Fluctuating Kinetic Theory: A Poissonian Stochastic Boltzmann Equation
Abstract
We introduce a nonlinear Poissonian fluctuating Boltzmann equation whose noise encodes both the fluctuations and the path large-deviation rate function of the underlying hard-sphere gas. Unlike the Gaussian-noise equations commonly studied in fluctuating hydrodynamics, the present equation raises a new difficulty: a Poisson random measure produces jumps that may destroy the nonnegativity of the solution. To address this problem, we construct a finite-dimensional coarse-grained jump process and prove its well-posedness and nonnegativity. For each fixed mesh, this process satisfies a good path large-deviation principle. We then fix a regularized velocity cutoff and study the asymptotic behavior of the discrete rate functions as the mesh is refined. On a class of biased regular paths, their limit is the corresponding cutoff Boltzmann large-deviation rate function associated with the hard-sphere gas. This establishes the consistency of the coarse-grained fluctuating Boltzmann model with the underlying particle system at the level of path large deviations.
Related
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BibTeX
@misc{wu2026fluctuating,
title = {{Fluctuating Kinetic Theory: A Poissonian Stochastic Boltzmann Equation}},
author = {Zhengyan Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06482},
eprint = {2609.06482},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06482},
}
RIS
TY - UNPB
TI - Fluctuating Kinetic Theory: A Poissonian Stochastic Boltzmann Equation
AU - Zhengyan Wu
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06482
UR - https://arxiv.org/abs/2609.06482
AB - We introduce a nonlinear Poissonian fluctuating Boltzmann equation whose noise encodes both the fluctuations and the path large-deviation rate function of the underlying hard-sphere gas. Unlike the Gaussian-noise equations commonly studied in fluctuating hydrodynamics, the present equation raises a new difficulty: a Poisson random measure produces jumps that may destroy the nonnegativity of the solution. To address this problem, we construct a finite-dimensional coarse-grained jump process and prove its well-posedness and nonnegativity. For each fixed mesh, this process satisfies a good path large-deviation principle. We then fix a regularized velocity cutoff and study the asymptotic behavior of the discrete rate functions as the mesh is refined. On a class of biased regular paths, their limit is the corresponding cutoff Boltzmann large-deviation rate function associated with the hard-sphere gas. This establishes the consistency of the coarse-grained fluctuating Boltzmann model with the underlying particle system at the level of path large deviations.
ER -