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.
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BibTeX
@misc{vu2026exact,
title = {{Exact Fluctuation-Response Relations for Underdamped Langevin Dynamics}},
author = {Tan Van Vu and Van Tuan Vo and Ruicheng Bao and Keiji Saito},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.20013},
eprint = {2608.20013},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.20013},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact Fluctuation-Response Relations for Underdamped Langevin Dynamics
AU - Tan Van Vu
AU - Van Tuan Vo
AU - Ruicheng Bao
AU - Keiji Saito
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2608.20013
UR - https://arxiv.org/abs/2608.20013
AB - 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.
ER -