Hydrodynamics of solids

Abstract

The hydrodynamic equations for solids are derived from the connection between hydrodynamic variables (and linearized hydrodynamic modes) and continuous broken symmetries. A simple fluid has five hydrodynamic variables (and modes), one for each conservation law (mass, three components of momentum and energy). As pointed out by Martin, Pershan, and Parodi a crystal must have three additional hydrodynamic variables (and modes) because it does not possess the (threefold) continuous translational invariance of the underlying Hamiltonian of the system. Previous treatments of the hydrodynamics of solids gave only seven hydrodynamic modes. As suggested by Martin, Pershan, and Parodi, the additional mode which was omitted from other earlier treatments, is associated with vacancy diffusion. Previous treatments of vacancy diffusion failed to recognize its coupling with the other hydrodynamic variables. The recognition of the necessity of inclusion of vacancy diffusion leads to the identification of two tensor transport coefficients (in addition to the usual viscosity and thermal conductivity). One is associated with the vacancy flux while the other is connected with the cross effect of thermal diffusion of vacancies (or equivalently a heat flow in a vacancy concentration gradient). The linearized equations are solved for propagation along the [100], [110], and [111] directions in cubic crystals. For these directions the longitudinal equations are isomorphic with those in a binary mixture when vacancy concentration is identified with the concentration in the mixture. In these cases the identification of the additional mode with vacancy diffusion is most clear. It is suggested that this additional mode should appear in the spectrum of light scattered from such crystals.