Brownian-Like Motion of a One-Dimensional φ⁴ Kink and Fluctuation-Dissipation Theorem

Abstract

It was found that Brownian-like motion of a φ⁴ kink is induced by two mechanisms. One is a random walk whose basic steps are shifts of the kink due to collisions with thermally excited phonons. It was studied by Wada and Schrieffer. The other is an ordinary Brownian motion, where friction is caused by momentum transfers in the collision. It was studied by the present authors. The relation between the two is clarified with the help of fluctuation-dissipation theorem for the kink motion. In the static limit, the second mechanism is dominant and the Einstein relation D=k_BT⁄MΓ∝(k_BT)⁻¹ holds, where M is mass of the kink. When the frequency is not zero and the temperature is low enough, the first mechanism becomes dominant and the diffusion constant is proportional to (k_BT)². It is four times as large as the previous result.