Martingale theory for heat and phase-space contraction in heterogeneous diffusions
Abstract
We investigate martingale properties of heat dissipated by generic non-equilibrium overdamped Langevin dynamics with multiple dimensions and transport coefficients that may depend explicitly on time and/or space. We find quantitative criteria that establish that heat dissipation may exhibit transient martingale (time-conserved), submartingale (time increasing), and supermartingale (time decreasing) properties in Langevin dynamics with drift coefficients with spatial non-linear dependencies. Furthermore, we reveal a tight link between the heat statistics at stopping (e.g. first-passage) times with that of a functional quantifying the phase-space contraction in the system dynamics. The theoretical results are used to predict the statistical properties of the heat dissipated by a spherical microscopic particle subject to gravity and double-layer forces and hydrodynamically interacting with a rigid wall, whose validity is confirmed by numerical simulations. These results extend the scope of stochastic energetics by revealing a non-trivial relation between heat fluctuations' statistical properties and the non-linear features of non-equilibrium processes.
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BibTeX
@misc{qin2026martingale,
title = {{Martingale theory for heat and phase-space contraction in heterogeneous diffusions}},
author = {Jing Qin and Nariya Uchida and Édgar Roldán},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20681},
eprint = {2609.20681},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20681},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Martingale theory for heat and phase-space contraction in heterogeneous diffusions
AU - Jing Qin
AU - Nariya Uchida
AU - Édgar Roldán
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20681
UR - https://arxiv.org/abs/2609.20681
AB - We investigate martingale properties of heat dissipated by generic non-equilibrium overdamped Langevin dynamics with multiple dimensions and transport coefficients that may depend explicitly on time and/or space. We find quantitative criteria that establish that heat dissipation may exhibit transient martingale (time-conserved), submartingale (time increasing), and supermartingale (time decreasing) properties in Langevin dynamics with drift coefficients with spatial non-linear dependencies. Furthermore, we reveal a tight link between the heat statistics at stopping (e.g. first-passage) times with that of a functional quantifying the phase-space contraction in the system dynamics. The theoretical results are used to predict the statistical properties of the heat dissipated by a spherical microscopic particle subject to gravity and double-layer forces and hydrodynamically interacting with a rigid wall, whose validity is confirmed by numerical simulations. These results extend the scope of stochastic energetics by revealing a non-trivial relation between heat fluctuations' statistical properties and the non-linear features of non-equilibrium processes.
ER -