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  • Letter
  • Open Access

Universal response inequalities beyond steady states via trajectory information geometry

Jiming Zheng* and Zhiyue Lu†

  • *Contact author: jiming@unc.edu
  • †Contact author: zhiyuelu@unc.edu
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Phys. Rev. E 112, L012103 – Published 25 July, 2025

DOI: https://doi.org/10.1103/scg2-qkxv

Abstract

Fluctuation-dissipation relations elucidate the response of near-equilibrium systems to environmental changes, with recent advances extending response theory to nonequilibrium steady states. However, a general response theory for systems evolving far from steady states has remained elusive. This letter presents a complete trajectory information geometric framework that generalizes response theory for nonstationary Markov processes. By constructing the full trajectory probability manifold and identifying a globally orthogonal coordinate system defined by transition rates, we derive a diagonal Fisher information metric that enables explicit calculations in this high-dimensional space. From the local metric structure, we obtain a Cramér-Rao-type inequality that bounds the linear response of arbitrary nonstationary observables. Furthermore, by analyzing the global geometry of this manifold, we derive a universal nonperturbative (nonlinear) response inequality in terms of geodesic length. This geometric framework reveals deep connections between dynamical activity, observable variance, and system sensitivity, and it encompasses or anticipates several recent results as special cases. Our approach offers new design principles for responsive behaviors in far-from-equilibrium systems.

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Phys. Rev. E 112, L012103– Published 25 July, 2025
  • Received 26 April 2024
  • Revised 28 May 2025
  • Accepted 27 June 2025
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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.

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