Thermodynamic Geometric Constraint on the Spectrum of Markov Rate Matrices
(2025) Physical Review Letters — Vol. 135, n° 25 (2025)
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- Authors
Xu, Guo-Hua 
Kolchinsky, Artemy Ito, Sosuke
- 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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Citations
Xu, G.-H., Kolchinsky, A., Delvenne, J.-C., & Ito, S. (2025). Thermodynamic Geometric Constraint on the Spectrum of Markov Rate Matrices. Physical Review Letters, 135(25). https://doi.org/10.1103/z4t2-18cx (Original work published 2025)