Phase boundary dynamics in a one-dimensional non-equilibrium lattice gas

We study dynamics of a phase boundary in a one-dimensional lattice gas, which is initially put into a non-equilibrium configuration and then is let to evolve in time by particles performing nearest-neighbor random walks constrained by hard-core interactions. Initial non-equilibrium configuration is characterized by an $S$-shape density profile, such that particles density from one side of the origin (sites $X \leq 0$) is larger (high density phase, HDP) than that from the other side (low-density phase, LDP). We suppose that all the lattice gas particles, except for the rightmost particle of the HDP, have symmetric hopping probabilities. The rightmost particle of the HDP, which determines the position of the phase separating boundary, is subject to a constant force $F$, oriented towards the HDP; in our model this force mimics an effective tension of the phase separating boundary. We find that, in the general case, the mean displacement $\bar{X(t)}$ of the phase boundary grows with time as $\bar{X(t)} = \alpha(F) t^{1/2}$, where the prefactor $\alpha(F)$ depends on $F$ and on the initial densities in the HDP and LDP. We show that $\alpha(F)$ can be positive or negative, which means that depending on the physical conditions the HDP may expand or get compressed. In the particular case when $\alpha(F) = 0$, i.e. when the HDP and LDP coexist with each other, the second moment of the phase boundary displacement is shown to grow with time sublinearly, $\bar{X^2(t)} = \gamma t^{1/2}$, where the prefactor $\gamma$ is also calculated explicitly. Our analytical predictions are shown to be in a very good agreement with the results of Monte Carlo simulations.