Lattice models of disorder with order

Abstract

This paper describes the use of simple lattice models for studying the properties of structurally disordered systems like glasses and granulates. The models considered have crystalline states as ground states, finite connectivity, and are not subject to constrained evolution rules. After a short review of some of these models, the paper discusses how two particularly simple kinds of models, the Potts model and the exclusion models, evolve after a quench at low temperature to glassy states rather than to crystalline states. I IntroductionIn recent years, there is growing interest in systems commonly found in disordered structural configurations such as glasses and dense granulates. Clearly different in many respects, these two classes of materials share in common the property of displaying stable states which are very far from what would be expected from equilibrium considerations [1,2]. As a consequence of this property other features emerge, like slow relaxation and response functions.Models of disorder are generally based on the presence of quenched, or a priori, disorder which takes the system far from ordered configurations. Only recently it has been observed that lattice models capable of ordering can also well reproduce many properties of disordered systems. On the other hand glass formers in nature usually have crystalline states as ground states, but this does not prevent them from being frequently found in some glassy state. The situation for a granular system is in general different, insofar as irregular grains inhibit the evolution of ordered states. Nevertheless even in the case of identical beads it is practically impossible to arrange them into an ordered fashion merely by supplying the energy to them.The aim of this paper is to briefly review some lattice models employed in the description of the slow dynamics of glasses and dense granular matter, which do not possess any a priori disorder nor long range interactions. We shall not discuss in detail the models, but just recall some of the main results, addressing the interested reader to the bibliography. More attention will be drawn to some recent observations regarding the condition for the emergence of glassy phases in some models of this kind.The paper is structured as follows: in Section II exclusion models and hard particle models are briefly introduced together with their employment in irreversible dynamics; Section III is devoted to their use in the description of dense granular matter and related results; Section IV describes spin models which have been recently shown to exhibit glassy states and associated slow dynamics; finally in Sec. V some new ideas are illustrated on when and why glassy states are generated in the place of crystalline states in some lattice models. II Exclusion and hard particle modelsFlory [3] modeled the irreversible deposition of dimers on a one-dimensional surface by the random sequential adsorption (RSA) of particles on a lattice. In this model, particles are placed one at time at randomly chosen positions of a ...

Zero-temperature coarsening in the 2d Potts model

“…This continuous evolution stops because the system gets trapped in a disordered state and escape from this state is slow because the process is thermally activated [21]. When the final temperature is zero, it was noticed previously that the ground state may not be reached [22,23]. It was further suggested that for q ≥ d + 1, glassy evolution can occur [20,21,24,25] because the system gets pinned in a disordered state and the ultimate relaxation to equilibrium proceeds via thermally activated processes [20,21], [26]- [28].…”