Polymer Reptation — de Gennes

Entangled polymer diffusing through a tube — Rouse modes, MSD ~ t^{1/2} → t crossover
80
0.080
1.20

Polymer Chain + Primitive Path Tube

COM displacement
R_g
Mode 1 amp

Mean-Squared Displacement (log-log)

Rouse (free): MSD ~ t
Entangled (Reptation):
t < τ_e: Rouse ~ t^{1/2}
τ_e < t < τ_R: ~ t^{1/4}
τ_R < t < τ_d: ~ t^{1/2}
t > τ_d: free diff ~ t
D_rep ~ 1/N²