Relaxation dynamics of an elastic string in random media.

We investigate numerically the relaxation dynamics of an elastic string in two-dimensional random media by thermal fluctuations starting from a flat configuration. Measuring spatial fluctuations of its mean position, we find that the correlation length grows in time asymptotically as xi approximately (ln t)1/chi . This implies that the relaxation dynamics is driven by thermal activations over random energy barriers which scale as EB(l) approximately l;chi with a length scale l . Numerical data strongly suggest that the energy barrier exponent chi is identical to the energy fluctuation exponent chi=1/3 . We also find that there exists a long transient regime, where the correlation length follows a power-law dynamics as xi approximately t1/z with a nonuniversal dynamic exponent z . The origin of the transient scaling behavior is discussed in the context of the relaxation dynamics on finite ramified clusters of disorder.