Exact Fluctuation-Response Relations for Underdamped Langevin Dynamics

Abstract

Thermodynamic uncertainty relations connect current fluctuations to dissipation and, in certain settings, can emerge from underlying fluctuation-response relations. However, in underdamped dynamics, conventional mean-current uncertainty relations can fail, and the underlying connection between fluctuations, response, and dissipation remains elusive. Here, we uncover this connection by deriving an exact finite-time fluctuation-response equality for underdamped Langevin dynamics, which is valid for arbitrary time-dependent driving and general additive observables. The equality yields sharp response bounds and a variational characterization of the dynamically generated variance. Choosing the perturbation along the irreversible probability flow, we obtain an experimentally accessible friction-response thermodynamic uncertainty relation that is saturable at any finite observation time. It provides both an exact variational principle for entropy production and a weak-noise coherence-dissipation bound for underdamped limit cycles. We further show that the conventional uncertainty factor of velocity-resolved currents can decay exponentially with dissipation, even in driven free diffusion, whereas the friction-response factor retains its universal lower bound. These results establish response, rather than the mean current itself, as the quantity directly linking fluctuations and dissipation in underdamped dynamics.