We investigate the dynamics of a one-dimensional asymmetric exclusion process with Langmuir kinetics and a fluctuating wall. At the left boundary, particles are injected onto the lattice; from there, the particles hop to the right. Along the lattice, particles can adsorb or desorb, and the right boundary is defined by a wall particle. The confining wall particle has intrinsic forward and backward hopping, a net leftward drift, and cannot desorb. Performing Monte Carlo simulations and using a moving-frame finite segment approach coupled to mean field theory, we find the parameter regimes in which the wall acquires a steady state position. In other regimes, the wall will either drift to the left and fall off the lattice at the injection site, or drift indefinitely to the right. Our results are discussed in the context of non-equilibrium phases of the system, fluctuating boundary layers, and particle densities in the lab frame versus the frame of the fluctuating wall.
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