Mode-Coupling Theory: Glass Transition

MCT predicts a two-step relaxation of the density autocorrelation function F(q,t). As temperature approaches T_c from above, a plateau (the nonergodicity parameter f_q) develops and the alpha-relaxation time diverges. Below T_c, the system is arrested.

Temperature

T/T_c1.20
Plateau f_q0.00
τ_α (log)—
PhaseLiquid
Schematic MCT model:
F(q,t) follows the MCT equation with memory kernel:
M(t) = V² F²(q,t)

Two-step relaxation:
• Fast β-relaxation: caging, F → f_q
• Slow α-relaxation: cooperative escape

Critical behavior:
f_q jumps discontinuously at T_c (type-B transition)
τ_α ~ (T - T_c)^{-γ}
Von Schweidler law: F ~ f_q - B·t^b
α-relaxation: Kohlrausch function exp[-(t/τ)^β]