Thermodynamic correlation inequalities for finite times and transients

Abstract Recently, a thermodynamic bound on correlation times was formulated in Dechant et al (2023 Phys. Rev. Lett. 131 167101), showing how the decay of correlations in Langevin dynamics is bounded by short-time fluctuations and dissipation. Whereas these original results only address very long observation times in steady-state dynamics, we here generalize the respective inequalities to finite observations and general initial conditions. We utilize the connection between correlations and the fluctuations of time-integrated density functionals and generalize the direct stochastic calculus approach from Dieball and Godec (2023 Phys. Rev. Lett. 130 087101) which paves the way for further generalizations. We address the connection between short and long time scales, as well as the saturation of the bounds via complementary spectral-theoretic arguments. Motivated by the spectral insight, we formulate all results also for complex-valued observables.