External Random Force Research Articles

Directed motion generated by heat bath nonlinearly driven by external noise

Based on the heat bath system approach where the bath is nonlinearly modulated by an external Gaussian random force, we propose a new microscopic model to study directed motion in the overdamped limit for a nonequilibrium open system. Making use of the coupling between the heat bath and the external modulation as a small perturbation, we construct a Langevin equation with multiplicative noise- and space-dependent dissipation and the corresponding Fokker–Planck–Smoluchowski equation in the overdamped limit. We examine the thermodynamic consistency condition and explore the possibility of observing a phase-induced current as a consequence of state-dependent diffusion and, necessarily, nonlinear driving of the heat bath by the external noise.

Stochastic cooperativity in non-linear dynamics of genetic regulatory networks

Two major approaches are known in the field of stochastic dynamics of genetic regulatory networks (GRN). The first one, referred here to as the Markov Process Paradigm (MPP), places the focus of attention on the fact that many biochemical constituents vitally important for the network functionality are present only in small quantities within the cell, and therefore the regulatory process is essentially discrete and prone to relatively big fluctuations. The Master Equation of Markov Processes is an appropriate tool for the description of this kind of stochasticity. The second approach, the Non-linear Dynamics Paradigm (NDP), treats the regulatory process as essentially continuous. A natural tool for the description of such processes are deterministic differential equations. According to NDP, stochasticity in such systems occurs due to possible bistability and oscillatory motion within the limit cycles. The goal of this paper is to outline a third scenario of stochasticity in the regulatory process. This scenario is only conceivable in high-dimensional, highly non-linear systems, and thus represents an adequate framework for conceptually modeling the GRN. We refer to this framework as the Stochastic Cooperativity Paradigm (SCP). In this approach, the focus of attention is placed on the fact that in systems with the size and link density of GRN (∼25 000 and ∼100, respectively), the confluence of all the factors which are necessary for gene expression is a comparatively rare event, and only massive redundancy makes such events sufficiently frequent. An immediate consequence of this rareness is ‘burstiness’ in mRNA and protein concentrations, a well known effect in intracellular dynamics. We demonstrate that a high-dimensional non-linear system, despite the absence of explicit mechanisms for suppressing inherent instability, may nevertheless reside in a state of stationary pseudo-random fluctuations which for all practical purposes may be regarded as a stochastic process. This type of stochastic behavior is an inherent property of such systems and requires neither an external random force as in the Langevin approach, nor the discreteness of the process as in MPP, nor highly specialized conditions of bistability as in NDP, nor bifurcations with transition to chaos as in low-dimensional chaotic maps.

On the Asymptotic Behaviour of Deterministic and Stochastic Volterra Integro--Differential Equations

This thesis examines a question of stability in stochastic and deterministic systems with memory, and involves studying the asymptotic properties of Volterra integro-differential equations. The type of stability that has been established for this class of equations is important in a variety of real-world problems which involve feedback from the past, and are subject to external random forces. These include modelling endemic diseases, and more particularly the modelling of inefficient financial markets. The theine of the thesis is to subject a dynamical system with memory to increasingly trong and unpredictable external noise. Firstly, a fundamental deterministic Volterra quation is considered. Necessary and sufficient conditions for the solution to approach nontrivial limit are known. A strengthened version of these conditions is shown to be necessary and sufficient for exponential convergence to a nontrivial limit. Next, a Volterra equation with a fading stochastic perturbation is studied. Two types f stochastic convergence are considered: mean square and almost sure convergence. Conditions re found which ensure that the solution converges to a non-equilibrium random imit. Moreover, the rate at which this limit is approached is established. In the mean quare case, necessary and sufficient conditions on the resolvent, kernel and noise are determined o ensure this rate of convergence. In the almost sure case, the same conditions re found to be sufficient; furthermore, it is shown that the conditions on the resolvent and he kernel are necessary. A correspoilding result was also found to hold for a more general lass of weakly singular kernels. As in the deterministic case, necessary and sufficient onditions for the solution to converge exponentially fast to its limit are found. Finally, a stochastic Volterra equation with constant noise intensity is considered. This ives rise to the process analogous to Brownian motion, which has applications to mathematical inance. It can be shown that the increments of the process converge to a stationary tatistical distribution, which is Gaussian distributed. The conditions under which uch convergence can take place are completely characterised. In fact, a solution of a orresponding Volterra equation with infinite memory is shown to have exactly stationary ncrements which match the limiting distributions of the increments of solutions.

