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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 $ξ\sim (\ln t)^{1/\tildeχ}$. This implies that the relaxation dynamics is driven by thermal activations over random energy barriers which scale as $E_B(\ell) \sim \ell^{\tildeχ}$ with a length scale $\ell$. Numerical data strongly suggest that the energy barrier exponent $\tildeχ$ 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 $ξ\sim t^{1/z}$ 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.