Universal short-time motion of a polymer in a random environment: Analytical calculations, a blob picture, and Monte Carlo results

Using a recently established renormalization group approach [U. Ebert, J. Stat. Phys. (to be published)], we analyze the center-of-mass motion of a polymer in a Gaussian disordered potential. While in the long-time limit normal diffusion is found, we concentrate here on shorter times. We discuss the general structure of the relevant crossover scaling function and evaluate it quantitatively in three dimensions to one-loop order. We identify a universal short-time regime, where the chain length dependence of the center-of-mass motion is Rouse-like, while the time dependence is nontrivial. Motion in this regime can be interpreted in terms of a blob picture: A "time blob" defines an additional intrinsic length scale of the problem. The short-time dependence of the center-of-mass motion over several decades approximates a power law with an effective exponent that continuously depends on disorder (and also weakly on the time interval). We furthermore present the results of a simulation measuring the motion of a (pearl necklace) chain in Gaussian disorder in three dimensions. We find full agreement between theory and numerical experiment. The characteristic behavior found in these simulations closely resembles the results of some previous simulations aimed at seeing reptation. This suggests that such work was strongly influenced by energetic disorder or entropic traps