Weak Noise Limit Research Articles

Kramers Rate for Thermal Plus Dichotomous Noise Applied to Ratchets.

The Kramers escape problem for an overdamped Brownian particle subjected to both dichotomous and thermal noise is studied. In the limit of weak thermal noise an asymptotically exact rate formula is obtained that is valid for general potentials and arbitrary correlation times of the dichotomous noise. Our predictions are verified by accurate numerical results and their consequences with regards to nonequilibrium potentials, resonant activation, and Brownian motors are discussed.

Oscillatory Behavior of the Rate of Escape through an Unstable Limit Cycle.

Suppose a two-dimensional dynamical system has a stable attractor that is surrounded by an unstable limit cycle. If the system is additively perturbed by white noise, the rate of escape through the limit cycle will fall off exponentially as the noise strength tends to zero. By analyzing the associated Fokker--Planck equation we show that in general the weak-noise escape rate is non-Arrhenius: it includes a factor that is periodic in the logarithm of the noise strength. The presence of this slowly oscillating factor is due to the nonequilibrium potential of the system being nondifferentiable at the limit cycle. We point out the implications for the weak-noise limit of stochastic resonance models.

Action principle in nonequilibrium statistical dynamics.

We introduce a variational method for approximating distribution functions of dynamics with a ``Liouville operator'' L^, in terms of a nonequilibriumaction functional for two independent (left and right) trial states. The method is valid for deterministic or stochastic Markov dynamics and for stationary or time-dependent distributions. A practical Rayleigh-Ritz procedure is advanced, whose inputs are a finitely parametrized Ansatz for the trial states, leading to a ``parametric action'' for their evolution. The Euler-Lagrange equations of the action principle are Hamiltonian in form (generally noncanonical). This permits a simple identification of fixed points as critical points of the parametric Hamiltonian. We also establish a variational principle for low-order statistics, such as mean values and correlation functions, by means of the least effective action. The latter is a functional of the given variable, which is positive and convex as a consequence of H\older realizability inequalities. Its value measures the ``cost'' for a fluctuation from the average to occur and in a weak-noise limit it reduces to the Onsager-Machlup action. In general, the effective action is shown to arise from the nonequilibrium action functional by a constrained variation. This result provides a Rayleigh-Ritz scheme for calculating just the desired low-order statistics, with internal consistency checks less demanding than for the full distribution. \textcopyright{} 1996 The American Physical Society.

The Effects of Adding Noise During Backpropagation Training on a Generalization Performance

We study the effects of adding noise to the inputs, outputs, weight connections, and weight changes of multilayer feedforward neural networks during backpropagation training. We rigorously derive and analyze the objective functions that are minimized by the noise-affected training processes. We show that input noise and weight noise encourage the neural-network output to be a smooth function of the input or its weights, respectively. In the weak-noise limit, noise added to the output of the neural networks only changes the objective function by a constant. Hence, it cannot improve generalization. Input noise introduces penalty terms in the objective function that are related to, but distinct from, those found in the regularization approaches. Simulations have been performed on a regression and a classification problem to further substantiate our analysis. Input noise is found to be effective in improving the generalization performance for both problems. However, weight noise is found to be effective in improving the generalization performance only for the classification problem. Other forms of noise have practically no effect on generalization.

Noise-induced effects on a non-linear oscillator

We examine various phenomena induced by white Gaussian random perturbations in the response of non-linear dynamical systems. In the first part of this work digital and analog experiments are conducted on a simple single-degree-of-freedom oscillator with a piecewise linear restoring force and harmonic forcing. They reveal that small noise perturbations can give rise to large deviations of the response which ultimately lead to transitions between the coexisting attractors of the unperturbed system. These transitions are analyzed probabilistically by determining the mean time spent by the trajectories to exit from the basin of a given attractor. By determining the relationship between mean first-exit time and noise intensity, it is found that each attractor can be characterized by an activation energy which yields a measure of its relative stability. We also find that, even in the case of a single attractor, weak noise can induce large excursions to sets of the state space (chaotic semi-attractor) which are otherwise globally repelling in the absence of noise. In the second part of this work, some results obtained numerically are shown to be predicted theoretically by the use of asymptotic analyses of the randomly perturbed response of dynamical systems in the limit of weak noise. These techniques provide a generalization of the notion of potential to non-potential, non-equilibrium systems. In particular, the notion of activation energy is verified theoretically, and its determination may be possible without massive computer simulations.

Optimum paths for systems subject to internal noise

We formulate the stochastic dynamics of a particle subject to internal non-white (coloured) noise in terms of path-integrals. In the simplest case, where the noise is exponentially correlated, the weak-noise limit is characterised by optimum paths which are given by third-order differential equations. In contrast to system subjects to white noise or external coloured noise, the overdamped limit for these systems is singular. We analyse the origin of this behaviour. The whole formalism is generalised to more general noise processes and the essential features are shown to be similar to the exponentially correlated case.

