Glass Transition: Mode Coupling Theory

Mode Coupling Theory (MCT) predicts a dynamical glass transition at a critical packing fraction φ_c. The intermediate scattering function F(q,t) = ⟨ρ_q(0)ρ_{-q}(t)⟩/S(q) describes density fluctuation decay. For φ<φ_c (ergodic liquid): F(q,t)→0 (complete relaxation). For φ≥φ_c (non-ergodic glass): F(q,t)→f_q>0 — the non-ergodicity parameter (plateau). Near φ_c the decay shows two steps: fast β-relaxation (cage rattling) then slow α-relaxation (cage escape), with α-relaxation time diverging as τ_α ∝ (φ_c−φ)^{−γ}.