Deriving the second law of thermodynamics and exploring its boundaries
Abstract
The second law of thermodynamics still lacks a general proof applicable to classical and quantum systems across broad ranges of degrees of nonequilibrium, particle interactions, and time scales. In this paper, we show that from Boltzmann's assumption of equal a priori probabilities, the second law naturally emerges via the phase-space continuity equation for the macrostate-level probability f defined over fixed Ω-intervals, where Ω is the number of microstates. Further analysis suggests a more general assumption underlying the second law: f increases monotonically with Ω, i.e., ∂ f/∂ Ω> 0. Based on these findings, the second law can be readily derived for fully chaotic systems. The derivation does not rely on dynamical details, highlighting the statistical nature of entropy increase. For the dilute hard-sphere gas model considered in the H-theorem, the second law may also be established through a simplified approach that requires the Boltzmann equation but does not use the H-function. In contrast, if the positive correlation between f and Ω breaks down, i.e., ∂ f/∂ Ω≤ 0 (e.g., because of local nonchaoticity), the second law of thermodynamics may not apply: entropy can decrease spontaneously with no energetic penalty, and the system can reach an intrinsic-nonequilibrium steady state.
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BibTeX
@misc{qiao2026deriving,
title = {{Deriving the second law of thermodynamics and exploring its boundaries}},
author = {Yu Qiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.28765},
eprint = {2607.28765},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.28765},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Deriving the second law of thermodynamics and exploring its boundaries
AU - Yu Qiao
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2607.28765
UR - https://arxiv.org/abs/2607.28765
AB - The second law of thermodynamics still lacks a general proof applicable to classical and quantum systems across broad ranges of degrees of nonequilibrium, particle interactions, and time scales. In this paper, we show that from Boltzmann's assumption of equal a priori probabilities, the second law naturally emerges via the phase-space continuity equation for the macrostate-level probability f defined over fixed Ω-intervals, where Ω is the number of microstates. Further analysis suggests a more general assumption underlying the second law: f increases monotonically with Ω, i.e., ∂ f/∂ Ω> 0. Based on these findings, the second law can be readily derived for fully chaotic systems. The derivation does not rely on dynamical details, highlighting the statistical nature of entropy increase. For the dilute hard-sphere gas model considered in the H-theorem, the second law may also be established through a simplified approach that requires the Boltzmann equation but does not use the H-function. In contrast, if the positive correlation between f and Ω breaks down, i.e., ∂ f/∂ Ω≤ 0 (e.g., because of local nonchaoticity), the second law of thermodynamics may not apply: entropy can decrease spontaneously with no energetic penalty, and the system can reach an intrinsic-nonequilibrium steady state.
ER -