$d$, thermal equilibrium has been reached, and the line has a self affine shape where the structure function is time independent and approximately follows the theoretically expected scaling for an elastic line, $R \propto (d k_{\rm B} T/{\cal T})^{\zeta}$ with $\zeta = 0.5$ [@Nattermann1992_PRA]. The apparent deviations from slope 0.5 on the double-logarithmic plot can be accounted for by noting that the effective line energy of the dislocation decreases on small scales. Setting ${\cal T} \propto \ln(d/b)$ produces a good representation of the data.

The overall behavior of $R(d,t)$ can be described by the equation $$R(d,t) = \left(\frac{d k_{\rm B} T}{{\cal T}}\right)^{\zeta} \Phi\left(\frac{\cal T}{B} \frac{t}{d^z}\right)$$ with the roughness exponent $\zeta = 0.5$ and the dynamic exponent $z=2$ for the elastic line under thermal forces. The scaling function $\Phi$ has the following properties: $\Phi(x) \to \Phi_{\infty}\; {\rm for}\; x \to \infty, \Phi(x) \propto x^{\zeta/z}\;
{\rm for}\; x \to 0$. This implies that, for sufficiently large $d$, the structure function approaches an approximately $d$ independent value $R_{\infty} \propto T^{\zeta} B^{-\zeta/z} {\cal T}^{\zeta(1-z)/z} t^{\zeta/z}$. With $\zeta = 0.5$, $z = 2$, and a linear temperature dependency $B \propto T$, we find that $R_{\infty} \propto T {0.25} t^{0.25}$. Figure \[fig:rtimetemp\] shows that the time- 