We prove that the center of mass of a sine-Gordon kink in contact with a thermal reservoir approaches equilibrium by undergoing a Brownian motion in the limit that kBTEₖ where Eₖ is the rest energy of the kink. Our method consists of introducing a collective variable Hamiltonian for the kink system in which the center of mass of the kink is a canonical variable. Next we use standard projection-operator techniques to derive the equation of motion of the distribution function of the center of mass of the kink. Then we show that in the limit kBTEₖ the distribution function satisfies the Fokker-Planck equation of Brownian motion.
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Marchesoni et al. (1987) studied this question.