Random walk with resetting in a 1D chain

TitleRandom walk with resetting in a 1D chain
Publication TypeJournal Article
Year of Publication2020
AuthorsChristophorov, LN
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
Date Published8/2020

If the classical model of random walks is added with the stochastic resetting to the starting point, then the whole process acquires new nontrivial features. In particular, there appears a non-equilibrium steady state. In addition, the mean first passage time (which is infinite in the absence of restarts) becomes finite and can be optimized by choosing a proper mean intermittence frequency r. It is shown that, in the case of random walks on the nodes of a one-dimensional chain, these effects essentially differ from their analogs within the classical continuous diffusion model. In particular, the asymptotes of the dependences of stationary node populations on r change from exponential to power ones. Similar qualitative and quantitative distinctions take place for the mean first passage time as well. In the case of a finite chain, the interesting effect of emergence and disappearance of a possibility of the minimization of this time, depending on the distance to a defined target, shows up.

Keywordsfirst passage time, low-dimensional lattices, random walk, stochastic resetting

1. Reuveni, S., Urbakh, M. & Klafter J. (2014). Role of substrate unbinding in Michaelis-Menten enzymatic reactions. Proc. Natl. Acad. Sci. USA, 111, pp. 4391-4396. https://doi.org/10.1073/pnas.1318122111
2. Christophorov, L. N. (2019). Influence of substrate unbinding on kinetics of enzymatic catalysis. Dopov. Nac. akad. nauk Ukr., No. 1, pp. 40-46 (in Ukrainian). http://doi.org/10.15407/dopovidi2019.01.040
3. Evans, M. R. & Majumdar, S. N. (2011). Diffusion with stochastic resetting. Phys. Rev. Lett., 106, pp. 160-601. https://doi.org/10.1103/PhysRevLett.106.160601
4. Lu, H. P., Xun, L. & Xie, X. S. (1998). Single-molecule enzymatic dynamics. Science, 282, pp. 1877-1882. https://doi.org/10.1126/science.282.5395.1877
5. Reuveni, S. (2016). Optimal stochastic restart renders fluctuations in first passage times universal. Phys. Rev. Lett., 116, pp. 170601. https://doi.org/10.1103/PhysRevLett.116.170601
6. Majumdar, S. N., Pal, A. & Schehr, G. (2020). Extreme value statistics of correlated random variables: A pedagogical review. Phys. Reports, 840, pp. 1-32. https://doi.org/10.1016/j.physrep.2019.10.005
7. Bateman, H. (1954). Tables of integral transforms. V. 1, p. 182. New York: McGraw-Hill.
8. Christophorov, L. N. (2020). On the velocity of enzymatic reactions in Michaelis-Menten-like schemes (ensemble and single-molecule versions). Ukr. J. Phys., 65, pp. 412-418. https://doi.org/10.15407/ujpe65.5.412
9. Christophorov, L. N., Zagorodny, A. G. (2017). Peculiarities of migration and capture of a quantum particle in a chain with traps. Chem. Phys. Lett., 682, pp. 77-81. http://dx.doi.org/10.1016/j.cplett.2017.06.010