First-Passage-Time Asymmetry for Biased Run-and-Tumble Processes

TL;DR

This work demonstrates that first-passage-time duality for biased run-and-tumble processes is generally violated at finite interval widths but can be restored asymptotically when GC symmetry holds in the associated current large-deviation sense. By combining exact analytical results (via Montroll’s defect technique) with renewal-process arguments and comprehensive numerics for discrete RnT models, the authors reveal how hidden tumbling dynamics bias FPT statistics between symmetric barriers. They introduce and compare robust asymmetry measures, notably Kullback-Leibler divergence and SNR, and show how these measures capture finite-size effects and distinct scaling regimes (including fL collapse in ballistic versus diffusive runs). The findings illuminate how underlying microdynamics imprint on macroscopic FPTs, offering potential inference paths for active matter systems and guiding future extensions to more general renewal or higher-dimensional settings.

Abstract

We explore first-passage phenomenology for biased active processes with a renewal-type structure, focusing in particular on paradigmatic run-and-tumble models in both discrete and continuous state spaces. In general, we show there is no equality between distributions of conditional first-passage times to symmetric barriers positioned in and against the bias direction. However, we give conditions for such a duality to be restored asymptotically (in the limit of a large barrier distance) and highlight connections to the Gallavotti-Cohen fluctuation relation and the method of images. Our general trajectory arguments of first-passage-time distributions for asymmetric run-and-tumble processes to escape from an interval of arbitrary width are supported by exact analytical results, which we derive extending Montroll's defect technique. Furthermore, we quantify the degree of violation of first-passage duality using Kullback-Leibler divergence and signal-to-noise ratios associated with the first-passage times to the two barriers. We reveal an intriguing dependence of such measures of first-passage asymmetry on the underlying often hidden tumbling dynamics which may inspire inference techniques based on first-passage-time statistics in active systems.

Figures (11)

• Figure 1: FPT illustration for the position of a single particle on the symmetric interval $[-L, L]$. As a representative example, we use the discrete run-and-tumble (RnT) model defined in section \ref{['ss:RnT']}, with parameters: $L=8$, $p=0.7$, $p'=0.75$, and $f=0.05$. (a) The process stops at a stochastic time $T$ given by the earliest time when the particle reaches either of two absorbing boundaries, $+L$ or $-L$. The solid zig-zag blue and orange lines are sample trajectories escaping the interval via the boundaries $+L$ and $-L$ at times $\mathcal{T}_+$ and $\mathcal{T}_-$, respectively. (b) Normalized distributions $P_+(\mathcal{T})$ (blue filled circles) and $P_-(\mathcal{T})$ (orange filled squares), associated with $\mathcal{T}_+$ and $\mathcal{T}_-$ respectively are displayed with symbols corresponding to simulations from $10^8$ trajectories. The lines are a guide to the eye which join discrete points given by the exact analytical expressions obtained from numerically inverting the $z$-transforms of $P_+(\mathcal{T})$ and $P_-(\mathcal{T})$eq:F-z, see section \ref{['s:fpt-theory']} for further details. Here, $\langle T \rangle_+\approx 19$, $\langle T \rangle_- \approx 20$, $\mathrm{Var}_+(T) \approx 134$, $\mathrm{Var}_-(T) \approx 156$, which indicates that the FPT duality given by e:duality does not hold in this example.
• Figure 2: (a) Sketch of the RnT model defined by the transition probabilities (\ref{['eq:pbeg']})--(\ref{['eq:pend']}). A random walker jumps at discrete times between states represented as lattice sites (red and blue circles), here with open boundary conditions along $x$. Each state is characterized by two variables: its position along the $x$-axis and the internal preferred-direction variable which is indicated by the position along the $y$-axis: $y=+1$ (or loosely '$y=+$') in the upper lane (blue) and $y=-1$ (or loosely '$y=-$') in the lower lane (red). Here, $f$ is the tumble probability while $p$ and $q=1-p$ are the probabilities that after a tumble $y=+1$ and $y=-1$ respectively. Furthermore, $p'$ and $q'=1-p'$ denote the probabilities of jumps in and against the preferred direction, with the preferred direction as right in the upper lane and left in the lower lane. (b,c) Sample RnT trajectories with segments shown in different colours to indicate preferred direction: blue for $Y_t=+1$ and red for $Y_t=-1$. In (b) effective tumbles are visible ($p'=1,p=0.6,f=0.25$); movements to the right (left) occur only when the particle is in the upper (lower) lane, which makes the preferred direction identifiable from the particle motion. In (c) effective tumbles are hidden ($p'=0.75, p=0.7, f=0.25$); a particle may move to the right or left in any lane, which renders the preferred direction a genuinely hidden variable.
• Figure 3: Probability-preserving operations on a representative RW trajectory: (a) original trajectory, (b) time reversal, (c) spatial translation, (d) step reordering.
• Figure 4: FPT asymmetry for the RnT model with visible tumbles (VT, $p=0.6$, $p'=1$) exiting a symmetric interval. As a function of tumbling probability $f$, we show (a) KLD $D_\mathrm{KL} (P_- || P_+)$ [see eq:kld] and (b) signal-to-noise ratio $\textrm{SNR}$ [see e:hatrho] associated with the first-passage times to reach the boundary against and in the direction of the bias. Exact analytical results are solid lines while estimates from numerical simulations are discrete data points; different colours and symbols represent different boundary positions $L$ (see legend). Here and in all subsequent figures, simulation data were obtained from $10^9$ trajectories with a cut-off at time 2000 and points only displayed if at least $10^4$ trajectories are absorbed at the negative barrier. The analytical results are obtained by numerically inverting abate_numerical_1992 the $z$-transforms of the probabilities $P_{\pm}(\mathcal{T})$ [derived in section \ref{['s:fpt-theory']}, see eq:F-z] and then applying eq:kld to get the KLD or computing moments to obtain the ${\rm SNR}$ via e:hatrho.
• Figure 5: FPT asymmetry for the RnT model with hidden tumbles (HT, $p=0.7$, $p'=0.75$) exiting a symmetric interval. As a function of tumbling probability $f$, we show (a) KLD $D_\mathrm{KL} (P_- || P_+)$ [see eq:kld] and (b) signal-to-noise ratio $\textrm{SNR}$ [see e:hatrho] associated with the first-passage times to reach the boundary against and in the direction of the bias. Exact analytical results are solid lines while estimates from numerical simulations are discrete data points; different colours and symbols represent different boundary positions ($L=4,8,16,32$). The analytical results are obtained by numerically inverting abate_numerical_1992 the $z$-transforms of the probabilities $P_{\pm}(\mathcal{T})$ [derived in section \ref{['s:fpt-theory']}, see eq:F-z] and then applying eq:kld to get the KLD or computing moments to obtain the ${\rm SNR}$ via e:hatrho.