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Local Resetting in a Bidirectional Transport System

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Abstract

Inspired by different stochastic mechanisms, such as the two-sided motion of ribosomes seen during the initiation of mRNA translation, which is backed by their decay, we investigate a totally asymmetric simple exclusion process with open boundaries in a bidirectional setting where two oppositely charged species of particles move opposite to each other and locally reset to the respective entry site. The steady-state characteristics, such as density profiles and phase diagrams, are investigated theoretically under the mean-field framework. The introduction of resetting into the system produces non-trivial effects in the form of two novel asymmetric phases that appear in the phase diagram. The system possesses several different combinations of symmetric phases as well as asymmetric phases for different resetting rates. A rich behavior is observed in the system, emphasizing the occurrence of spontaneous symmetry-breaking phenomena even in the small resetting regime. Moreover, the significance of the resetting rate is analyzed on the domain wall, and it is found that one of the stationary phases with a localized domain wall vanishes for a substantial resetting rate. Due to the interaction of both species at the boundaries, the consequences of the resetting dynamics on the boundary densities are also investigated. All the findings, including finite-system size, are thoroughly validated by the Monte Carlo simulations.

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Author notes

  1. N. Bhatia and A. K. Gupta have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, 140001, Punjab, India

    Nikhil Bhatia & Arvind K. Gupta

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  1. Nikhil Bhatia
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Correspondence to Arvind K. Gupta.

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Appendix A

Appendix A

This section provides an alternative approach to obtaining the stationary state density solution corresponding to each species with the help of Eq. (18). The primary advantage of utilizing this scheme is its reduced complexity in contrast to the computation of conditions for the existence of asymmetric phases discussed in Sect. 4.2. Secondly, this scheme can be readily extended if the model is generalized to incorporate additional dynamics where analytical approaches may not be applicable. The estimate of the (±) species of particle density is represented numerically by the notation \(\rho _{i,n}^{\pm }\) at \(i^{\text {th}}\) lattice site, \(n^{\text {th}}\) time step and in the limit n \(\rightarrow \infty \), the stationary-state solution is assured. The finite-difference equation for the continuum density evolution Eq. (12) in the bulk utilizing the forward-in-time and central-in-space (FTCS) scheme is given by:

$$\begin{aligned}{} & {} \rho _{i,n+1}^{+}=\rho _{i,n}^{+}+ \triangle t^{'}\Bigg (\frac{\epsilon }{2} \Big (\frac{\rho _{i+1,n}^{+}-2\rho _{i,n}^{+}+\rho _{i-1,n}^{+}}{\triangle x^2}\Big )+ \Big (\frac{\rho _{i+1,n}^{+}-\rho _{i-1,n}^{+}}{2\triangle x}\Big ) (2\rho _{i,n-1}^{+})\nonumber \\{} & {} -R(1-\rho _1^{+}-\rho _1^{-})\rho _{i,n}^{+}\Bigg ), \end{aligned}$$
(A1)

where the spatial (\(\triangle x=1/L\)) and temporal (\(\triangle t^{'}\)) grid spacing variables adhere to the condition \(\triangle t^{'}/\triangle x^2\le 1\) to ensure the stability of the scheme against small perturbations. As postulated in the model, explicit determination of boundary conditions in the continuum limit is unattainable because interactions between species and particle resetting from the bulk exclusively occur at the boundaries. At boundaries, we employ Eq. (5) and Eq. (6) to derive finite-difference scheme provided by:

$$\begin{aligned} \rho _{1,n+1}^{+}=\rho _{1,n}^{+}+L \triangle t^{'} \bigg ((1-\rho _{1,n}^{+}-\rho _{1,n}^{-})\Big (\alpha + r\displaystyle \sum _{i=2}^L \rho _{i,n}^{+} \Big )- \rho _{1,n}^{+}(1-\rho _{2,n}^{+})\bigg ), \end{aligned}$$
(A2)

and

$$\begin{aligned} \rho _{L,n+1}^{+}=\rho _{L,n}^{+}+L \triangle t^{'} \bigg ( \rho _{L-1,n}^{+}(1-\rho _{L,n}^{+}) -\rho _{L,n}^{+}\Big (\beta +r(1-\rho _{1,n}^{+}-\rho _{1,n}^{-})\Big )\bigg ). \end{aligned}$$
(A3)

Similarly, comparable finite-difference equations for the negative particles may also be constructed by simply switching the boundary conditions.

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Bhatia, N., Gupta, A.K. Local Resetting in a Bidirectional Transport System. J Stat Phys 191, 79 (2024). https://doi.org/10.1007/s10955-024-03298-5

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