Predicting how systems respond to external perturbations far from equilibrium remains a fundamental challenge across physics, chemistry, and biology. We present a unified response framework for stochastic Markov dynamics that integrates linear and nonlinear perturbations. Our formalism expresses nonlinear responses of observables in terms of the covariance between the observable and a nonlinear conjugate variable. The nonlinear conjugate variable is subject to the complete Bell polynomial form and is determined by the stochastic entropy production. In addition, the fluctuation-response inequalities are also derived for nonlinear responses, unraveling the general trade-off relations between nonlinear response and systems' fluctuations far from equilibrium. The validity of our theory is verified by the numerical results from a symmetric exclusion process. By unifying and extending nonequilibrium linear response theories, our approach can provide principled design rules for sensitive, adaptive synthetic, and biological networks.
Zheng et al. (Tue,) studied this question.