  We study the sine-Gordon kink diffusion at finite temperature in the
overdamped limit. By means of a general perturbative approach, we calculate the
first- and second-order (in temperature) contributions to the diffusion
coefficient. We compare our analytical predictions with numerical simulations.
The good agreement allows us to conclude that, up to temperatures where
kink-antikink nucleation processes cannot be neglected, a diffusion constant
linear and quadratic in temperature gives a very accurate description of the
diffusive motion of the kink. The quadratic temperature dependence is shown to
stem from the interaction with the phonons. In addition, we calculate and
compute the average value $<\phi(x,t)>$ of the wave function as a function of
time and show that its width grows with $\sqrt{t}$. We discuss the
interpretation of this finding and show that it arises from the dispersion of
the kink center positions of individual realizations which all keep their
width.
