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Unified Linear Fluctuation-Response Theory Arbitrarily Far from Equilibrium

Jiming Zheng, Zhiyue Lu

TL;DR

This work addresses the challenge of predicting system response in far-from-equilibrium dynamics beyond the fluctuation-dissipation theorem. It introduces an exact linear response equality for arbitrary Markov processes, decomposing the response into edge-wise correlations via the dynamical discrepancies Θ_{ij} and the trajectory score Λ, with Λ[Xτ] = ∑_{i≠j} (∂ ln R_{ij}/∂λ) Θ_{ij}. By connecting this decomposition to trajectory Fisher information I(λ) and Fluctuation-Response Inequalities, the authors derive high-dimensional Cramér–Rao bounds and establish equivalences and monotonicity properties for multi-parameter and multi-observable cases. Numerically, the framework estimates response properties from unperturbed trajectories, outperforming finite-difference approaches and enabling efficient analysis of large networks and biological systems far from equilibrium.

Abstract

Understanding how systems respond to external perturbations is a fundamental challenge in physics, particularly for non-equilibrium and non-stationary processes. The fluctuation-dissipation theorem provides a complete framework for near-equilibrium systems, and various bounds have recently been reported for specific non-equilibrium regimes. Here, we present an exact response equality for arbitrary Markov processes that decompose system response into spatial correlations of local dynamical events. This decomposition reveals that response properties are encoded in correlations between transitions and dwelling times across the network, providing a natural generalization of the fluctuation-dissipation theorem and recently developed non-equilibrium linear response relations. Our theory unifies existing response bounds, extends them to time-dependent processes, and reveals fundamental monotonicity properties of the tightness of multi-parameter response inequalities. Beyond its theoretical significance, this framework enables efficient numerical evaluation of response properties from sampling unperturbed trajectories, offering significant advantages over traditional finite-difference approaches for estimating response properties of complex networks and biological systems far from equilibrium.

Unified Linear Fluctuation-Response Theory Arbitrarily Far from Equilibrium

TL;DR

This work addresses the challenge of predicting system response in far-from-equilibrium dynamics beyond the fluctuation-dissipation theorem. It introduces an exact linear response equality for arbitrary Markov processes, decomposing the response into edge-wise correlations via the dynamical discrepancies Θ_{ij} and the trajectory score Λ, with Λ[Xτ] = ∑_{i≠j} (∂ ln R_{ij}/∂λ) Θ_{ij}. By connecting this decomposition to trajectory Fisher information I(λ) and Fluctuation-Response Inequalities, the authors derive high-dimensional Cramér–Rao bounds and establish equivalences and monotonicity properties for multi-parameter and multi-observable cases. Numerically, the framework estimates response properties from unperturbed trajectories, outperforming finite-difference approaches and enabling efficient analysis of large networks and biological systems far from equilibrium.

Abstract

Understanding how systems respond to external perturbations is a fundamental challenge in physics, particularly for non-equilibrium and non-stationary processes. The fluctuation-dissipation theorem provides a complete framework for near-equilibrium systems, and various bounds have recently been reported for specific non-equilibrium regimes. Here, we present an exact response equality for arbitrary Markov processes that decompose system response into spatial correlations of local dynamical events. This decomposition reveals that response properties are encoded in correlations between transitions and dwelling times across the network, providing a natural generalization of the fluctuation-dissipation theorem and recently developed non-equilibrium linear response relations. Our theory unifies existing response bounds, extends them to time-dependent processes, and reveals fundamental monotonicity properties of the tightness of multi-parameter response inequalities. Beyond its theoretical significance, this framework enables efficient numerical evaluation of response properties from sampling unperturbed trajectories, offering significant advantages over traditional finite-difference approaches for estimating response properties of complex networks and biological systems far from equilibrium.
Paper Structure (15 sections, 1 theorem, 47 equations, 2 figures)

This paper contains 15 sections, 1 theorem, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Linear Response Relation for Stochastic Systems
  3. Markov Jump Processes
  4. Spatial Correlation Analysis
  5. From Linear Response Equality to Fluctuation-Response Inequalities
  6. Trajectory Fisher Information and FRI
  7. Multi-dimensional Cramér-Rao Inequality
  8. Numerical Efficiency
  9. Numerical Comparison on a Three-state Markov Network
  10. Numerical Comparison on a Large Markov Network
  11. Conclusion
  12. Equivalence between two types of multi-dimensional Cramér-Rao inequalities
  13. Information Monotonicity
  14. Information Monotonicity for Adding Parameters
  15. Information Monotonicity for Global Parameter Decomposition

Key Result

Lemma 1

(Block Matrix Inverse) Given a block matrix if $D$ is invertible, define the Schur complement of $D$ as $\Phi$ is invertible if and only if $\Sigma$ is invertible. Its inverse is

Figures (2)

  • Figure 1: Four sets of kinetic Monte Carlo simulations on a three-state Markov system with the rate matrix in \ref{['eq: rate matrix']}. Each point and curve is obtained from trajectory averages. The traditional finite-difference method is repeated 30 times for each subfigure to show its convergence issue. The number of trajectories $N_\text{traj}$ is $8000$ for (a) and (c), and is $10^6$ for (b) and (d). The $\Delta \lambda$ in the finite-difference method is $0.1$ for (a) and (b), and is $1$ for (c) and (d).
  • Figure 2: A Markov network graph with $100$ states, with its transition rates randomly generated with values between $10$ and $100$. This system's kinetic Monte Carlo simulations are performed with the same initial probability (randomly generated). (a) The input edge (control) is in red, and the output edge (observable) is in blue. (b) In each realization, we perform the finite-difference method (orange dots) and our method (blue curves) for the sampling size $N_{\text{traj}} = 2\times 10^{5}$. We repeat these realizations $15$ times to illustrate the variances. The parameter difference for the traditional finite-difference method is $\Delta \lambda = 1$.

Theorems & Definitions (1)

  • Lemma 1