At the macroscale, transport processes in crystalline solids generate irreversibility. These dissipative effects must be included in the set of hydrodynamic equations and a key question is to derive them from the underlying microscopic dynamics of the particles. Contrary to fluids, crystals manifest long-range order by the spatial periodicity of their atomic structure. The breaking of the three-dimensional continuous group of spatial translations implies the existence of three slow modes in addition to the five slow modes arising from the fundamental conservation laws of mass, energy, and linear momentum. Using a systematic approach based on the local-equilibrium and originally developed for normal fluids, a statistical-mechanical derivation of the hydrodynamic equations, including the Goldstone modes, is introduced. The set of dissipative hydrodynamic equations is obtained as well as expressions for the transport coefficients, in terms of Green-Kubo and Einstein-Helfand formulas. The entropy production rate is shown to be non-negative, in agreement with the second law of thermodynamics. The dispersion relations of the eight hydrodynamic modes are investigated for a cubic crystal, and the transport coefficients are computed numerically from the Einstein-Helfand formulas using a molecular dynamics simulation of hard spheres.

Contact: Sébastien Fumeron