Short Time Behavior in de Gennes' Reptation Model

To establish a standard for the distinction of reptation from other modes of polymer diffusion, we analytically and numerically study the displacement of the central bead of a chain diffusing through an ordered obstacle array for times $t<\mathrm{}O\left(N^2\right)$. Our theory and simulations agree quantitatively and show that the second moment approaches the $t^{1/4}$ power law (often viewed as a signature of reptation) only after a very long transient and only for long chains ( $N>\mathrm{}100\mathrm{}$). Our analytically solvable model furthermore predicts a very short transient for the fourth moment. This is verified by computer experiment.