Abstract
A new approach to the microscopic treatment of a system of interacting phonons is developed based on techniques used by Mori to describe density fluctuations in liquids. Introduction of a matrix of retarded Greens functions allows a description of dynamic cooperative phenomena especially suited to investigate coupling of displacement and energy density correlations in the hydrodynamic domain. The general method is applied in conjunction with perturbation theory to the model of an anharmonic crystal. The usual expressions for frequency shift and damping of thermal phonons are reproduced. For long wavelengths and low frequencies a three-variable theory, which includes the fluctuations in the energy density, leads to expressions expected from phenomenological equations. This derivation is direct and avoids complicated treatment of hydrodynamic singularities of other microscopic theories. Finally, it is shown that a four-variable theory can account for second sound and its coupling to first sound.
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Meier, P.F. Displacement and energy density correlations in solids. Phys cond Matter 17, 17–36 (1973). https://doi.org/10.1007/BF01464263
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DOI: https://doi.org/10.1007/BF01464263