The Symmetric and Fully Distributed Solution to a Generalized Dining Philosophers Problem: An Analogue of the Coupling Theory of Relaxations in Complex Correlated Systems

Abstract

In the past ten years we as well as our coworkers^[1–4] have made various attempts to understand irreversible processes (i.e. relaxations) in complex correlated systems (CCS’s). The latter includes structural relaxation of supercooled liquids and glasses, motions of segments or entire chains in dense entangled polymers, ionic conductivity relaxations in vitreous ionic conductors with large concentration of mobile ions, etc. The common characteristic of these is that the elementary units responsible for the relaxation process are correlated with or coupled to each other. An elementary unit can no longer relax independently as if the others were not present. The loss of independence is caused by the mutual constraints between the elementary units which requires the relaxation process to be highly cooperative. This many-body problem in irreversible statistical mechanics is extremely difficult to solve. At present an ab initio, first principle and parameter free theory for any relaxation process in a realistic CCS is not available. Nevertheless, much progress has been made by less ambitious approaches which use physical principles to derive the general rules of cooperative relaxations in CCS’s. These approaches, whenceforth summarily referred to as the coupling theory, are not parameter free but provide a whole host of predictions that can be and have been verified. Many research opportunities remain in the quest for a better understanding of cooperative relaxation processes. We are constantly on the lookout for alternative methods to solve this problem. In this contribution we shall present a solution based on a recent advance in computer science to solve a similar problem.