EXACT SOLUTIONS AND DYNAMICS OF GLOBALLY COUPLED OSCILLATORS

We analyze mean-field models of synchronization of phase oscillators with singular couplings and subject to external random forces. They are related to the Kuramoto–Sakaguchi model. Their probability densities satisfy local partial differential equations similar to the porous medium, Burgers and extended Burgers systems depending on the degree of singularity of the coupling. We show that porous medium oscillators (the most singularly coupled) do not synchronize and that (transient) synchronization is possible only at zero temperature for Burgers oscillators. The extended Burgers oscillators have a nonlocal coupling first introduced by Daido and they may synchronize at any temperature. Exact expressions for their synchronized phases and for Daido's order function are given in terms of elliptic functions.

Identification of stiffness, dissipation and input parameters of randomly excited non-linear systems: Capability of restricted potential models (RPM)

A dynamic identification technique in the time domain for time invariant systems under random external forces is presented. This technique is based on the use of the class of restricted potential models (RPM), which are characterized by a non-linear stiffness and a special form of damping, that is a product of the input power spectral density (PSD) matrix and the velocity gradient of a non-linear function of the total mechanical energy. By applying It o ^ stochastic differential calculus and by specific analytical manipulations, some algebraic equations, depending on the response statistics and on the mechanic parameters that characterize RPM, are obtained. These equations can be used for the dynamic identification of the above mechanic parameters once the response statistics of the system to be identified are evaluated. The proposed technique allows one to identify single-degree-of-freedom or multi-degrees-of-freedom systems in the case of unmeasurable input. Further, the probabilistic characteristics of the external forces can be completely estimated in terms of PSD matrix.

Stochastic nonhomogeneous incompressible Navier–Stokes equations

We construct solutions for 2- and 3-D stochastic nonhomogeneous incompressible Navier–Stokes equations with general multiplicative noise. These equations model the velocity of a mixture of incompressible fluids of varying density, influenced by random external forces that involve feedback; that is, multiplicative noise. Weak solutions for the corresponding deterministic equations were first found by Kazhikhov [A.V. Kazhikhov, Solvability of the initial and boundary-value problem for the equations of motion of an inhomogeneous viscous incompressible fluid, Soviet Phys. Dokl. 19 (6) (1974) 331–332; English translation of the paper in: Dokl. Akad. Nauk SSSR 216 (6) (1974) 1240–1243]. A stochastic version with additive noise was solved by Yashima [H.F. Yashima, Equations de Navier–Stokes stochastiques non homogènes et applications, Thesis, Scuola Normale Superiore, Pisa, 1992]. The methods here extend the Loeb space techniques used to obtain the first general solutions of the stochastic Navier–Stokes equations with multiplicative noise in the homogeneous case [M. Capiński, N.J. Cutland, Stochastic Navier–Stokes equations, Applicandae Math. 25 (1991) 59–85]. The solutions display more regularity in the 2D case. The methods also give a simpler proof of the basic existence result of Kazhikhov.

Ergodicity for a class of Markov processes and applications to randomly forced PDE'S. II

The paper is devoted to studying the problem of ergodicity for the complex Ginzburg--Landau (CGL) equation perturbed by an external random force. We show that the conditions of a simple general result established in [22] are fulfilled for the equation in question. As a consequence, we prove that the corresponding family of Markov processes has a unique stationary distribution, which possesses a mixing property. The result of this paper was announced in the joint work with Sergei Kuksin [14].

