equations (PDEs). These stochastic equations describe the dynamics of penalised nematic liquid crystals under the influence of stochastic external forces. Firstly, we prove the existence of a global weak solution (in the sense of both stochastic analysis and PDEs). Secondly, we show the pathwise uniqueness of the solution in a 2D domain. In contrast to several works in the deterministic setting we replace the Ginzburg–Landau function mathbb {1}_{|{mathbf {n}}|le 1}(|{mathbf {n}}|^2-1){mathbf {n}} by an appropriate polynomial f({mathbf {n}}) and we give sufficient conditions on the polynomial f for these two results to hold. Our third result is a maximum principle type theorem. More precisely, if we consider f({mathbf {n}})=mathbb {1}_{|d|le 1}(|{mathbf {n}}|^2-1){mathbf {n}} and if the initial condition {mathbf {n}}_0 satisfies |{mathbf {n}}_0|le 1, then the solution {mathbf {n}} also remains in the unit ball.

On the existence and uniqueness of solution to a stochastic simplified liquid crystal model

We study in this article a stochastic version of a 2D simplified Ericksen-Leslie systems, which model the dynamic of nematic liquid crystals under the influence of stochastic external forces. We prove the existence and uniqueness of strong solution. The proof relies on a new formulation of the model proposed in [ 19 ] as well as a Galerkin approximation

The Role of Forcing and Eddy Viscosity Variation on the Log-Layer Mismatch Observed in Wall-Modeled Large-Eddy Simulations

We investigate the role of eddy viscosity variation and the effect of zonal enforcement of the mass flow rate on the log-layer mismatch problem observed in turbulent channel flows. An analysis of the mean momentum balance shows that it lacks a degree-of-freedom (DOF) when eddy viscosity is large, and the mean velocity conforms to an incorrect profile. Zonal enforcement of the target flow rate introduces an additional degree-of-freedom to the mean momentum balance, similar to an external stochastic forcing term, leading to a significant reduction in the log-layer mismatch. We simulate turbulent channel flows at friction Reynolds numbers of 2000 and 5200 on coarse meshes that do not resolve the viscous sublayer. The second-order turbulence statistics agree well with the direct numerical simulation benchmark data when results are normalized by the velocity scale extracted from the filtered velocity field. Zonal enforcement of the flow rate also led to significant improvements in skin friction coefficients.

Salton Sea: An ecosystem in crisis

Salton Sea (a destination resort) is to be saved from being converted into a skeleton-filled wasteland. The critical amount of water flowing into the sea to maintain its level and salinity has been diverted since January 2018. This will lead to shrinking volumes and increasing salinities. Ecological consequences and public health impacts of altered conditions will be phenomenal. We design and analyze a minimal eco-epidemiological model to figure out future journey of this sea; a way station for fish-eating migratory birds. The mathematical model has been assembled in terms of prey-predator interaction. The salient feature of the proposed model is its seasonally varying contact rate which represents rate of conversion of susceptible fishes into infectives. We have analytically investigated the global stability, disease persistence and periodic solutions of the proposed model system. Susceptible prey-induced periodic solution is globally asymptotically stable when [Formula: see text], otherwise unstable and hence disease persists for [Formula: see text].Global stability and Hopf bifurcation (HB) analysis help us extract parameter values to explore the dynamical behavior of the model system. Two-dimensional parameter scans and bifurcation diagrams reveal that the model displays propensity towards chaotic dynamics, which is associated with extinction-sized population densities. In the presence of stochastic external forces, this implies extinction of most of the fish species. This, in turn, suggests that resident birds will have to migrate to other destinations. The fish-eating migratory birds will be forced to switch over to invertebrates. Ecological consequences and public health impacts of this transition would be severe. Conservation groups are solicited to draw attention of the government to avert this impounding danger. It is important to plan for an ecosystem-wide transition such that impacts on birds and on human inhabitants living adjacent to the shrinking and salinizing sea are minimized.

Nonlinear Trends and Nonstationary Oscillations as Extracted From Annual Accumulated Precipitation at Mexico City

AbstractExtracted nonstationary oscillations and nonlinear trends of precipitation and sea surface temperature (SST) data reveal that rainfall variability in Mexico City is mainly composed by a long‐term positive trend, a multidecadal oscillation highly correlated with the Atlantic Multidecadal Oscillation (AMO), and year‐to‐year variability. The precipitation trend, lasting for more than a century, cannot be attributed to global warming or urbanization alone; rather, it can be thought of as part of a natural oscillation composed of alternating wet‐dry anomalies with a period of a couple of centuries, as past evidence indicates. To further test the dependence of the AMO‐related component, yearly SST time series were derived from a simplified model of the atmosphere‐ocean system forced by white noise. The simulated SST time series exhibits AMO‐like variability entirely consistent with the observed one, implying that North Atlantic SST multidecadal variability can be seen as the integrated response of surface ocean layers to external stochastic atmospheric forcing.

Steady-state work fluctuations for an overdamped Brownian particle driven by external stochastic force

The steady-state work fluctuations for an overdamped Brownian particle driven by an external stochastic force are analyzed by using the Langevin approach. The first two moments of the work distribution are given analytically, both are proportional to the time duration of the work injection. The first two moments determine a unique Gaussian function, and the deviations of the probability density functions (PDFs) of work from the Gaussian functions are measured. The tails of the PDFs are shown to approximate the Gaussian functions. In the limit of weak external force and large time interval, the PDF can be approximated by the Gaussian form and the steady-state fluctuation theorem is held.