The Hidden Second Law of Thermodynamics behind the Boltzmann-Grad Limit
Abstract
We show that the Boltzmann-Grad (BG) limit, Nεd−1=α=const, is not a neutral scaling condition but already encodes the Second Law of Thermodynamics, resolving Loschmidt's paradox. Newton's First Law is the bridge: applying the BG condition locally to two adjacent cells at different temperatures, we decompose their evolution into a free-streaming period δtf, governed by inertia alone, during which molecules cross the interface ballistically and still carry their cell of origin's velocity distribution; and a collisional adjustment period δta, governed by Newton's Second Law, during which the locally mixed, non-Gaussian population relaxes toward a new Gaussian equilibrium. Using the one-dimensional Gaussian velocity distribution, we show that isotropy alone produces non-zero net molecular and momentum flux across the interface, directed from the warmer to the cooler cell -- the microscopic origin of the pressure gradient driving the system toward uniformity. Treating relaxation as a sum of random collisional increments, the Central Limit Theorem fixes the collision count n* required for convergence, giving δta ∼ n* ℓ/v̅. This decomposition resolves Loschmidt's paradox: his reversal acts on a fixed, identifiable set of molecules, while the temperature field Boltzmann's equation tracks is carried by an ever-changing local population; the step relating the two -- convergence under the Central Limit Theorem -- is itself a coarse-graining that discards microscopic information by construction, recoverable only at an unavoidable thermodynamic cost, by Landauer's principle. Temporal irreversibility is present from the moment the BG framework, and the coarse-graining it requires, are adopted -- not generated during derivation.
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BibTeX
@misc{wu2026hidden,
title = {{The Hidden Second Law of Thermodynamics behind the Boltzmann-Grad Limit}},
author = {Zhaohua Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.25056},
eprint = {2608.25056},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25056},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Hidden Second Law of Thermodynamics behind the Boltzmann-Grad Limit
AU - Zhaohua Wu
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2608.25056
UR - https://arxiv.org/abs/2608.25056
AB - We show that the Boltzmann-Grad (BG) limit, Nε^d−1=α=const, is not a neutral scaling condition but already encodes the Second Law of Thermodynamics, resolving Loschmidt's paradox. Newton's First Law is the bridge: applying the BG condition locally to two adjacent cells at different temperatures, we decompose their evolution into a free-streaming period δt_f, governed by inertia alone, during which molecules cross the interface ballistically and still carry their cell of origin's velocity distribution; and a collisional adjustment period δt_a, governed by Newton's Second Law, during which the locally mixed, non-Gaussian population relaxes toward a new Gaussian equilibrium. Using the one-dimensional Gaussian velocity distribution, we show that isotropy alone produces non-zero net molecular and momentum flux across the interface, directed from the warmer to the cooler cell -- the microscopic origin of the pressure gradient driving the system toward uniformity. Treating relaxation as a sum of random collisional increments, the Central Limit Theorem fixes the collision count n^* required for convergence, giving δt_a ∼ n^* ℓ/v̅. This decomposition resolves Loschmidt's paradox: his reversal acts on a fixed, identifiable set of molecules, while the temperature field Boltzmann's equation tracks is carried by an ever-changing local population; the step relating the two -- convergence under the Central Limit Theorem -- is itself a coarse-graining that discards microscopic information by construction, recoverable only at an unavoidable thermodynamic cost, by Landauer's principle. Temporal irreversibility is present from the moment the BG framework, and the coarse-graining it requires, are adopted -- not generated during derivation.
ER -