Nonlinear response relations and fluctuation-response inequalities for nonequilibrium stochastic systems

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.