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

Abstract

The aim of this paper is to develop a simple and efficient method for observing the fluctuating spectral density of subdiffusive Brownian motion in an overdamped periodic potential for exploring the subdiffusive property in frequency domain. Based on the general frame of linear response theory for subdiffusive fractional Fokker–Planck equation systems, an explicit relation between fluctuating spectral density and linear dynamical susceptibility is deduced, and then a method of moments based on the expansion of trigonometric functions is proposed for calculating the linear dynamic susceptibility. With the linear dynamic susceptibility available, the fluctuating spectral density is finally obtained. The numerical results demonstrate that subdiffusion weakens coherent oscillations in the periodic system, but enhances aperiodic components. Our observation embodies the fact of the Mittag–Leffler residence time distribution with an infinite mean in the subdiffusive process from the frequency domain.

A Study on Stochastic Resonance in Biased Subdiffusive Smoluchowski Systems within Linear Response Range

“…We remark that for the unbiased potential in Refs. [19][20], a seven-term fractional differential recurrent relation was involved; while our recurrent relation (15) has six terms more. Therefore, the numerical procedure here can be seen as an extension towards the biased potential.…”

Coherence resonance in subdiffusive fractional Klein-Kramers periodic potential systems without a bifurcation precursor

“…There is not any kind of bifurcation or excitability existing in the system under study, so the investigation provides a new evidence for CR. The physical mechanism of CR in the underdamped system lies in that noisecontrolled nonzero coherent characteristic frequency peak undergoes a nontrivial evolution, which is an essential difference from the overdamped counterpart [27]. Noting that normal diffusive or subdiffusive Brownian motion in periodic potentials has a fundamental importance in many physical problems, it is expected that the CR disclosed in this letter can find use in those physical and engineering…”

“…Among the numerous ways such as fractional Brownian motion [18], generalized diffusion equation [19], continuous time random walk method [20,21], and generalized Langevin equation [22] for modeling anomalous diffusion, it is well known that time fractional Fokker-Planck equation approach is more easy in analytic or numeric (a) E-mail: kangyanmei2002@yahoo.com.cn treatment. In this regard, various numerical techniques including direct simulation [23], matrix continued-fraction method [24,25], subordination process simulation [26] and method of moments [27] have been proposed for investigating the diffusion and spectral properties of subdiffusive processes.…”

“…Considering underdamped Brownian motion in periodic potentials has much importance in modeling problems related to superionic conductor, Josephson tunneling junction, phase-locked loop etc. [28], the purpose of this letter is to extend the method of moments from a subdiffusive fractional periodic Smoluchowski equation system [27] to a subdiffusive time fractional periodic Klein-Kramers (KK) equation system for investigating the corresponding fluctuating spectral density, and then with the fluctuating spectral density available, to clarify whether CR occurs in normal diffusion and how subdiffusion has an effect on it.…”

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

“…Additionally, for the sake of comparison, we will also revisit the linear response characteristic of the system (II) [11]. Although equation ( 3) is obtained under somewhat unrealistic physical assumption, the linear response characteristics of the system (II) have been shown beneficial for accessing the spectral statistic of spontaneous fluctuations involved in the corresponding time-independent counterpart [37][38][39].…”