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

Abstract

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.

Cavity Approach to the Random Solid State

“…Thus, the foregoing combinatorics continues to apply, and we arrive at the formula for the probability of having exactly k bonds with the infinite cluster given in Eq. (5). As a consequence, we obtain the foregoing result for the fraction of the infinite cluster Q, Eq.…”

“…[4,5]. Thus, it was already known [6] that (as is typical for mean-field theories) Q obeys a transcendental equation, in this case…”

Towards finite-dimensional gelation

“…The main idea is now to interpret this expression as the partition function of a fluid of N "effective molecules", each consisting of n + 1 particles, cf. [20]. We simplify the notation by introducing a d(n + 1)-dimensional vector Ri = ( R 0 i , ..., R n i ) for the position vectors of the "constituents" of molecule i.…”

Heterogeneous Solids and the Micro/Macro Connection: Structure and Elasticity in Architecturally Complex Media as Emergent Collective Phenomena

“…It is worth noting that there are other, perhaps more physical, ways of introducing replicas, which come to light once one sees the kinds of mathematical detectors (known as order parameters) that are useful in the setting of random solids [4] and other systems, such as spin glasses [8], that undergo phase transitions to 'randomly frozen' states. But the key point I wish to stress here is this: the configurations of the original system involve the coordinates of the many constituent particles, each in three dimensions, and they give rise, say, to physical fields describing densities that 'live' on three-dimensional space, just as the displacement, strain, and stress fields of conventional elasticity theory do.…”