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

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.