Bistable flow driven by coloured gaussian noise: A critical study

Abstract

A one-dimensional bistable flow driven by additive, exponentially correlated Gaussian noise is considered. The small relaxation time Fokker-Planck approximations, widely used in the recent literature, are derived and possible shortcomings of those approximation schemes are discussed. In particular, it is pointed out that higher order nonFokker-Planck type contributions are generally of the same order as the Fokker-Planck terms. In principle, those contributions cannot be neglected if the global behavior of the probability solutions is to be described accurately. The result for the activation rate (Arrhenius factor), as evaluated from the approximative Fokker-Planck schemes, does not coincide in leading order in the correlation time z of the noise with a computer simulation of the rate at low noise level. This result indicates that the wings of the stationary probability /~(x) are in leading order in ~ not recovered correctly from the approximative Fokker-Planck schemes. Some implications of our study for adiabatic elimination procedures are also discussed.

Stochastic resonance in bistable systems with nonlinear dissipation and multiplicative noise: A microscopic approach

“…Refs. [6,7]. However, they exhibit no signs of the SR against τ , which is in contrast with the result in mutually related by the FDR.…”

“…In particular, the τ dependence of SR in the item (ii) is different even qualitatively from previous results: a monotonous decrease of SNR with τ [6][7][8] and the minimum of SNR at intermediate τ [9]. It would be interesting to apply the present approach to a study of relevant phenomena in bistable systems such as the first passage time and resonant activation.…”

“…(39) reduces to the Markovian Langevin equation given bÿ In previous studies [6,7], the overdamped Langevin equation given bẏ…”

“…In Sec. IV, we examine a validity of the Markovian Langevin equations with the local approximation, which have been widely employed in previous studies on SR [4,[6][7][8]. Sec.…”

“…This is in contrast with previous studies [30][31][32] which show that the stationary PDF of the Langevin model for the harmonic potential is Gaussian or non-Gaussian, depending on magnitudes of additive and multiplicative noises. We expect that when the generalized CL model with a nonlinear coupling [20][21][22]29] is applied to a bistable system, properties of the calculated SR may be rather different from those having been derived by the phenomenological Langevin models [4,[6][7][8][9][10][11][12][13][14]. It is the purpose of the present paper to make a detailed study on SR of an open bistable system described by the microscopic, generalized CL model with nonlinear nonlocal dissipation and state-dependent diffusion [29].…”