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.

• Received 8 May 2020
• Accepted 21 September 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.043123

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society