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In discrete-state Markovian systems, many important properties ofcorrelations functions and relaxation dynamics depend on the spectrum of therate matrix. Here we demonstrate the existence of a universal trade-off betweenthermodynamic and spectral properties. We show that the entropy productionrate, the fundamental thermodynamic cost of a nonequilibrium steady state,bounds the difference between the eigenvalues of a nonequilibrium rate matrixand a reference equilibrium rate matrix. Using this result, we derivethermodynamic bounds on the spectral gap, which governs autocorrelations timesand the speed of relaxation to steady state. We also derive thermodynamicbounds on the imaginary eigenvalues, which govern the speed of oscillations. Weillustrate our approach using a simple model of biomolecular sensing.

Thermodynamic bounds on spectral perturbations, with applications to oscillations and relaxation dynamics