Browsing by Subject "Stochastic differential equations"

Now showing 1 - 9 of 9

Open Access
Better stability with measurement errors
(Springer New York LLC, 2016-06) Argun, A.; Volpe, G.
Often it is desirable to stabilize a system around an optimal state. This can be effectively accomplished using feedback control, where the system deviation from the desired state is measured in order to determine the magnitude of the restoring force to be applied. Contrary to conventional wisdom, i.e. that a more precise measurement is expected to improve the system stability, here we demonstrate that a certain degree of measurement error can improve the system stability. We exemplify the implications of this finding with numerical examples drawn from various fields, such as the operation of a temperature controller, the confinement of a microscopic particle, the localization of a target by a microswimmer, and the control of a population.
Open Access
Dynamical effects of noise on nonlinear systems
(2014) Duman, Özer
Randomness and nonlinear dynamics consitute the most essential part of many events in nature. Therefore, a better and comprehensive understanding of them is a crucial step in describing natural phenomena as well as the prospect of predicting their future outcome. Besides the interest from a fundamental point of view, it is also useful in a wide variety of applications requiring delicate and careful use of energy. Especially recent advances in micro- and nano-scale technology requires harnessing the underlying noise itself as it is relatively hard to exert forces without damaging the system at that scale. The main aim of this work is to study the effects of noise on nonlinear dynamics. We show that the interplay between noise, nonlinearity and nonequilibrium conditions leads to a finite drift with the potential to change the dynamics of the system completely in a predictable and tunable fashion. We report that the noise-induced drift disrupts the phase space of a 2-D nonlinear system by shifting the fixed point by a finite amount which may result in dramatic alterations over the temporal behavior of the system. We track such alterations to several multi-dimensional model systems from ecology, soft matter and statistical physics. In a 2-D ecological model describing two species competing for the same resource, it is found that the system switches between coexistence and extinction states depending on the shift due to the noise-induced drift whereas for an aggregate of Brownian particles, it is shown that noise-induced drift selectively shifts the probability distribution in certain geometries which can be used in the realization of a microparticle sorter in the mould of Feynman ratchets. In the case of the aggregate consisting of microswimmers, tunable anomalous diffusion depending on the confinement length is reported.
Open Access
Noise-induced drift in stochastic differential equations with arbitrary friction and diffusion in the smoluchowski-kramers limit
(Springer, 2012) Hottovy, S.; Volpe, G.; Wehr, J.
We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e. g. Brownian motion. We study the limit where friction effects dominate the inertia, i. e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Itô stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α∈ℝ. Interestingly, in addition to the classical Itô (α=0), Stratonovich (α=0. 5) and anti-Itô (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α∉[0,1] arise. Our findings are supported by numerical simulations. © 2012 Springer Science+Business Media, LLC.
Open Access
Numerical simulation of Brownian particles in optical force fields
(SPIE, 2013) Volpe, G.; Volpe, Giovanni
Optical forces can affect the motion of a Brownian particle. For example, optical tweezers use optical forces to trap a particle at a desirable position. Using more complex force fields it is possible to generate more complex configurations. For example, by using two optical traps placed next to each other, it is possible to obtain a bistable potential where a particle can jump between the two potentials with a characteristic time scale. In this proceeding, we discuss a simple finite difference algorithm that can be used to simulate the motion of a Brownian particle in a one-dimensional field of optical forces.
Open Access
Numerical simulation of optically trapped particles
(SPIE, 2014) Volpe, G.; Volpe, Giovanni
Some randomness is present in most phenomena, ranging from biomolecules and nanodevices to financial markets and human organizations. However, it is not easy to gain an intuitive understanding of such stochastic phenomena, because their modeling requires advanced mathematical tools, such as sigma algebras, the Itô formula and martingales. Here, we discuss a simple finite difference algorithm that can be used to gain understanding of such complex physical phenomena. In particular, we simulate the motion of an optically trapped particle that is typically used as a model system in statistical physics and has a wide range of applications in physics and biophysics, for example, to measure nanoscopic forces and torques.
Open Access
Optimal control for a class of partially observed bilinear stochastic systems
(IEEE, 1990) Dabbous, Tayel E.
An alternative formulation is presented for a class of partially observed bilinear stochastic control problems which is described by three sets of stochastic differential equations: one for the system to be controlled, one for the observer, and one for the control process which is driven by the observation process. With this formulation, the stochastic control problem is converted to an equivalent deterministic identification problem of control gain matrices. Using standard variation arguments, the necessary conditions of optimality on the basis of which the optimal control parameters can be determined are obtained.
Open Access
An SDE approximation for stochastic differential delay equations with state-dependent colored noise
(Polymat Publishing, 2016-11) McDaniel, A.; Duman, Ö.; Volpe, G.; Wehr, J.
We consider a general multidimensional stochastic differential delay equation (SDDE) with state-dependent colored noises. We approximate it by a stochastic differential equation (SDE) system and calculate its limit as the time delays and the correlation times of the noises go to zero. The main result is proven using a theorem about convergence of stochastic integrals by Kurtz and Protter. It formalizes and extends a result that has been obtained in the analysis of a noisy electrical circuit with delayed state-dependent noise, and may be used as a working SDE approximation of an SDDE modeling a real system where noises are correlated in time and whose response to noise sources depends on the system's state at a previous time.
Open Access
Simulation of active Brownian particles in optical potentials
(SPIE, 2014) Volpe, G.; Gigan, S.; Volpe, Giovanni
Optical forces can affect the motion of a Brownian particle. For example, optical tweezers use optical forces to trap a particle at a desirable position. Unlike passive Brownian particles, active Brownian particles, also known as microswimmers, propel themselves with directed motion and thus drive themselves out of equilibrium. Understanding their motion in a confined potential can provide insight into out-of-equilibrium phenomena associated with biological examples such as bacteria, as well as with artificial microswimmers. We discuss how to mathematically model their motion in an optical potential using a set of stochastic differential equations and how to numerically simulate it using the corresponding set of finite difference equations.
Open Access
Spatial measurement of spurious forces with optical tweezers
(SPIE, 2013) Bordeu, I.; Volpe, Giovanni; Staforelli, J. P.
The study of diffusion in a crowded and complex environment, such as inside a cell or within a porous medium, is of fundamental importance for science and technology. Combining blinking holographic optical tweezers and sub-pixel video microscopy permits one to study Brownian motion in confined geometries. In this work, in particular, we have studied the Brownian motion of two colloidal particles interacting hydrodynamically with each other. The proximity between the two microspheres induces a space-dependence in the particles diffusion coefficient and, therefore, a spurious drift. We measure this drift and evaluate the magnitude of the spurious force associated with it. We present the optoelectronic tools employed in the experiment and we discuss the experimental results.