A test for two Fokker-Planck modellings of a master equation

The kind of two Fokker-Planck equation modellings of a non-equilibrium system, defined by a two-variable master equation, is studied. We compute explicitly their associated stationary potentials in the weak noise limit, and we compare them with the exact one coming from the master equation. We Find that a recently proposed Fokker-Planck equation is far better than the other coming from the commonly used Kramers-Moyal expansion.

Use of Hamiltonian mechanics in systems driven by colored noise.

The evaluation of the path-integral representation for stochastic processes in the weak-noise limit shows that these systems are governed by a set of equations which are those of a classical dynamics. We show that, even when the noise is colored, these may be put into a Hamiltonian form which leads to better insights and improved numerical treatments. We concentrate on solving Hamilton's equations over an infinite time interval, in order to determine the leading order contribution to the mean escape time for a bistable potential. The paths may be oscillatory and inherently unstable, in which case one must use a multiple shooting numerical technique over a truncated time period in order to calculate the infinite time optimal paths to a given accuracy. We look at two systems in some detail: the underdamped Langevin equation driven by external exponentially correlated noise and the overdamped Langevin equation driven by external quasi-monochromatic noise. We deduce that caustics, focusing and bifurcation of the optimal path are general features of all but the simplest stochastic processes.

Surmounting fluctuating barriers: A simple model in discrete time.

The escape rate is calculated for a piecewise linear map in the presence of two additive weak noises: Gaussian white noise (thermal noise) and Ornstein-Uhlenbeck noise (barrier fluctuations). A transition state theory yields the exact escape rate in the weak noise limit. By including the dominant finite-noise corrections we find very good agreement with numerical simulations. Of particular interest is the behavior of the escape rate k(\ensuremath{\tau}) as a function of the correlation time \ensuremath{\tau} of the Ornstein-Uhlenbeck noise. We show that the qualitative behavior of k(\ensuremath{\tau}) at any fixed thermal noise intensity is the same as in the absence of thermal noise. Whether k(\ensuremath{\tau}) exhibits a local maximum (resonant activation) is determined by the detailed \ensuremath{\tau} dependence of the colored-noise intensity: e.g., resonant activation is found (at a correlation time of order 1) if the integral autocorrelation of the colored noise is kept \ensuremath{\tau} independent but not if the variance is kept \ensuremath{\tau} independent. In the continuous-time limit neither of these cases shows resonant activation.

Weak-noise limit of stochastic resonance.

The autocorrelation function of a periodically driven bistable system is considered in the limit of weak noise. We predict a broad window in the Fourier spectrum with a universal (power law) decay of the spectral density, accompanied by dips of finite width at the second and fourth harmonics of the driving frequency. For higher driving frequencies the dips are replaced by peaks of finite though extremely narrow width. Analytical results are confirmed by numerical studies and are used to analyze previous experimental data.

Escape problem for irreversible systems.

The problem of noise-induced escape from a metastable state arises in physics, chemistry, biology, systems engineering, and other areas. The problem is well understood when the underlying dynamics of the system obey detailed balance. When this assumption fails many of the results of classical transition-rate theory no longer apply, and no general method exists for computing the weak-noise asymptotic behavior of fundamental quantities such as the mean escape time. In this paper we present a general technique for analyzing the weak-noise limit of a wide range of stochastically perturbed continuous-time nonlinear dynamical systems. We simplify the original problem, which involves solving a partial differential equation, into one in which only ordinary differential equations need be solved. This allows us to resolve some old issues for the case when detailed balance holds. When it does not hold, we show how the formula for the asymptotic behavior of the mean escape time depends on the dynamics of the system along the most probable escape path. We also present results on short-time behavior and discuss the possibility of [ital focusing] along the escape path.

TIME-DEPENDENT SOLUTION OF THE FOKKER-PLANCK EQUATION OF NONLINEAR NONPOTENTIAL SYSTEMS

The time-dependent problem of Fokker-Planck equation of the nonlinear systems without detailed balance is solved in the weak noise limit by using the Ω-expansion of the Green function. We further reveal the possibility of the dimension reduction and some scaling relations in the relaxation of the system from the unstable state to the stable state.

Potential of systems subjected to weak noise with large correlation time.

Colored-noise-driven bistable systems are considered in the weak-noise limit D\ensuremath{\rightarrow}0 and long-correlation-time limit \ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\infty}. The analytic solution of the two-dimensional Fokker-Planck equation is worked out. The theoretical result is compared with a numerical simulation.

