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Thermodynamic Geometric Constraint on the Spectrum of Markov Rate Matrices

Guo-Hua Xu1,*, Artemy Kolchinsky2, Jean-Charles Delvenne3,4, and Sosuke Ito1,5

  • *Contact author: guohua.xu@ubi.s.u-tokyo.ac.jp
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Phys. Rev. Lett. 135, 257102 – Published 18 December, 2025

DOI: https://doi.org/10.1103/z4t2-18cx

Abstract

The spectrum of Markov generators encodes physical information beyond simple decay and oscillation, which reflects irreversibility and governs the structure of correlation functions. In this Letter, we prove an ellipse theorem that provides a universal thermodynamic geometric constraint on the spectrum of Markov rate matrices. The theorem states that all eigenvalues lie within a specific ellipse in the complex plane. In particular, the imaginary parts of the spectrum, which indicate oscillatory modes, are bounded by the maximum thermodynamic force associated with individual transitions. This spectral bound further constrains the initial short-time behavior of correlation functions between two arbitrary observables. Finally, we compare our result with a previously proposed conjecture, which remains an open problem and warrants further investigation.

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Phys. Rev. Lett. 135, 257102– Published 18 December, 2025
  • Received 25 July 2025
  • Accepted 19 November 2025
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