Nonlinear Response Identities and Bounds for Nonequilibrium Steady States

Understanding how systems respond to external perturbations is fundamental to statistical physics. For systems far from equilibrium, a general framework for response remains elusive. While progress has been made on the linear response of nonequilibrium systems, a theory for the nonlinear regime under finite perturbations has been lacking. Here, building on a novel connection between response and mean first-passage times in continuous-time Markov chains, we derive a comprehensive theory for the nonlinear response to archetypal local perturbations. We establish an exact identity that universally connects the nonlinear response of any observable to its linear counterpart via a simple scaling factor. This identity directly yields universal bounds on the response magnitude. Furthermore, we establish a universal bound on response resolution -- an inequality constraining an observable's change by its intrinsic fluctuations -- thereby setting a fundamental limit on signal-to-noise ratio. These results provide a rigorous and general framework for analyzing nonlinear response far from equilibrium, which we illustrate with an application to transcriptional regulation.