of the different Fourier modes are independent random variables, we can deduce, in the limit $L \to \infty$ the scaling of the power spectrum of the line as $P(k) \propto q^{-a}$ where $a = 2$. Using general results for self affine manifolds [@Schmittbuhl1995_PRE], this implies that the line shape represents, in the limit $L \to \infty$, a self affine fractal with roughness exponent $\zeta = (a-1)/2 = 0.5$, i.e., it is equivalent to the graph of a random walk. We will investigate later to which extent this result is correct for dislocations.

To investigate the wandering dynamics, we consider an initially straight elastic line moving under the influence of thermal forces in an over-damped manner with a drag coefficient per unit length $B$. For a nearly straight line of line energy ${\cal T}$, the wandering dynamics under the influence of thermal Langevin forces is described by the annealed Edwards-Wilkinson equation $$B \frac{d y}{d t} = {\cal T}\frac{\partial^2 y}{\partial x^2} + f_T (x,t)
\label{eq:aedwilk}$$ where $f_T$ is a Gaussian random force with the correlation function $$\langle f_T(x,t) f_T(x',t') \rangle = 2 k_{\rm B} T B \delta(x-x') \delta (t-t').$$ The linearity of the governing equation allows for analytical solution in Fourier space, 