Revisiting Kinetic Monte Carlo Algorithms for Time-dependent Processes: from open-loop control to feedback control

TL;DR

A single-step-trajectory probability analysis to time-dependent stochastic systems and a novel feedback-controlled kinetic Monte Carlo algorithm that accurately captures the dynamics systems controlled by an external signal based on the measurements of the system's state are presented.

Abstract

Simulating stochastic systems with feedback control is challenging due to the complex interplay between the system's dynamics and the feedback-dependent control protocols. We present a single-step-trajectory probability analysis to time-dependent stochastic systems. Based on this analysis, we revisit several time-dependent kinetic Monte Carlo (KMC) algorithms designed for systems under open-loop-control protocols. Our analysis provides an unified alternative proof to these algorithms, summarized into a pedagogical tutorial. Moreover, with the trajectory probability analysis, we present a novel feedback-controlled KMC algorithm that accurately captures the dynamics systems controlled by external signal based on measurements of the system's state. Our method correctly captures the system dynamics and avoids the artificial Zeno effect that arises from incorrectly applying the direct Gillespie algorithm to feedback-controlled systems. This work provides a unified perspective on existing open-loop-control KMC algorithms and also offers a powerful and accurate tool for simulating stochastic systems with feedback control.

Figures (4)

• Figure 1: Approximating open-loop protocol control. a) Graphical representation of both (i) piece-wise linear rate function and ii) piece-wise step-function rates. The black curve is the actual negative escape rate $R_xx(t)$. The approximation breaks the time into multiple windows. Each window $w_i$ starting at time $\tau_i$ is represented by shaded regions, and the colored straight lines represent the approximated piece-wise negative escape rate. b) Cartoon of one possible trajectory of the system. Here the trajectory within the step of simulation is shown by the bold black lines; previous and future trajectory are shown by light gray lines. This simulation step is conditioned on the starting time $t_{\rm last}$ and state $x$, and proposes to jumped to a new state $x'$ at time $t^*>t_k$.
• Figure 2: Closed-loop control protocol. a) Graphical representation of a series of scheduled measurements occurring at $\tau_{i-1}, \tau, \cdots, \tau_{i+k}$. b) Two types of single-step trajectories illustrated in the piece-wise step-function feedback control: i) the next event occurs prior the the next scheduled measurement evolved using the determined protocol based on the last measurement $R_{xx(\lambda_a})$ or ii) event occurs after the $k$ additional measurements, and the rate is projected by fixing the future measurements to be state $x$ resulting in control signal being $\lambda_b$, and the escape rate is $R_{xx}(\lambda_b)$.
• Figure 3: Two-state refrigerator through periodic feedback control. a) i) Design of energy landscape of two-state system in two different conditions and ii) feedback control decision occurring at each measurement time. b) Time-averaged thermodynamic properties of the refrigerator including (left) information rate and (right) entropy production rate computed under different feedback measurement frequencies. To demonstrate the artificial Zeno effect, both the correct feedback controlled KMC method (correct) and an naively implemented direct KMC method (incorrect) are shown. c) System's thermodynamic efficiency ($\eta\leq 1$) calculated under different feedback control frequencies with both the correct and the incorrect algorithms.
• Figure 4: Diffusion along a 1D lattice. a) (left panel) Schematic of the diffusion processes with jumping rates to the right and left direction denoted as $r^{A/B}_+$ and $r^{A/B}_-$. The control condition is denoted by $A$ and $B$. (right panel) Periodic control protocol with period $2 \tau$. b) Simulated diffusion trajectories using (top panel) averaged rate dynamics in traditional KMC, (center panel) individual rate dynamics using incorrectly applied KMC due timescale mismatch between $\tau$ and time for each move $\Delta t$, (bottom panel) individual rate dynamics correctly simulated using the KMC described herein.