Spatial inhomogeneity and thermodynamic chaos.

We present a coherent approach to the competition between thermodynamic states in spatially inhomogeneous systems, such as the Edwards-Anderson spin glass with a fixed coupling realization. This approach explains and relates chaotic size dependence, “dispersal of the metastate”, and for replicas: non-independence, symmetry breaking, and overlap (non-)self-averaging. The connection between the existence of many thermodynamic states and the phenomena of non-self-averaging (NSA) and replica symmetry breaking (RSB) has long been a central topic of research in disordered systems. These phenomena play a key role in Parisi’s solution [1, 2, 3] of the infinite-ranged Sherrington-Kirkpatrick (SK) Ising spin glass model [4], and have been discussed in many other contexts, e.g., short-ranged spin glasses [5, 6], random field XY models [7], random manifolds [8], heteropolymers [9], and impure superconductors in magnetic fields [10]. Another aspect of the competition between thermodynamic states, introduced by the authors in Ref. [11], is “chaotic size dependence” (CSD) which, unlike NSA, manifests itself for a fixed realization of the disorder. An important issue is whether (and in what sense) the many novel features of Parisi’s solution can apply to more realistic models, such as the Edwards-Anderson (EA) nearestneighbor Ising spin glass [12]. In a previous paper [13], it was shown that this “SK picture”, as conventionally understood, is not valid. In particular, the overlaps of the thermodynamic states for coupling realization J do not depend on J . This leads us to approach all basic phenomena as accessible for a fixed realization. Although we focus here on disordered systems, this approach is applicable to the more general setting of inhomogeneous systems. We shall see that CSD is one aspect of a phenomenon, “dispersal of the metastate”, which is closely connected with both NSA and RSB. The metastate is a natural ensemble, i.e., a probability measure, on the (pure or mixed) thermodynamic states of the system [14]. At high temperature, it is a δ-function on a single state; we call this a non-dispersed Partially supported by the National Science Foundation under grant DMS-95-00868. Partially supported by the U.S. Department of Energy under grant DE-FG03-93ER25155.