Random solids and random solidification: What can be learned by exploring systems obeying permanent random constraints?

TL;DR

This paper reviews recent theoretical developments in the study of systems with permanent random constraints that undergo a phase transition to a random solid state, highlighting similarities and differences with the glass transition.

Contribution

It introduces new results on equilibrium correlations and susceptibilities signaling the transition, analyzed through Gaussian and renormalization-group methods.

Findings

Random solidification is an equilibrium phase transition.

Correlation functions and susceptibilities can signal the transition.

Analysis includes Gaussian approximation and renormalization-group approach.

Abstract

In many interesting physical settings, such as the vulcanization of rubber, the introduction of permanent random constraints between the constituents of a homogeneous fluid can cause a phase transition to a random solid state. In this random solid state, particles are permanently but randomly localized in space, and a rigidity to shear deformations emerges. Owing to the permanence of the random constraints, this phase transition is an equilibrium transition, which confers on it a simplicity (at least relative to the conventional glass transition) in the sense that it is amenable to established techniques of equilibrium statistical mechanics. In this Paper I shall review recent developments in the theory of random solidification for systems obeying permanent random constraints, with the aim of bringing to the fore the similarities and differences between such systems and those exhibiting the conventional glass transition. I shall also report new results, obtained in collaboration with Weiqun Peng, on equilibrium correlations and susceptibilities that signal the approach of the random solidification transition, discussing the physical interpretation and values of these quantities both at the Gaussian level of approximation and, via a renormalization-group approach, beyond.