Stochastic Langevin equations: Markovian and non-Markovian dynamics

Abstract

Non-Markovian stochastic Langevin-like equations of motion are compared to their corresponding Markovian (local) approximations. The validity of the local approximation for these equations, when contrasted with the fully nonlocal ones, is analyzed in detail. The conditions for when the equation in a local form can be considered a good approximation are then explicitly specified. We study both the cases of additive and multiplicative noises, including system-dependent dissipation terms, according to the fluctuation-dissipation theorem.

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

“…It has been shown that the stationary probability distribution function (PDF) is independent of magnitudes of additive and multiplicative noises and of the relaxation time of colored noise [20][21][22]29], although the response to applied input depends on noise parameters [29]. 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 consider a system of a Brownian particle coupled to a bath consisting of N -body uncoupled oscillators, which is described by the generalized CL model [20][21][22]29],…”

“…By using the generalized CL model including a nonlinear system-bath coupling, we may obtain the non-Markovian Langevin equation with multiplicative noise which preserves the FDR [18,19]. The nature of nonlinear dissipation and multiplicative noise has been recently explored with renewed interest [20][21][22]. Nonlinear dissipation and multiplicative noise have been recognized as important ingredients in several fields such as mesoscopic scale systems [23,24] and ratchet problems [25][26][27][28].…”

“…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].…”

“…In Sec. II, we briefly summarize the non-Markovian Langevin equation derived from the generalized CL model with OU colored noise [20][21][22]29].…”