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 kB T per mode. The translational mode (zero mode) is found to carry an average energy of 1/2kBT. 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.
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Salerno et al. (1984) studied this question.
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