non-singular dislocation self interaction kernel [@Cai2006_JMPS; @Arsenlis2007_MSMSE], both for an isolated dislocation and for a periodic array of rigidly coupled dislocations as implied in the present simulations by our boundary conditions.

![\[fig:rougheningT\]Top: time evolution of the structure function $R(d,t)$, simulation at $T$=300K; dashed line: $R \propto
    (d/{\cal T}(d)^{0.5}$, average over 10 simulations; bottom: scaled structure function $w = R (k_{\rm B}T)^{-0.5} B^{0.25} {\cal T}^{0.25} t^{-0.25}$ as function of scaled displacement $u = d({\cal T} t/B)^{-0.5}$; data for different temperatures as shown in legend, data at each temperature for times $t=0.05ns, t= 0.5ns, t=3ns$.](rougheningT.pdf){width="3.3in"}

In a next step, we investigate the transient dynamics before the dislocation has reached thermal equilibrium, starting from an initially straight dislocation line configuration which, under the influence of thermal fluctuations, gradually develops a self affine shape $y(x)$. We statistically characterize this process by the structure factor $R(d,t)$ defined as: $$\label{equ:3}
R(d,t)=\left\langle \left\langle \left| y(x,t)-y(x+d,t)\right|\right\rangle_{x}\right\rangle _{N}$$ where $\left\langle \right\rangle_N $ means average over an ensemble of $N$ simulations and $\left\langle \right\rangle_x$ the average over all $x$ positions for which $y(x+d,t)$ can be computed. The time evolution of $R(d,t)$ is shown in Figure \[fig:rougheningT\] for $T=300$K. For a given time, the function $R(d,t)$ exhibits two regimes: At small 