Real Time Dynamics of Soliton Diffusion

We study the non-equilibrium dynamics of solitons in model Hamiltonians for Peierls dimerized quasi-one dimensional conducting polymers and commensurate charge density wave systems. The real time equation of motion for the collective coordinate of the soliton and the associated Langevin equation is found in a consistent adiabatic expansion in terms of the ratio of the optical phonon or phason frequency to the soliton mass. The equation of motion for the soliton collective coordinate allows to obtain the frequency dependent soliton conductivity. In lowest order we find that although the coefficient of static friction vanishes, there is dynamical dissipation represented by a non-Markovian dissipative kernel associated with two-phonon processes. The correlation function of the noise in the quantum Langevin equation and the dissipative kernel are related by a generalized quantum fluctuation dissipation relation. To lowest adiabatic order we find that the noise is gaussian, additive and colored. We numerically solve the equations of motion in lowest adiabatic order and compare to the Markovian approximation which is shown to fail both in the $\phi^4$ and the Sine Gordon models even at large temperatures.