A methodology for efficiently sampling the conformation space of molecular structures

Motivated by recently developed computational techniques for studying protein flexibility, and their potential applications in docking, we propose an efficient method for sampling the conformational space of complex molecular structures. We focus on the loop closure problem, identified in the work of Thorpe and Lei (2004 Phil. Mag. 84 1323–31) as a primary bottleneck in the fast simulation of molecular motions. By modeling a molecular structure as a branching robot, we use an intuitive method in which the robot holds onto itself for maintaining loop constraints. New conformations are generated by applying random external forces, while internal, attractive forces pull the loops closed. Our implementation, tested on several model molecules with low number of degrees of freedom but many interconnected loops, gives promising results that show an almost four times speed-up on the benchmark cube-molecule of Thorpe and Lei.

Critical Condition Expectancy of Flange Beam at the Loss of Flat Form of Stability with Riveted Flanges under the Influence of Random Forces and Initial Flexures

The functioning of multipart flange beam with riveted flanges, which are loaded with random external forces, is examined. Prior to elastic assembling the beam had random initial flexures of the flanges. We obtained relations for determination of probability of flange beam critical condition by the loss of flat form of flexure under the influence of random shear forces, which occur at the welding points between web plate and flanges of the beam. The examples for determination of probability of critical condition of flange beam under the influence of random initial flexures of the flanges are provided.

Transverse vibrations of rotating shafts: Probability density and first-passage time of whirl radius

Transverse vibrations are considered for a single mass/two-degrees-of-freedom rotating shaft with linear internal or “rotating” damping and nonlinear external damping. The shaft is excited by external random forces. Analysis of resulting random vibrations is based on stochastic averaging method which yields separated (in the linear approximation) equations for complex amplitudes of forward and backward whirling motions. The former of these motions is shown to be dominant at rotation speeds in the vicinity of the instability threshold. Using this approximation an analytical solution is obtained for probability density of squared radius of the shaft's whirl. This solution can be used to detect on-line shaft's instability from its observed response. Solution is also obtained for expected time for reaching given level by the squared whirl radius of the shaft.

Law of large numbers and central limit theorem for randomly forced PDE's

We consider a class of dissipative PDE's perturbed by an external random force. Under the condition that the distribution of perturbation is sufficiently non-degenerate, a strong law of large numbers (SLLN) and a central limit theorem (CLT) for solutions are established and the corresponding rates of convergence are estimated. It is also shown that the estimates obtained are close to being optimal. The proofs are based on the property of exponential mixing for the problem in question and some abstract SLLN and CLT for mixing-type Markov processes.

Symmetry breaking induced by random fluctuations for Bose-Einstein condensates in a double-well trap

This paper is devoted to the study of the dynamics of two weakly-coupled Bose-Einstein condensates confined in a double-well trap and perturbed by random external forces. Energy diffusion due to random forcing allows the system to visit symmetry-breaking states when the number of atoms exceeds a threshold value. The energy distribution evolves to a stationary distribution which depends on the initial state of the condensate only through the total number of atoms. This loss of memory of the initial conditions allows a simple and complete description of the stationary dynamics of the condensate which randomly visits symmetric and symmetry-breaking states.

Anomalous scaling of a passive scalar advected by the Navier-Stokes velocity field: Two-loop approximation

The field theoretic renormalization group and operator-product expansion are applied to the model of a passive scalar quantity advected by a non-Gaussian velocity field with finite correlation time. The velocity is governed by the Navier-Stokes equation, subject to an external random stirring force with the correlation function proportional to delta(t- t')k(4-d-2epsilon). It is shown that the scalar field is intermittent already for small epsilon, its structure functions display anomalous scaling behavior, and the corresponding exponents can be systematically calculated as series in epsilon. The practical calculation is accomplished to order epsilon2 (two-loop approximation), including anisotropic sectors. As for the well-known Kraichnan rapid-change model, the anomalous scaling results from the existence in the model of composite fields (operators) with negative scaling dimensions, identified with the anomalous exponents. Thus the mechanism of the origin of anomalous scaling appears similar for the Gaussian model with zero correlation time and the non-Gaussian model with finite correlation time. It should be emphasized that, in contrast to Gaussian velocity ensembles with finite correlation time, the model and the perturbation theory discussed here are manifestly Galilean covariant. The relevance of these results for real passive advection and comparison with the Gaussian models and experiments are briefly discussed.

