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 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.
Highlights
Many systems of practical and scientific relevance are of an intrinsic stochastic nature with properties dominated by fluctuations, e.g., colloidal particles, protein folding networks, molecular motors, or stochastic heat engines [1]
For perturbations of equilibrium systems, we develop an exact theoretical framework for analyzing the nonlinear response of coarse-grained observables to time-dependent perturbations, using a path-integral formalism
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
Summary
Many systems of practical and scientific relevance are of an intrinsic stochastic nature with properties dominated by fluctuations, e.g., colloidal particles, protein folding networks, molecular motors, or stochastic heat engines [1]. This is the case, for example, if a driving protocol acts on the unknown degree of freedom Such questions in relation to entropy production, work, and other thermodynamic notions in stochastic processes have been analyzed under coarse-graining, both theoretically [23,24,31,32] and experimentally [30,33]. Coarse-graining the path integrals yields coarse-grained path weights, including entropy production, and the more difficult time-symmetric part of the corresponding weights, from which the second-order response can be found These formal expressions can be used in practice, e.g., via an extrapolation scheme. We show how to measure the second-order response for any protocol from linear perturbations with one step only, thereby greatly facilitating the measurement This concept is illustrated and verified below in an analytically solvable jump process and in simulations of the two-dimensional (2D) Ising model
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