  A run-and-tumble particle in a one dimensional box (infinite potential well)
is studied. The steady state is analytically solved and analyzed, revealing the
emergent length scale of the boundary layer where particles accumulate near the
walls. The mesoscopic steady state entropy production rate of the system is
derived from coupled Fokker-Planck equations with a linear reaction term,
resulting in an exact analytic expression. The entropy production density is
shown to peak at the walls. Additionally, the derivative of the entropy
production rate peaks at a system size proportional to the length scale of the
accumulation boundary layer, suggesting that the behavior of the entropy
production rate and its derivatives as a function of the control parameter may
signify a qualitative behavior change in the physics of active systems, such as
phase transitions.
