Resetting dynamics in a system with quenched disorder
Although resetting has widespread applicability, applying it to the dynamics in the presence of spatial quenched disorder, which is essential in many physical problems, is challenging. In this study, we consider a well-known one-dimensional model of particle hopping on a lattice with quenched disorder in the form of site-dependent hopping probabilities, drawn from a power-law distribution, and apply the resetting formalism. As a physical example, we recast the growth dynamics of microtubules with sudden catastrophic disassembly events as a resetting dynamics. We consider two distinct regimes for growth dynamics: a strongly biased case and a less biased case. Motivated by experimental results, we take a Gamma distribution for the resetting time. Our results show that occasional disassembly events are crucial for the experimentally observed distribution of reset (or catastrophe) lengths. We also analyze steady-state distributions under different resetting protocols-resetting to the initial position versus a random site. We also investigate the distribution of first-passage times to a fixed distance following reset. Finally, by considering other resetting probability distributions, we identify a regime where the mean displacement grows as slowly as $\log^2 t$. We also elucidate the role of disorder in the system properties under the resetting dynamics. Our study paves the way to treat the dynamics of complex physical systems using resetting.