Thermodynamic Geometric Constraint on the Spectrum of Markov Rate Matrices

TL;DR

This paper establishes a universal geometric constraint on the spectrum of Markov rate matrices, linking eigenvalues to thermodynamic forces and providing insights into correlation functions and system irreversibility.

Contribution

It proves an ellipse theorem that bounds the spectrum of Markov generators in the complex plane based on thermodynamic principles, a novel geometric constraint.

Findings

Eigenvalues lie within a specific ellipse in the complex plane.

Imaginary parts of eigenvalues are bounded by thermodynamic forces.

Spectral bounds influence short-time correlation behaviors.

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 work, 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.