The Binary-ternary Boltzmann Equation Near Equilibrium

Abstract

The classical Boltzmann equation, formulated by Boltzmann in 1872, describes the statistical behavior of a dilute gas in a non-equilibrium state. While the global well-posedness of the classical Boltzmann equation has been extensively studied and is relatively well-understood, this model becomes inadequate when the gas is no longer sufficiently dilute. Therefore, it is necessary to investigate higher-order extensions of the Boltzmann equation that account for multi-particle interactions. Recently, Ampatzoglou and Pavlović provided the rigorous derivation of both the ternary [\textit{Comm. Math. Phys.} \textbf{387} (2021), no. 2, 793-863.] and the binary-ternary [\textit{Forum Math. Sigma} \textbf{13} (2025), Paper No. e52, 95 pp.] Boltzmann equations from a classical particle system, yielding an explicit form of the ternary collision operator. However, the global existence near equilibrium for the binary-ternary Boltzmann equation remains an interesting and fundamental question, a gap that this paper seeks to address as its primary contribution.

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BibTeX RIS

BibTeX

@misc{hu2026binaryternary,
  title = {{The Binary-ternary Boltzmann Equation Near Equilibrium}},
  author = {Xianshen Hu and Linjie Xiong and Zhihong Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22980},
  eprint = {2609.22980},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22980},
}

RIS

TY  - UNPB
TI  - The Binary-ternary Boltzmann Equation Near Equilibrium
AU  - Xianshen Hu
AU  - Linjie Xiong
AU  - Zhihong Xu
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.22980
UR  - https://arxiv.org/abs/2609.22980
AB  - The classical Boltzmann equation, formulated by Boltzmann in 1872, describes the statistical behavior of a dilute gas in a non-equilibrium state. While the global well-posedness of the classical Boltzmann equation has been extensively studied and is relatively well-understood, this model becomes inadequate when the gas is no longer sufficiently dilute. Therefore, it is necessary to investigate higher-order extensions of the Boltzmann equation that account for multi-particle interactions. Recently, Ampatzoglou and Pavlović provided the rigorous derivation of both the ternary [\textit{Comm. Math. Phys.} \textbf{387} (2021), no. 2, 793-863.] and the binary-ternary [\textit{Forum Math. Sigma} \textbf{13} (2025), Paper No. e52, 95 pp.] Boltzmann equations from a classical particle system, yielding an explicit form of the ternary collision operator. However, the global existence near equilibrium for the binary-ternary Boltzmann equation remains an interesting and fundamental question, a gap that this paper seeks to address as its primary contribution.
ER  -

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