Motion of damped sine-Gordon kinks in the presence of thermal fluctuations

The motion of damped sine-Gordon kinks in the presence of thermal fluctuations is studied to first order in $k_B$T by using a singular perturbation expansion. The change in the average shape of the kink is determined as well as its mobility. These results are compared with the earlier results of Kaup on the overdamped sine-Gordon equation, and it is found that the heavily damped limit of the damped sine-Gordon kink is not the same as that of the overdamped sine-Gordon kink. In particular, we find the change in the shape of a heavily damped sine-Gordon kink to be approximately one-half that of an overdamped sine-Gordon kink at comparable temperatures. We also verify that the singular perturbation method gives the correct results for the average kinetic energies of kinks and phonons. Lastly, we also evaluate the low-temperature correlation function for the damped sine-Gordon field and the diffusion coefficient for a soliton.