Phonons and solitons in the "thermal" sine-Gordon system

Standard methods of stochastic processes are used to study the coupling of the sine-Gordon system with a heat reservoir. As a result we find thermal phonons with an average energy of $k_B T$ per mode. The translational mode (zero mode) is found to carry an average energy of $\frac{1}{2}{k}_{B}T$. This last value is just the energy in the Brownian motion of a thermal soliton. These results are in agreement with those obtained by the use of a statistical-mechanical description of a dilute soliton gas. Connection of the above results with Josephson junctions and the extension of the analysis to more general equations is also discussed.