External Stochastic Force Research Articles

Optimal control for two-dimensional stochastic second grade fluids

This article deals with a stochastic control problem for certain fluids of non-Newtonian type. More precisely, the state equation is given by the two-dimensional stochastic second grade fluids perturbed by a multiplicative white noise. The control acts through an external stochastic force and we search for a control that minimizes a cost functional. We show that the Gâteaux derivative of the control to state map is a stochastic process being the unique solution of the stochastic linearized state equation. The well-posedness of the corresponding stochastic backward adjoint equation is also established, allowing to derive the first order optimality condition.

Stochastic sensitivity technique in a persistence analysis of randomly forced population systems with multiple trophic levels

Motivated by important ecological applications we study how noise can reduce a number of trophic levels in hierarchically related multidimensional population systems. A nonlinear model with three trophic levels under the influence of external stochastic forcing is considered as a basic conceptual example. We analyze a probabilistic mechanism of noise-induced extinction of separate populations in this “prey-predator-top predator” system. We propose a new general mathematical approach for the estimation of the proximity of equilibrium regimes of this stochastic model to hazardous borders where abrupt changes in dynamics of ecological systems can occur. Our method is based on the stochastic sensitivity function technique and visualization method of confidence domains. Constructive abilities of this mathematical approach are demonstrated in the analysis of different scenaria of noise-induced reducing of the number of trophic levels.

Comparison of numerical advection schemes in two-dimensional turbulence simulation

Abstract The influence of numerical approximations on statistical characteristics of modelled two-dimensional turbulence sustained by a stochastic external forcing is studied. The ability of various finite-difference and semi-Lagrangian schemes to reproduce reliably the dual energy and enstrophy cascades for coarse spatial resolution is tested. It is also studied how the requirement of preserving invariants inherent to a two-dimensional ideal fluid is important relative to numerical schemes. The results of calculations with high spatial resolution were taken as a reference solution. The choice of studied schemes was motivated by their use in atmosphere and ocean numerical models, in particular, in the Institute of Numerical Mathematics climate model (INMCM) and semi-Lagrangian absolute vorticity (SLAV) model of medium-range weather forecast. The importance of conservation laws for integral vorticity and enstrophy is revealed in the numerical experiments with a small-scale external forcing.

STATIONARY RESPONSES OF GEAR SYSTEMS WITH COMBINED PERIODICALLY TIME-VARYING STIFFNESS AND STOCHASTIC EXTERNAL EXCITATIONS

The stationary dynamic response of a geared system with time-varying mesh stiffness subjected to stationary stochastic external excitation is examined in this paper. To this end, a spur gear pair model with periodical time-varying mesh stiffness is extended to include stochastic external force excitations. These random forces are introduced as second-order stationary and ergodic processes. In order to compute the stationary response, a very efficient method, called the iterative spectral method is used. This method is derived in the frequency domain and provides the explicit power spectral density of the response. The PSD response is expressed as a function of the bispectrum which is the bilinear Fourier transform of the bitemporal impulse response of the parametric system.

Delayed-feedback chimera states: Forced multiclusters and stochastic resonance

A nonlinear oscillator model with negative time-delayed feedback is studied numerically under external deterministic and stochastic forcing. It is found that in the unforced system complex partial synchronization patterns like chimera states as well as salt-and-pepper–like solitary states arise on the route from regular dynamics to spatio-temporal chaos. The control of the dynamics by external periodic forcing is demonstrated by numerical simulations. It is shown that one-cluster and multi-cluster chimeras can be achieved by adjusting the external forcing frequency to appropriate resonance conditions. If a stochastic component is superimposed to the deterministic external forcing, chimera states can be induced in a way similar to stochastic resonance, they appear, therefore, in regimes where they do not exist without noise.

