Cluster Derivation of the Parisi Scheme for Disordered Systems

Abstract

We propose a general quantitative scheme in which systems are given the freedom to sacrifice energy equi-partitioning on the relevant time-scales of observation, and have phase transitions by separating autonomously into ergodic sub-systems (clusters) with different characteristic time-scales and temperatures. The details of the break-up follow uniquely from the requirement of zero entropy for the slower cluster. Complex systems, such as the Sherrington-Kirkpatrick model, are found to minimise their free energy by spontaneously decomposing into a hierarchy of ergodically equilibrating degrees of freedom at different (effective) temperatures. This leads exactly and uniquely to Parisi's replica symmetry breaking scheme. Our approach, which is somewhat akin to an earlier one by Sompolinsky, gives new insight into the physical interpretation of the Parisi scheme and its relations with other approaches, numerical experiments, and short range models. Furthermore, our approach shows that the Parisi scheme can be derived quantitatively and uniquely from plausible physical principles.