Macroscopic tunneling of Cooper pairs of electrons in a system of two superconductors separated by an insulator was modeled by Caldeira and Leggett by quantum-mechanical motion of a particle through the potential barrier. Friction (conductivity of the Josephson contact in experiments) was modeled by a one-dimensional phonon thermostat that interacts with the particle. Then the tunneling exponent was expressed in terms of the friction coefficient determined by the thermostat parameters. Alternatively, we propose a direct introduction of the friction coefficient into a classical action in the tunneling exponent using the method of imaginary time. As a result, the numerical coefficient of the classical friction contribution to the tunneling exponent is obtained.
V. P. Krainov (Wed,) studied this question.