Proc. SPIE. 5845, Noise in Complex Systems and Stochastic Dynamics III
KEYWORDS: Signal to noise ratio, Particles, Diffusion, Complex systems, Interference (communication), Differential equations, Stochastic processes, Stereolithography, Systems modeling, Correlation function
We analyze periodically driven bistable systems by two different approaches. The first approach is a linearization of the stochastic Langevin equation of our system by the response on small external force. The second one is based on the Gaussian approximation of the kinetic equations for the cumulants. We obtain with the first approach the signal power amplification and output signal-to-noise ratio for a model piece-wise linear bistable potential and compare with the results of linear response approximation. By using the second approach to a bistable quartic potential, we obtain the set of nonlinear differential equations for the first and the second cumulants.
We investigate an overdamped Brownian motion in symmetric periodic potential switching by Markovian dichotomous noise between two configurations. Second configuration differs from the first one by the half of spatial period displacement when the maxima of a potential profile become the minima and vice versa. We establish a validity of the formula for effective diffusion constant previously obtained in the case of fixed periodic potentials and, thus, reduce the problem to calculation of mean first-passage times (MFPTs). At the same time the MFPT in this formula should be substituted for the semi-sum of MFPTs regarding two initial configurations of potential profile. A set of equations for MFPTs in flipping potentials can be solved for the sawtooth periodic potential. As a result, after cumbersome calculations we obtain the exact complex expression for effective diffusion coefficient of Brownian particles in such a medium which is valid for arbitrary mean frequency of potential switchings. We detect the acceleration of diffusion in comparison with the case of fixed sawtooth potential profile as it was demonstrated for a symmetric periodic potential modulating by external Gaussian white noise.
We investigate an overdamped Brownian particle moving in: (a) a dichotomously fluctuating metastable potential; (b) a random fluctuating periodic potential. For piece-wise linear potential we obtain for case (a) the exact average lifetime and the mean first passage time as a function of the potential parameters, the noise intensity and the mean frequency of switchings of the dichotomous noise. We find noise enhanced stability (NES) in the system investigated. The parameter regions of the fluctuating potential where NES effect can be observed are analytically derived. For case (b) we consider a symmetric periodic potential modulated by white noise. We obtain for such a potential the same relationship between effective diffusion coefficient of Brownian particles and the mean first-passage time, discovered previously for fixed periodic potential (see ref. 3). The phenomenon of diffusion acceleration in comparison with free particle case has been found for arbitrary potential profile. The effective diffusion coefficients for sawtooth, sinusoidal and piecewise parabolic potentials are calculated in closed analytical form.
We show that the increment of generalized Wiener process (random process with stationary and independent increments) has the properties of a random value with infinitely divisible distribution. This enables us to write the characteristic function of increments and then to obtain the new formula for correlation of the derivative of generalized Wiener process (non-Gaussian white noise) and its arbitrary functional. IN the context of well-known functional approach to analysis of nonlinear dynamical systems based on a correlation formulae for nonlinear stochastic functionals, we apply this result for derivation of generalized Fokker-Planck equation for probability density. We demonstrate that the equation obtained takes the form of ordinary Fokker-Planck equation for Gaussian white noise and, at the same time, transforms in the fractional diffusion equation in the case of non-Gaussian white noise with stable distribution.