Exact solution of a boundary tumbling particle system in one dimension

We derive the fully time-dependent solution to a run-and-tumble model for particle which has tumbling restricted boundaries of one-dimensional interval. This is achieved through field-theoretic perturbative framework by exploiting an elegant underlying structure perturbation theory. calculate densities, currents and variance as well characteristics boundary tumbling. The analytical findings, in agreement with Monte-Carlo simulations, show how densities are linked scale diffusive fluctuations at boundaries. generality our approach suggests it could be readily applied similar problems described Fokker-Planck equations containing localised reaction terms.