  In this paper we extend the encounter-based model of diffusion-mediated
surface absorption to the case of an unbiased run-and-tumble particle (RTP)
confined to a finite interval $[0,L]$ and switching between two constant
velocity states $\pm v$ at a rate $\alpha$. The encounter-based formalism is
motivated by the observation that various surface-based reactions are better
modeled in terms of a reactivity that is a function of the amount of time that
a particle spends in a neighborhood of an absorbing surface, which is specified
by a functional known as the boundary local time. The effects of surface
reactions are taken into account by identifying the first passage time (FPT)
for absorption with the event that the local time crosses some random threshold
$\widehat{\ell}$. In the case of a Brownian particle, the local time $\ell(t)$
is a continuous non-decreasing function of the time $t$. Taking
$\Psi(\ell)\equiv \P[\widehat{\ell}>\ell]$ to be an exponential distribution,
$\Psi[\ell]=\e^{-\kappa_0\ell}$, is equivalent to imposing a Robin boundary
condition with a constant rate of absorption $\kappa_0$. One major difference
in the encounter-based model of an RTP is that the boundary local time
$\ell(t)$ is a now a discrete random variable that counts the number of
collisions of the RTP with the boundary. Given this modification, we show that
in the case of a geometric distribution $\Psi(\ell)=z^{\ell}$,
$z=1/(1+\kappa_0/v)$, we recover the RTP analog of the Robin boundary
condition. This allows us to solve the boundary value problem (BVP) for the
joint probability density for particle position and the local time, and thus
incorporate more general models of absorption based on non-geometric
distributions $\Psi(\ell)$.
