Exact Fluctuation-Response Relations for Underdamped Langevin Dynamics

Abstract

Thermodynamic uncertainty relations connect current fluctuations to dissipation and are often rooted in fluctuation-response principles. In underdamped dynamics, however, conventional mean-current uncertainty relations can fail, while the underlying connection between fluctuations, response, and dissipation remains elusive. Here we uncover this structure by deriving an exact finite-time fluctuation-response equality for underdamped Langevin dynamics, 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 gives a friction-response thermodynamic uncertainty relation that can be saturated at any finite observation time, leading to an exact variational principle for the total entropy production. We further show that the conventional uncertainty factor of velocity-resolved currents can decay exponentially with dissipation even in driven free diffusion, while 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.