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Ambegaokar and Tewordt approximation assume BCS model BCS theory BCS type Boltzman equation canonical momentum chemical potential component conservation contribution Cooper pair critical temperature D(ll Debye phonons decrease dependence derivation difference variables distribution function driving terms electron-phonon interaction electron-phonon system electrons Eliashberg equilibrium expression external field F(ll F+(ll Fermi surface Fourier transform frequency G(ll gauge transformation Green's function Hartree-Fock impurity scattering integral k-mv k,co k,oo k+mv _ k+mV k+mV Kadanoff lead Let us consider lifetime limit is valid low energy neglect normal metal pairs particle perturbation phonon distribution Phys quasiparticle limit reduced conductivity renormalize the mass result scattering processes scattering terms scattering-in terms scattering-out self-energy Simpson's rule spectral function strong-coupling materials strong-coupling theory superfluid equation superfluid flow superfluid velocity thermal conductivity thermodynamic equilibrium transport equations transport phenomena weak-coupling materials zero Zl Z2