Perturbation of the equilibrium for a totally asymmetric stick process in one dimension

Abstract

We study the evolution of a small perturbation of the equilibrium of a totally asymmetric one-dimensional interacting system. The model we take as an example is Hammersley's process as seen from a tagged particle, which can be viewed as a process of interacting positive-valued stick heights on the sites of $\mathbf{Z}$. It is known that under Euler scaling (space and time scale $n$) the empirical stick profile obeys the Burgers equation. We refine this result in two ways. If the process starts close enough to equilibrium, then over times $n^\nu$ for $1 \le \nu < 3$, and up to errors that vanish in hydrodynamic scale, the dynamics merely translates the initial stick configuration. In particular, on the hydrodynamic time scale, diffusive fluctuations are translated rigidly. A time evolution for the perturbation is visible under a particular family of scalings:over times $n_{\nu}, 1 < \nu < 3/2$, a perturbation of order $n^{1-\nu}$ from equilibrium follows the inviscid Burgers equation. The results for the stick model are derived from asymptotic results for tagged particles in Hammersley's process.