Run-and-tumble particle in one-dimensional potentials: mean first-passage time and applications

TL;DR

This work develops a general, exact framework to compute the mean first-passage time (MFPT) of a one-dimensional run-and-tumble particle (RTP) moving in an arbitrary external potential, using backward Fokker-Planck equations. By deriving a closed second-order ODE for the MFPT and expressing the two state MFPTs in terms of the derivative, the authors classify potential landscapes into four phases based on turning-point structure and provide explicit MFPT formulas, often in hypergeometric form, for representative potentials like a double-well and a log-potential. They illustrate the power of the method with three applications: a generalized active Kramers’ law for barrier crossing, the trapping time inside a harmonic well, and a comparison of search strategies between resetting and potential-driven RTP, showing resetting often yields a more efficient search. The results reveal rich, nontrivial MFPT behavior in active systems, distinguishing them from passive Brownian particles and offering exact benchmarks for active matter in confining landscapes.

Abstract

We study a one-dimensional run-and-tumble particle (RTP), which is a prototypical model for active system, moving within an arbitrary external potential. Using backward Fokker-Planck equations, we derive the differential equation satisfied by its mean first-passage time (MFPT) to an absorbing target, which, without any loss of generality, is placed at the origin. Depending on the shape of the potential, we identify four distinct ``phases'', with a corresponding expression for the MFPT in every case, which we derive explicitly. To illustrate these general expressions, we derive explicit formulae for two specific cases which we study in detail: a double-well potential and a logarithmic potential. We then present different applications of these general formulae to (i) the generalization of the Kramer's escape law for an RTP in the presence of a potential barrier, (ii) the ``trapping'' time of an RTP moving in a harmonic well and (iii) characterizing the efficiency of the optimal search strategy of an RTP subjected to stochastic resetting. Our results reveal that the MFPT of an RTP in an external potential exhibits a far more complex and, at times, counter-intuitive behavior compared to that of a passive particle (e.g., Brownian) in the same potential.

Figures (14)

• Figure 1: Left panel: We show a force such that $|f(x)|<v_0$ which correspond to phase I. Right panel: in phase II, the force is bounded such that $f(x)<-v_0$. The arrows on the dotted lines show the direction of the velocity of the RTP in state $\sigma=\pm$. If the arrow is directed to the right (left), the velocity is positive (negative) in this region.
• Figure 4: Consider a potential with a local minimum located at $x_{\text{min}}>0$ and a local maximum at the origin $x=0$. We want to estimate the average time needed for a particle to escape from this local minimum. This is given by the mean first-passage time to the origin. For a diffusive particle, it is simply proportional to $\exp(\Delta V/D)$ where $\Delta V$ is the barrier height and $D$ the diffusion coefficient. However, for an RTP, we show that the MFPT is approximated by $\exp(\Delta W/D)$ where $\Delta W$ is the height of an "active external potentia" given in Eq. (\ref{['activekramers']}).
• Figure 5: We show here the different behaviours of a force $f(x) = -\alpha(x-1)$ deriving from a double well potential $V(x) = \alpha/2\, \left(|x|-1\right)^2$. The little arrows represent the sign of the velocity in the two states $\sigma = \pm 1$ of the RTP. If the arrow is directed toward the right, it is positive. If it is directed toward the left, the velocity is negative. All these cases are discussed in Section \ref{['doublewellsection']}.
• Figure 6: Plot of the MFPT for an RTP in a double well $V(x)=\alpha/2\, (|x|-1)^2$ when $0<\alpha<v_0$. The blue line corresponds to the expression for $\tau_\gamma(x_0)$ given in Eq. (\ref{['MFPTDW1']}), while the red (respectively the green) line represents $\tau_\gamma^-(x_0)$ in Eq. (\ref{['minusMFPTDW']}) (respectively $\tau_\gamma^+(x_0)$ in Eq. (\ref{['plusMFPTDW']})). The dots represent results of our numerical simulations. On the left panel, we show a plot of the MFPT with respect to the tumble rate $\gamma$ and fixed values of the parameters $\alpha = 1$, $v_0= 2$, and $x_0 = 1$. As discussed in the text, $\tau_\gamma(x_0)$ and $\tau_\gamma^+(x_0)$ exhibit a minimum at an optimal value $\gamma = \gamma_{\rm opt}$ (see also Ref. letter). On the right panel, we show a plot of the MFPT as a function of the initial position $x_0$, with $\alpha = 1$, $v_0= 2$, and $\gamma = 1.1$.
• Figure 7: Plot of the MFPT for an RTP in a double well potential $V(x) = \alpha/2\, (|x|-1)^2$, when $\alpha<-v_0$. The blue line represents the MFPT $\tau_\gamma(x_0)$ given in Eq. (\ref{['TAUdwsecondformul2']}), while the red and green lines correspond to $\tau_\gamma^-(x_0)$ from Eq. (\ref{['dw_E3']}) and $\tau_\gamma^+(x_0)$ from Eq. (\ref{['dw_E4']}), respectively. The dots on the graph represent results from numerical simulations. The left panel displays the MFPT as a function of the tumble rate $\gamma$, while keeping the parameters fixed at $\alpha = -1$, $v_0 = 0.5$, and $x_0 = 0.1$. On the right panel, we plot the MFPT as a function of the initial position $x_0$, with $\alpha = -1$, $v_0 = 0.5$, and $\gamma = 1$.