Exact Fluctuation-Response Relations for Underdamped Langevin Dynamics

What the math gives to ML

The paper identifies an exact finite-time fluctuation-response equality for underdamped Langevin processes, including time-dependent driving and perturbations that alter both drift and diffusion. Its transferable asset is that response to a physically matched perturbation, rather than the mean current itself, controls fluctuations and dissipation through a variational characterization of dynamically generated variance. In neural-network training, this suggests calibrating optimizer noise and damping from measured response-covariance statistics, while using violations of the equality as diagnostics for excessive gradient noise or discretization instability.