RESONANT ACTIVATION DRIVEN BY STRONGLY NON-GAUSSIAN NOISES

The constructive role of non-Gaussian random fluctuations is studied in the context of the passage over the dichotomously switching potential barrier. Our attention focuses on the interplay of the effects of independent sources of fluctuations: an additive stable noise representing non-equilibrium external random force acting on the system and a fluctuating barrier. In particular, the influence of the structure of stable noises on the mean escape time and on the phenomenon of resonant activation (RA) is investigated. By use of the numerical Monte Carlo method it is documented that the suitable choice of the barrier switching rate and random external fields may produce resonant phenomenon leading to the enhancement of the kinetics and the shortest, most efficient reaction time.

Dislocation internal friction at random external loadings

Dislocation internal friction under random external forces acting upon a dislocation was investigated. Different types of random forces (telegraph, impulse, others) were considered. Inertial properties of dislocation and effects of potential (parabolic) crystal relief were taken into account. Analytical expressions for decrement were derived and its dependencies on a degree of correlation of random force, dislocation and medium parameters were analyzed.

Classical brackets for dissipative systems

We show how to write a set of brackets for the Langevin equation, describing the dissipative motion of a classical particle, subject to external random forces. The method does not rely on an action principle, and is based solely on the phenomenological description of the dissipative dynamics as given by the Langevin equation. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism.

New classical brackets for dissipative systems

A set of brackets for classical dissipative systems, subject to external random forces, are derived. The method is inspired by the old procedure found by Peierls, for deriving the canonical brackets of conservative systems, starting from an action principle. It is found that an adaptation of Peierls' method is applicable also to dissipative systems, when the friction term can be described by a linear functional of the coordinates, as is the case in the classical Langevin equation, with an arbitrary memory function. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism.

Kinetics and hydrodynamics of long-wave fluctuations under external random force

A universal phenomenological approach to study evolution of long-wave fluctuations caused by the external random force is developed. The Furutzu–Novikov theorem is generalized for non-Gaussian processes. General kinetic and hydrodynamic equations that include the effect of long-wave fluctuations are derived for the Gaussian distribution of the random force. The nonequilibrium entropy is determined in the presence of fluctuations in the system and the H-theorem is proved. It is shown that the Obukhov–Kolmogorov turbulence spectra can be found from the hydrodynamic equations of long-wave fluctuations obtained in this paper.

Characterization of the anticipated synchronization regime in the coupled FitzHugh–Nagumo model for neurons

We characterize numerically the regime of anticipated synchronization in the coupled FitzHugh–Nagumo model for neurons. We consider two neurons, coupled unidirectionally (in a master–slave configuration), subject to the same random external forcing and with a recurrent inhibitory delayed connection in the slave neuron. We show that the scheme leads to anticipated synchronization, a regime in which the slave neuron fires the same train of pulses as the master neuron, but earlier in time. We characterize the synchronization in the parameter space (coupling strength, anticipation time) and introduce several quantities to measure the degree of synchronization.

Amplitude-independent dislocation internal friction under random external actions

The amplitude-independent dislocation absorption (internal friction) is investigated under the joint action of constant and random external forces on the dislocation. The action of random forces of different types are considered with due regard for the inertial properties of the dislocation and the effect of the internal (parabolic) potential relief of the crystal. The dependences of the internal friction on the degree of correlation of random forces and the parameters of the dislocation and the medium are obtained.