Jensen-Mercer two-sided bounds for entropy production rate
Abstract
Bounds on entropy production convert limited observations into constraints on hidden dissipation. The Jensen bound expresses a lower bound on the total and subsystem entropy production rates of continuous stochastic systems in terms of mean coordinate rates alone. We show that the same convexity structure yields a complementary upper bound, inspired by Mercer's reflection inequality. For the quadratic entropy-production functional of overdamped Langevin dynamics, Mercer's reflection inequality is identical to the sharp Bhatia-Davis variance bound and coincides with the convex dual of the smallest affine majorant of the functional over the range of the local current. The two bounds yield a two-sided sandwich controlled solely by the mean drift and the extrema of the current, organized into a hierarchy of optimal bounds in which each further observation tightens the ceiling. The thermodynamic uncertainty relation supplies an independent fluctuation floor. We also treat position-dependent diffusion, underdamped dynamics and the tilted periodic potential, for which the weak-barrier series is obtained in closed form and the strong-barrier regime is solved asymptotically. We check the bounds numerically on the Brownian motor.
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BibTeX
@misc{guo2026jensenmercer,
title = {{Jensen-Mercer two-sided bounds for entropy production rate}},
author = {Xiao-Kan Guo and Zhiqiang Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34112},
eprint = {2610.34112},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34112},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Jensen-Mercer two-sided bounds for entropy production rate
AU - Xiao-Kan Guo
AU - Zhiqiang Huang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34112
UR - https://arxiv.org/abs/2610.34112
AB - Bounds on entropy production convert limited observations into constraints on hidden dissipation. The Jensen bound expresses a lower bound on the total and subsystem entropy production rates of continuous stochastic systems in terms of mean coordinate rates alone. We show that the same convexity structure yields a complementary upper bound, inspired by Mercer's reflection inequality. For the quadratic entropy-production functional of overdamped Langevin dynamics, Mercer's reflection inequality is identical to the sharp Bhatia-Davis variance bound and coincides with the convex dual of the smallest affine majorant of the functional over the range of the local current. The two bounds yield a two-sided sandwich controlled solely by the mean drift and the extrema of the current, organized into a hierarchy of optimal bounds in which each further observation tightens the ceiling. The thermodynamic uncertainty relation supplies an independent fluctuation floor. We also treat position-dependent diffusion, underdamped dynamics and the tilted periodic potential, for which the weak-barrier series is obtained in closed form and the strong-barrier regime is solved asymptotically. We check the bounds numerically on the Brownian motor.
ER -