Resetting dynamics in a system with quenched disorder

Abstract

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.

Figures (10)

• Figure 1: (a) Schematic representation of our model: the particle on site $i$ at time $t$ hops at time $t+1$ to either the right or left-neighboring sites with site-dependent probabilities $q_i$ and $p_i=1-q_i$, respectively. In addition, stochastic resetting events (red arrow) reset the particle instantaneously to the origin or to a previously visited site. (b) For a given realization of the disorder and an initial condition, the particle's trajectory exhibits alternating phases of growth and sudden resets to the origin. Here, $\tilde{\Delta}(t)$ represents the displacement of the particle with respect to the initial position.
• Figure 2: (a) Schematic of attachment-detachment kinetics of a tubulin unit in a mictotubule filament by probability $q_i$ and $p_i$, respectively. (b) Growth and catastrophe dynamics of microtubule length with time in the experiments of Ref. gardner2011depolymerizing. (c)The probability distribution function $P(p_i)$, defined over the interval $p_i \in [0.03, 0.1]$, shows a higher density of values near $p_i = 0.03$, indicating a strong bias in the lattice. (d) We plot the absolute displacement $\tilde{\Delta}(t)$ of a particle from its initial position with time for the biased case. Ten such trajectories are plotted together. (e) Probability distribution function, for the less biased case where the allowed values of $p_i \in [0.36,0.64]$. (f) Plot of absolute displacement $\tilde{\Delta}(t)$ as a function of time for ten representative trajectories in the presence of unbiased disorder.
• Figure 3: Displacement variation with and reset length distributions under stochastic resetting to the initial position. (a) Time evolution of particle displacement under stochastic resetting in the strongly-biased case ($p_i \in [0.03, 0.1]$). (b) Distribution of reset lengths for the strongly-biased case. The red dashed line indicates a Gamma fit with parameters $n = 3.00$ and $r = 0.0067$. (c) Displacement dynamics for the less-biased case ($p_i \in [0.36, 0.64]$) under stochastic resetting. (d) Reset length distribution for the less-biased case, fitted with a Gamma distribution (fit parameters: $n = 1.39$, $r = 0.04$).
• Figure 4: Displacement and reset length distributions under stochastic resetting to the uniform position. (a) Temporal evolution of particle displacement in the strongly biased regime ($p_i \in [0.03, 0.1]$) under stochastic resetting. (b) Corresponding reset length distribution in the strongly biased case, with a Gamma fit shown by the red dashed line ($n = 3.68$, $r = 0.0043$). (c) Displacement dynamics under stochastic resetting in the less biased regime ($p_i \in [0.36, 0.64]$). (d) Reset length distribution for the less biased case, fitted to a Gamma distribution with parameters $n = 1.76$ and $r = 0.029$.
• Figure 5: (a) Schematic representation of first passage times ($t_f$) for a particle at $d$. (b-e) Log-log plots of the first passage time distribution, $P(t_f)$, for $d=100$, shown for different resetting cases. In all the panels, we fit the initial part, i.e., the small time part, by a power-law $t_f^{a}$, and the long time part by $e^{-b t_f}$. (b) Less biased and resetting to the initial position: $a=-1.24$, $b=0.0048$ (c) Strongly biased and resetting to the initial position: $a=0.29$, $b=0.0039$(d) Less biased and resetting to a uniform position: $a=-1.29$, $b=0.0038$. (e) Less biased and resetting to a uniform position: $a=0.49$, $b=0.0040$.