The Spectral Zeta Function of Markov Generators

We introduce the spectral zeta function \(\zeta_L(s) = \mathrm{Tr}(L^{-s})\) of the generator \(L\) of a continuous-time Markov process as a universal diagnostic for complex dynamical systems. From a single analytic function of one complex variable \(s\), we extract four computable observables: (i) the mean first-passage time (\(s = 1\)), (ii) pathway entropy (a Shannon entropy over mode relaxation times), (iii) effective spectral dimension (via the Weyl law for eigenvalue asymptotics), and (iv) exponential purity of the relaxation (via the heat kernel trace). These four quantities characterize, respectively, the timescale, diversity, dimensionality, and simplicity of the system's dominant dynamics — from a single matrix.

We prove the formulas in general, establish their relationships, and validate all four on the Fokker–Planck generators of 14 protein domains from molecular dynamics (mdCATH), where the correlations reach \(r = 0.91\) (Weyl law) and \(r = 0.88\) (mean time formula). We then show how the same diagnostics apply without modification to credit portfolio dynamics, plasma confinement, epidemic spreading, quantum decoherence, and chemical reaction networks. The spectral zeta function provides a domain-agnostic language for comparing dynamical complexity across fields.