Analysis of noise-induced eruptions in a geyser model

Motivated by important geophysical applications we study a non-linear model of geyser dynamics under the influence of external stochastic forcing. It is shown that the deterministic dynamics is substantially dependent on system parameters leading to the following evolutionary scenaria: (i) oscillations near a stable equilibrium and a transient tendency of the phase trajectories to a spiral sink or a stable node (pre-eruption regime), and (ii) fast escape from equilibrium (eruption regime). Even a small noise changes the system dynamics drastically. Namely, a low-intensity noise generates the small amplitude stochastic oscillations in the regions adjoining to the stable equilibrium point. A small buildup of noise intensity throws the system over its separatrix and leads to eruption. The role of the friction coefficient and relative pressure in the deterministic and stochastic dynamics is studied by direct numerical simulations and stochastic sensitivity functions technique.

RETRACTED ARTICLE: Global strong solution to the three-dimensional stochastic incompressible magnetohydrodynamic equations

The three-dimensional incompressible magnetohydrodynamic equations with stochastic external forces are considered. First the existence and uniqueness of local strong solution to the stochastic magnetohydrodynamic equations are proved when the external forces satisfy some conditions. The proof is based on the contraction mapping principle, stopping time and stochastic estimates. The strong solution is a weak solution for the fluid variables with a given complete probability space and a given Brownian motion. Then, the global existence of strong solutions in probability is established if the initial data are sufficiently small, and the noise is multiplicative and non-degenerate.

Existence of the weak solutions for the compressible magnetohydrodynamic flows driven by stochastic forcing

In this paper, we study the existence of weak solutions for the compressible magnetohydrodynamics flows driven by stochastic external forcing. Our method is based on solving the system for each fixed representative of the random variable and applying an abstract result on the measurability of multi-valued maps.

Limits to seaward expansion of mangroves: Translating physical disturbance mechanisms into seedling survival gradients

Mangroves are valuable coastal habitats that are globally under pressure due to climate change and coastal development. Small-scale physical disturbance by tidal inundation and wave-induced sediment dynamics has been described as the main bottlenecks to mangrove seedling establishment on exposed tidal flats. However, such biophysical bottlenecks remain poorly studied.Mangrove progradation at our study site at the Firth of Thames (New Zealand) has been described as rapid but disturbance limited. For this site, we apply a mechanistic model according to the ‘Window of Opportunity’ (WoO) concept, which analyses real time series of external forcing for disturbance free periods. The model was parameterized with manipulative experiments on seedling stability and monitoring data. The modelled inundation free periods for initial anchoring were validated by a caging experiment with loose and tethered propagules.Although the time series of external forcing derived from monitoring data are simplified, our model confirms the absence of mangrove progradation due to failed recruitment events on the tidal flat since 1997. The model also shows, that a temporary reduction in external forcing would lead to a sudden progradation of the mangrove forest.WoO dynamics, where vegetation establishment requires temporarily benign conditions, may be of general importance to other ecosystems with stochastic external forcing. Understanding the biophysical interactions between vegetation and geomorphic processes is key to better manage and protect disturbance-driven ecosystems in times of changing wind pattern and accelerated sea-level rise.

Critical transitions in disturbance‐driven ecosystems: identifying Windows of Opportunity for recovery

Summary Vegetation recovery in disturbance‐driven ecosystems is difficult to predict. We demonstrate a concept to analyse time series for short‐term variability in external forcing that can identify potential events for sudden vegetation recovery in biogeomorphic ecosystems such as saltmarshes, mangroves, dunes or floodplains. Time series of external forcing (i.e. water level and wind speed) were analysed for ‘Windows of Opportunity’ (WoO), defined as disturbance‐free periods of a critical minimal duration directly following potential diaspore dispersal, which allow seedling establishment and can induce a sudden shift to a new persistent vegetation cover. Across different ecosystems, the minimal required WoO duration determines how many WoO events are available for seedling establishment. The distribution of WoO along an elevation gradient on riverbanks and in tidal systems, for example, is defined by the combination of the overall disturbance regime (e.g. seasonal vs. tidal flooding cycles) and the stochastic deviations from that regime (e.g. changes in weather conditions). Standardizing the WoO for the frequency of the regular disturbance regime shows that tidal and river systems have a similar relation between the required WoO length and the elevation suitable for establishment. WoO analysis correctly predicted a sudden vegetation recovery event in a saltmarsh case study. Synthesis. Time‐series analysis for ‘WoO’ offers an important tool towards predicting the establishment of vegetation cover in disturbance‐driven ecosystems and may have broader implications for understanding critical transitions in general. Quantifying the effects of stochastic external forcing on critical transitions in ecosystems is crucial for restoration efforts and to assess the effects of anthropogenic and global change.

