Coarse-grained second-order response theory

Abstract

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 a 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.