Comment on Eliashberg's free energy of a strong-coupling metal

We derive in the Migdal-Eliashberg approximation new formulas for the free energy, entropy, and specific heat of conduction electrons coupled to phonons. Analytic results in terms of standard functions are obtained for Einstein phonons, and the calculation for realistic phonon spectra is reduced to simple, weighted integrals over Eliashberg's spectral function ² F(). Our derivation requires a careful analysis of the range of validity of the Midgal-Eliashberg approximation which leads to some insight into the problem of lattice instabilities induced by a strong electron-phonon coupling.