Quantum-Limited Mirror-Motion Estimation

We experimentally demonstrate optomechanical motion and force measurements near the quantum precision limits set by the quantum Cramér-Rao bounds. Optical beams in coherent and phase-squeezed states are used to measure the motion of a mirror under an external stochastic force. Utilizing optical phase tracking and quantum smoothing techniques, we achieve position, momentum, and force estimation accuracies close to the quantum Cramér-Rao bounds with the coherent state, while the estimation using squeezed states shows clear quantum enhancements beyond the coherent-state bounds.

The Performance Characteristics of a Nano-thermoelectric Refrigerator Driven by an External Stochastic Force

We describe a single level quantum dot driven by an external stochastic force which works as a nano-thermoelectric refrigerator. Based on the model, expressions for the cooling rate, power input, and coefficient of performance (COP) are derived. The effects of the energy level and energy space on the refrigerator are revealed. The optimal performance characteristics are analyzed by numerical calculation. Furthermore, the practical operating regions of the refrigerator are determined.

Performance analysis of a Nano-thermoelectric engine driving by an external stochastic force

In this paper, we have studied a Nano-thermoelectric engine with a single energy level dot driving by an external stochastic force. Based on the model, expressions for the power output and efficiency of the engine are derived. The effects of the energy level ϵd and the energy space Δϵ=ϵu−ϵd on the performance of the engine are revealed. The performance characteristics of the engine are analyzed by a numerical calculation and graphic method. Furthermore, some important operating regions, including the energy level, the power output and the efficiency, are determined and evaluated optimally.

Numerical simulation of the stochastic damped improved Boussinesq equation

In this paper, we numerically investigate the dynamics of the solitary waves in the presence of damping and external stochastic forces for the improved Boussinesq equation. The hydrodynamical and Stokes damping will be considered respectively. Since the analytic solutions for the considered problems are not available, Monte Carlo method will be used in random space and finite volume element method will be used in physical space. Numerical results demonstrate that solitary wave profile is not strongly affected by the Gaussian white noise in our study. The average of the velocity of the solitary waves is very close to that for the deterministic case.

Non-Markovian stationary probability density for a harmonic oscillator in an electromagnetic field

We calculate the exact solution of the Fokker-Planck equation for the stationary-state probability density of a harmonic oscillator embedded in an electromagnetic field. The magnetic field is assumed to be a constant and the electric field an external stochastic force with the properties of a Gaussian and exponentially correlated noise (Ornstein-Uhlenbeck process). In this work, we first study the problem in the absence of the magnetic field, then we obtain the complete solution and corroborate that the latter reduces to the former when the magnetic field is suppressed.

Kinetic theory of nonequilibrium stochastic long-range systems: phase transition and bistability

We study long-range interacting systems driven by external stochastic forces that act collectively on all the particles constituting the system. Such a scenario is frequently encountered in the context of plasmas, self-gravitating systems, two-dimensional turbulence, and also in a broad class of other systems. Under the effect of stochastic driving, the system reaches a stationary state where external forces balance dissipation on average. These states have an invariant probability that does not respect detailed balance, and are characterized by non-vanishing currents of conserved quantities. In order to analyze spatially homogeneous stationary states, we develop a kinetic approach that generalizes the one known for deterministic long-range systems; we obtain a very good agreement between predictions from kinetic theory and extensive numerical simulations. Our approach may also be generalized to describe spatially inhomogeneous stationary states. We also report on numerical simulations exhibiting a first-order nonequilibrium phase transition from homogeneous to inhomogeneous states. Close to the phase transition, the system shows bistable behavior between the two states, with a mean residence time that diverges as an exponential in the inverse of the strength of the external stochastic forces, in the limit of low values of such forces.

