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Mutual linearity in and out of stationarity for Markov jump processes: A trajectory-based approach

Jiming Zheng* and Zhiyue Lu†

  • *Contact author: jiming@unc.edu
  • †Contact author: zhiyuelu@unc.edu
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Phys. Rev. E 114, 014126 – Published 15 July, 2026

DOI: https://doi.org/10.1103/tvqj-7ddn

Abstract

Nonequilibrium response theory is a fundamental framework for understanding how physical systems respond to perturbations. Recently, a mutual linearity has been discovered for Markov jump processes using linear algebra analysis. This mutual linearity states that two observables are linearly dependent on each other in the long-time limit when the transition rate of a single edge is altered. It has also been extended to nonstationary cases for current observables. In this work, we provide a trajectory-based derivation of mutual linearity utilizing the trajectory-level linear response theory. The trajectory approach allows us to generalize the mutual linearity to nonstationary relaxation dynamics for state observables and counting observables. Our results shed light on the fundamental response properties far from equilibrium and the trajectory-level origin of mutual linearity. Our trajectory-based approach makes it possible to generalize the mutual linearity to a broader class of systems, including diffusion processes and open quantum systems.

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Phys. Rev. E 114, 014126– Published 15 July, 2026
  • Received 7 April 2026
  • Accepted 1 July 2026
Article part of CHORUS
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