I
will derive the Green-Kubo expressions for linear transport
coefficients with
the aid of the Mori-Zwanzig projection operator
formalism. The
essential physical step in this procedure consists in
the identification
of the slow variables that are required for a closed
macroscopic
description of the dynamics. For solid systems, besides
energy and momentum
density, these include displacement fields,
describing the
deviations of particle positions from their equilibrium
averages. As a
result of this the shear modes
describing momentum
relaxation in fluids, are replaced by pairs of
transverse sound
modes. I will discuss the consequences of this for the
long time behavior
of the current-current time correlation functions
occurring in the
Green-kubo expressions.
As part of these
derivations I will discuss the proper equilibrium (or
metastable state)
description of solids under anisotropic and/or
nonuniform stress
and the slight differences in the common definitions
of hydrodynamic or
elastodynamic modes in fluids and solids
respectively, plus
the ways to relate these. Finally I will show how to
include processes
such as hopping diffusion in the Green-Kubo formalism.