Shift Equivalence of Measures and the Intrinsic Structure of Shocks in the Asymmetric Simple Exclusion Process

We investigate the properties of non-translation-invariant measures, describing particle systems on $mathbb{Z}$ , which are asymptotic to different translation invariant measures on the left and on the right. Often the structure of the transition region can only be observed from a point of view which is random—in particular, configuration dependent. Two such measures will be called shift equivalent if they differ only by the choice of such a viewpoint. We introduce certain quantities, called translation sums, which, under some auxiliary conditions, characterize the equivalence classes. Our prime example is the asymmetric simple exclusion process, for which the measures in question describe the microscopic structure of shocks. In this case we compute explicitly the translation sums and find that shocks generated in different ways—in particular, via initial conditions in an infinite system or by boundary conditions in a finite system—are described by shift equivalent measures. We show also that when the shock in the infinite system is observed from the location of a second class particle, treating this particle either as a first class particle or as an empty site leads to shift equivalent shock measures.