  We explain a method, inspired by control theory model reduction and
interpolation theory, that rigorously establishes the types of coarse graining
that are appropriate for systems with quadratic, generalized Hamiltonians. For
such systems, general conditions are given that establish when local coarse
grainings should be valid. Interestingly, our analysis provides a reduction
method that is valid regardless of whether or not the system is isotropic. We
provide the linear harmonic chain as a prototypical example. Additionally,
these reduction techniques are based on the dynamic response of the system, and
hence are also applicable to nonequilibrium systems.
