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 ξ∼ (lnt) 1/ χ∼. This implies that the relaxation dynamics is driven by thermal activations over random energy barriers which scale as EB (ℓ) ∼ ℓ χ∼ with a length scale ℓ. Numerical data strongly suggest that the energy barrier exponent χ∼ is identical to the energy fluctuation exponent χ=1/3. We also find that there exists a long transient regime, where the correlation length follows a power-law dynamics as ξ∼ 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.