Hydrodynamic Limits for One-dimensional Particle Systems with Moving Boundaries by L. Chayes

Overview and preliminaries. The study of interacting particle systems with conservation laws frequently revolves around hydrodynamic limits, in which the long-time, large-length scale behavior of a system is characterized by the evolution of the local density of the conserved quantity according to a partial differential equation. The typical setup consists of a particle system defined on a closed lattice with periodic boundary conditions or, perhaps, with w x a fixed particle density at the boundary. See, for example, 13 and references therein for an extensive discussion of these topics. In each of these cases, the dynamics of the particle system is prescribed by the transition rates, the initial condition and, if relevant, the boundary conditions. In particular, the internal dynamics of the particle systems does not affect the behavior of the boundaries. In this paper we study certain one-dimensional particle systems with exclusion dynamics and the additional feature that the region in which exclusion dynamics occurs is altered by the dynamics itself. The two basic examples can both be regarded as crude microscopic models of the dynamics of a liquid]solid system with an interface: the first case corresponds to the melting of a solid and the second to the freezing of a supercooled liquid. To describe the process of melting, consider the following particle system 4 N 4 with configurations in y1, 0, 1 , where L s yN, yN q 1, . . . , N y 1, N . N