  While linear response theory, manifested by the fluctuation dissipation
theorem, can be applied at any level of coarse graining, nonlinear response
theory is fundamentally of microscopic nature. For perturbations of equilibrium
systems, we develop an exact theoretical framework for analyzing the nonlinear
(second order) response of coarse grained observables to time-dependent
perturbations, using a path-integral formalism. The resulting expressions
involve correlations of the observable with coarse grained path weights. The
time symmetric part of these weights depends on the paths and perturbation
protocol in a complex manner; in addition, the absence of Markovianity prevents
slicing of the coarse-grained path integral. We show that these difficulties
can be overcome and the response function can be expressed in terms of path
weights corresponding to a single-step perturbation. This formalism thus leads
to an extrapolation scheme where measuring linear responses of coarse-grained
variables suffices to determine their second order response. We illustrate the
validity of the formalism with an exactly solvable four-state model and the
near-critical Ising model.
