  We present a microscopic derivation of the nonlinear fluctuating hydrodynamic
equation for the homogeneous crystalline solid from the Hamiltonian description
of a many-particle system. We propose a microscopic expression of the
displacement field that correctly generates the nonlinear elastic properties of
the solid and find the nonlinear mode-coupling terms in reversible currents
which are consistent with the phenomenological equation. The derivation relies
on the projection onto the coarse-grained fields including the displacement
field, the long-wavelength expansion, and the stationarity condition of the
Fokker-Planck equation.

Statistical Mechanics;Soft Condensed Matter