Brownian motion of a sine-Gordon kink

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 $k_{B}T\ll E_k$ where $E_k$ 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 $k_{B}T\ll E_k$ the distribution function satisfies the Fokker-Planck equation of Brownian motion.