Order-disorder criticality in classical particle systems

The nature of the glass transition is probably the most intriguing problem in condensed matter physics. In this talk, I present a semi-analytical method to study order-disorder criticality in classical particle systems. The method is based on the original idea of G. Faigel and co-workers (first published in 1979) of modelling amorphous solid structures as randomised crystals. Here, the generalisation of the idea is applied to the 1-dimensinal infinite Lennard-Jones system at zero temperature, where a mean-field critical point is found (in the expectation sense) between the periodic phase and the correlated Wiener sequence. It will be shown that re-fining the parameter(s) of the randomisation process results in more and more random configurations being closer and closer to mechanical equilibrium. In accordance with the concentration of measure phenomenon (M. Talagrand, Abel Prize 2024), this unexpected result suggests the existence of an "exact" solution where the randomised particle configurations are almost surely in mechanical equilibrium. To support the idea of order-disorder criticality in classical systems, the results of numerical simulations corresponding to finite temperatures will also be presented. At the end of the talk, the mathematical challenges associated with extending the methodology to higher spatial dimensions will be discussed.