The response of subdiffusive bistable fractional Fokker–Planck systems to rectangular signals

Abstract

We study the response of one-dimensional subdiffusive fractional Fokker–Planck systems with a general confining potential, when it is perturbed from its stationary state by a time-dependent non-sinusoidal driving force. Three types of rectangular driving signals have been investigated: a rectangular pulse, a periodic telegraph signal and a generalized telegraph signal with a fractional duty cycle. We derive analytic expressions for the linear response and the input energy in one period of the driving signal. In particular, for signals with a long period, we obtain several asymptotic results concerning the wave form of the response and the stochastic energetics. Numerical results for representative symmetric, as well as asymmetric, subdiffusive bistable systems are presented and discussed.

Observing fluctuating spectral density of subdiffusive overdamped Brownian particles in periodic potentials

“…1 to be a weak periodic telegraph signal [9] with period 2T , whose Fourier expansion is given by To confirm the accuracy of the long-time linear response calculated by the method of moments, in figures 1 and 2 we also show the linear response results simulated with the Box-Muller algorithm [30] for σ = 1 based on the corresponding Langevin equation ẋ = − sin x + ε(t) + ξ(t).…”

“…Since easier in analytic or numerical treatment than other modeling methods, fractional FPE has attracted extensive attention from physicians to mathematicians etc. As far as the time fractional FPE is concerned, several existing numerical schemes such as direct simulation [6], matrix continuation fractions [7], subordination process simulation [8], and perturbation analysis [9] have been proposed, but mainly concentrate on calculating the time-domain properties such as probability distributions or time history. Although a more recent scheme introduced in [10] aims at investigating the frequency property such as the fluctuating spectral density of fractional Klein-Kramers equations with a bistable potential, the related result for more complex potentials is still in a scarcity.…”

Linear response characteristics of time-dependent time fractional Fokker–Planck equation systems

“…Based on this consideration, we will focus on the linear response characteristic of time-dependent time fractional Fokker-Planck equation systems in this paper. When the external field is time modulated, to our best knowledge, there are two types of time-dependent time fractional Fokker-Planck equations in the existing literatures [26][27][28][29][30][31][32][33][34]. The first type (referred to as the time-fractional system (I)), which origins from the continuous limit of continuous time random walk with the Boltzmann jumping probabilities modulated by a time dependent external field [12,27,33,34], has the form…”