Power spectra of TASEPs with a localized slow site

Abstract

The totally asymmetric simple exclusion process (TASEP) with a localized defect is revisited in this article with attention paid to the power spectra of the particle occupancy N (t). Intrigued by the oscillatory behaviors in the power spectra of an ordinary TASEP in high/low density phase(HD/LD) observed by Adams et al. (2007 Phys. Rev. Lett. 99 020601), we introduce a single slow site with hopping rate q < 1 to the system. As the power spectrum contains time-correlation information of the particle occupancy of the system, we are particularly interested in how the defect affects fluctuation in particle number of the left and right subsystems as well as that of the entire system. Exploiting Monte Carlo simulations, we observe the disappearance of oscillations when the defect is located at the center of the system. When the defect is off center, oscillations are restored. To explore the origin of such phenomenon, we use a linearized Langevin equation to calculate the power spectrum for the sublattices and the whole lattice. We provide insights into the interactions between the sublattices coupled through the defect site for both simulation and analytical results.

Fluctuations in meta-population exclusion processes

“…This is motivated by the following results: (i) In the limit Ω = 1 the model reduces to the standard TASEP; (ii) As we will show, to lowest order in the expansion in Ω −1/2 (i.e. taking the limit Ω → ∞) the model reproduces the mean field equations usually written down for the TASEP, thus the model interpolates between the conventional TASEP and its mean field theory; (iii) Taking the expansion to next-to-leading order reproduces the Langevin dynamics proposed and studied in [24][25][26]; (iv) Crucially our approach is fully controlled, and allows one to state the limitations of the description in terms of Gaussian random processes; (v) The relevant coefficients in the Langevin (or Ornstein-Uhlenbeck) dynamics are derived from first principles, they do not need to be obtained by a fit from simulation data, our analysis may hence also serve as a starting point for a better understanding of the 'serious renormalization' of diffusion constants and noise strengths reported in [24]. We apply these methods first to the basic TASEP and then to what is referred to as the constrained TASEP [25,27].…”

“…Carrying out a systematic expansion in powers of Ω −1/2 one derives the deterministic limiting equations in the leading order of the expansion, and obtains a set of Langevin equations describing fluctuations about this deterministic limit in the sub-leading order of the cell-size expansion. Such Langevin equations are not new for the description of exclusion processes, they have for example been formulated and used in [24][25][26]. These existing studies however take a mostly phenomenological approach, we feel that the angle taken here provides a more systematic derivation of an effective Langevin dynamics from first principles, and using well-controlled expansion techniques.…”

“…We are therefore hopeful that the approach taken here might be useful to investigate other variants of the exclusion processes, for example twolane models (see e.g. [47]) or models with spatial heterogeneities and/or individual 'slow' sites [26]. Cell-size expansion techniques may also be considered for models * It should be noted that this agreement cannot generally be expected to hold very close to phase boundaries.…”

“…Such Langevin equations are not new for the description of exclusion processes. They have, for example, been formulated and used in [24]- [26]. These existing studies, however, take a mostly phenomenological approach.…”

“…The mathematics required to carry out exact analyses on the other hand is intricate, so that it is desirable to develop approaches which systematically improve on mean field descriptions, but do not require an overly involved mathematical apparatus. Fluctuation effects in ASEP models have for example been studied by an effective Langevin description in [23][24][25][26][27]. While successful in describing the spectral properties of fluctuations about the mean field theory the precise form of these equations is often not derived from first principles.…”