North Atlantic variability driven by stochastic forcing in a simple model

ABSTRACTThis study investigates the mechanisms driving North Atlantic (NA) variability using a simple model that incorporates the time evolution of interactive upper ocean temperature anomalies, horizontal (Gyre, Ψg) and vertical (meridional overturning circulation, Ψm) circulation. The model is forced with multicentury long synthetic time series of external stochastic forcing that captures key statistical properties of observations such as the range of fluctuations and persistence of processes. The simulated oceanic response may be viewed as a delayed response to a cumulative atmospheric forcing over an interval defined by the system damping properties. Depending on the choice of parameters, the model suggests either compensatory mechanism (Ψm and Ψg are anti-correlated) or amplification mechanism (Ψm and Ψg are positively correlated). The compensatory mechanism implies that an increase of heat supplied by an anomalously strong Ψg would be balanced by a decrease of heat provided by a weaker Ψm and vice versa. The amplification mechanism suggests that both Ψm and Ψg maintain the heat budget in the system compensating its damping properties. Some evidence for these mechanisms is found in a global climate model. Further investigations of NA variability mechanisms are important as they improve understanding of how the NA climate system functions.

Compressible fluid flows driven by stochastic forcing

We consider the Navier–Stokes system describing the motion of a compressible barotropic fluid driven by stochastic external forces. Our approach is semi-deterministic, based on solving the system for each fixed representative of the random variable and applying an abstract result on measurability of multi-valued maps. The class of admissible driving forces includes the (temporal) white noise and the random kicks, considered recently in the context of incompressible fluid models.

The formation of correlated states and the increase in barrier transparency at a low particle energy in nonstationary systems with damping and fluctuations

We consider peculiarities in the formation of a coherent correlated state (CCS) of a particle in a periodically modulated harmonic oscillator with damping for various types of stochastic perturbation. It is shown that in the absence of stochastic perturbation, an optimal relation exists between the damping parameter (damping coefficient) and the modulation depth, for which the “extrinsic” characteristics of the oscillator (amplitudes of “classical” oscillation and the momentum of a particle) remain unchanged, while the correlation coefficient rapidly increases from |r| = 0 to |r|max ≈ 1; this corresponds to a completely correlated coherent state. Under nonoptimal conditions, the formation of the CCS with a simultaneous increase in is accompanied by damping or excitation of the oscillator. It is shown that for a certain relation between the damping coefficient and the modulation depth, the presence of a stochastic external force acting on the nonstationary oscillator does not prevent the formation of a CCS with |r|max → 1. A fundamentally different effect is observed under a stochastic influence on the nonstationary frequency of the oscillator; this effect always limits the value of |r| at a level |r|max < 1; a CCR cannot be formed with an unlimited increase in its intensity, and |r|max → 0. The influence of the CCS formation on the averaged probability 〈D〉 of the tunnel effect (transparency of the potential barrier) is considered for a particle in an oscillator with damping both in the absence and in the presence of a stochastic force. It is shown using a specific example that complete clearing of the potential barrier and the increase in the barrier transparency from the initial value 〈Dr=0〉 = 10−80 to 〈D〉 ≈ 1 can occur over a comparatively short time interval in both these cases. These effects can be used to obtain highly efficient nuclear fusion at a low energy of interacting particles.

Stochastically driven single-level quantum dot: A nanoscale finite-time thermodynamic machine and its various operational modes

We describe a single-level quantum dot in contact with two leads as a nanoscale finite-time thermodynamic machine. The dot is driven by an external stochastic force that switches its energy between two values. In the isothermal regime, it can operate as a rechargeable battery by generating an electric current against the applied bias in response to the stochastic driving and then redelivering work in the reverse cycle. This behavior is reminiscent of the Parrondo paradox. If there is a thermal gradient the device can function as a work-generating thermal engine or as a refrigerator that extracts heat from the cold reservoir via the work input of the stochastic driving. The efficiency of the machine at maximum power output is investigated for each mode of operation, and universal features are identified.