Abstract
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 ${k}_{B}T\ensuremath{\ll}{E}_{k}$ 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 ${k}_{B}T\ensuremath{\ll}{E}_{k}$ the distribution function satisfies the Fokker-Planck equation of Brownian motion.
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