Effect of noise and perturbations on limit cycle systems

This paper demonstrates that the influence of noise and of external perturbations on a nonlinear oscillator can vary strongly along the limit cycle and upon transition from limit cycle to stationary point behaviour. For this purpose we consider the role of noise on the Bonhoeffer-van der Pol model in a range of control parameters where the model exhibits a limit cycle, but the parameters are close to values corresponding to a stable stationary point. Our analysis is based on the van Kampen approximation for solutions of the Fokker-Planck equation in the limit of weak noise. We investigate first separately the effect of noise on motion tangential and normal to the limit cycle. The key result is that noise induces diffusion-like spread along the limit cycle, but quasistationary behaviour normal to the limit cycle. We then describe the coupled motion and show that noise acting in the normal direction can strongly enhance diffusion along the limit cycle. We finally argue that the variability of the system's response to noise can be exploited in populations of nonlinear oscillators in that weak coupling can induce synchronization as long as the single oscillators operate in a regime close to the transition between oscillatory and excitatory modes.

Stochastic Landau equation with time-dependent drift.

The stochastic differential equation ${\mathrm{\ensuremath{\tau}}}_{0}$${\mathrm{\ensuremath{\partial}}}_{\mathit{t}}$A=\ensuremath{\epsilon}(t)A-${\mathit{g}}_{3}$${\mathit{A}}^{3}$+f\ifmmode\bar\else\textasciimacron\fi{}(t), where f\ifmmode\bar\else\textasciimacron\fi{}(t) is Gaussian white noise, is studied for arbitrary time dependence of \ensuremath{\epsilon}(t). In particular, cases are considered where \ensuremath{\epsilon}(t) goes through the bifurcation of the deterministic system, which occurs at \ensuremath{\epsilon}=0. In the limit of weak noise an approximate analytic expression generalizing earlier work of Suzuki [Phys. Lett. A 67, 339 (1978); Prog. Theor. Phys. (Kyoto) Suppl. 64, 402 (1978)] is obtained for the time-dependent distribution function P(A,t). The results compare favorably with a numerical simulation of the stochastic equation for the case of a linear ramp (both increasing and decreasing) and for a periodic time dependence of \ensuremath{\epsilon}(t). The procedure can be generalized to an arbitrary deterministic part ${\mathrm{\ensuremath{\partial}}}_{\mathit{t}}$A=D(A,t)+f\ifmmode\bar\else\textasciimacron\fi{}(t), but the deterministic equation may then have to be solved numerically.

Collective synchronization of pulse-coupled oscillators and excitable units

A phase model is proposed and studied for a population of integrate-and-fire-type oscillators and excitable units with pulsatile coupling. The units are subject to external noise, and their mutual connection is all-to-all. An evolution equation is derived for the phase distribution in a closed form. Its steady-state analysis in the limit of weak noise shows the possibility of collective bistability. The evolution equation is simplified drastically when the coupling as well as noise is sufficiently weak. This reduction of equation enables analytical calculation of various quantities. Among others, the stability of the steady distribution is analyzed explicity, and the onset of collective oscillations is described in terms of the Hopf bifurcation. It is also shown that the present model is improved in some respects by introducing absolute refractoriness.

Stationary and dynamic solutions of stochastic relaxation oscillators

Two different approaches are proposed to obtain explicit solutions for stochastic relaxation oscillator problems in the weak noise limit. The first method generalizes the idea of the cumulant expansion. It does not presuppose an analytical treatment of the deterministic motion. It is however restricted to the discussion of stationary situations. In the second method an adiabatic elimination of irrelevant variables allows for the computation of time dependent solutions. It can be carried through only if the deterministic limit cycle is known analytically. As special examples the stationary solutions of the stochastic van der Pol oscillator and time dependent solutions of a simple one dimensional model system have been obtained.

Two kinds of singularities in planar differential systems and the response to external forces.

A Lyapunov function is defined for two-dimensional differential systems. The relationship between the Lyapunov function and the stationary probability distribution of the corresponding Fokker-Planck equations in the weak-noise limit is analyzed. Based on the structure of the Lyapunov function we classify two types of singularities. The classification of different kinds of singularities is shown to be significant for the intrinsical dynamics of the deterministic flow. The bifurcation behavior of periodically forced planar systems that contain distinctive kinds of singularities in the autonomous cases is investigated. It is found that the responses of planar systems to the external force, i.e., the resulting bifurcation sequences, are substantially different if the trajectories circle around different types of singularities.

Path integrals and non-Markov processes. III. Calculation of the escape-rate prefactor in the weak-noise limit.

Path integrals and non-Markov processes. III. Calculation of the escape-rate prefactor in the weak-noise limit.

Steepest-descent approximation of stationary probability distribution of systems driven by weak colored noise.

One-dimensional systems driven by colored external noise are studied in the weak-noise limit. The steepest-descent approximation is employed to obtain the stationary probability distribution in two-dimensional space. The approach is expected to be applicable to a large class of two-dimensional Fokker-Planck equations with noninvertible diffusion. A systematical expansion to improve the steepest-descent approximation is suggested.