  The theory of fluctuating hydrodynamics has been an important tool for
analyzing macroscopic behavior in nonlinear lattices. However, despite its
practical success, its microscopic derivation is still incomplete. In this
work, we provide the microscopic derivation of fluctuating hydrodynamics, using
the coarse-graining and projection technique; the equivalence of ensembles
turns out to be critical. The Green-Kubo (GK) like formula for the bare
transport coefficients is presented in a numerically computable form. Our
numerical simulations show that the bare transport coefficients exist for a
sufficiently large but finite coarse-graining length in the infinite lattice
within the framework of the GK like formula. This demonstrates that the bare
transport coefficients uniquely exist for each physical system.
