Brownian motion of a domain wall and the diffusion constants

We have studied interactions between a domain wall and phonons in a one-dimensional-model system of a structurally unstable lattice with a double-well local potential and nearest-neighbor coupling. We find that a nonlinear effect in the interacting phonon amplitudes gives rise to a Brownian-like motion of isolated domain walls at low temperatures as well as higher harmonic generation of transmitted and reflected phonons. When there is a domain wall at rest, it was known that the linearized equation of motion has three types of independent solutions: "translation mode," "amplitude oscillation" of the domain wall, and propagating "phonons." In the second-order approximation, these modes interact. An incoming phonon produces a translation of the wall, giving rise to its Brownian motion. The magnitude of the translation is computed together with the amplitude and phase shift of the higher harmonics. We estimate the diffusion constant of walls, using the fluctuation-dissipation theorem and the thermal average over the phonons, to be $D=0.516\mathrm{}$ ${\omega }_0{l}^2{\left(\frac{k_{B}T}{{\mu }_0^2{\omega }_0^2}\right)}^2$, where $l$ is the lattice spacing and ${\omega }_0$ is the frequency of small oscillation of the ion (with mass $m$) around $u_0$, a minimum point of an isolated double-well potential.