A Program for Fluctuating Kinetic Theory
Abstract
A program for fluctuating kinetic theory is proposed in this paper to study nonlinear Lévy-type fluctuation corrections to kinetic PDEs in a unified way. The Vlasov--Fokker--Planck (VFP), Landau, and Boltzmann equations can each be decomposed into a kinetic transport term and a generalized gradient structure generated by a nonlinear dual pair of dissipation potentials. Going beyond the linear Onsager structure and the classical fluctuation--dissipation relation, we propose a new nonlinear Lévy-type fluctuation--dissipation relation that identifies the thermal fluctuations associated with this generalized gradient structure. On a regular tilted class, the canonical action of the generalized gradient structure agrees with the Freidlin--Wentzell-type action predicted by the resulting SPDE. This consistency gives thermodynamic support for the new nonlinear fluctuation--dissipation relation and leads to a master fluctuating kinetic equation. We further establish a sufficient condition on the nonlinear dual dissipation potential under which its centered cumulant admits a Lévy--Khintchine representation, and show that the resulting Lévy current can be realized through Gaussian white noise and a compensated Poisson random measure. As an application of this framework, an α-stable Lévy-type Dean-Kawasaki equation is derived. Furthermore, by rescaling the fluctuating Boltzmann equation under a distinguished relation between the noise intensity and the hydrodynamic scaling parameter, we prove that its Hamiltonian converges to the Hamiltonian of the Landau--Lifshitz--Navier--Stokes equations. This reveals a scaling relation between kinetic fluctuations and fluctuating hydrodynamics.
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BibTeX
@misc{wu2026program,
title = {{A Program for Fluctuating Kinetic Theory}},
author = {Zhengyan Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08247},
eprint = {2610.08247},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08247},
}
RIS
TY - UNPB
TI - A Program for Fluctuating Kinetic Theory
AU - Zhengyan Wu
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08247
UR - https://arxiv.org/abs/2610.08247
AB - A program for fluctuating kinetic theory is proposed in this paper to study nonlinear Lévy-type fluctuation corrections to kinetic PDEs in a unified way. The Vlasov--Fokker--Planck (VFP), Landau, and Boltzmann equations can each be decomposed into a kinetic transport term and a generalized gradient structure generated by a nonlinear dual pair of dissipation potentials. Going beyond the linear Onsager structure and the classical fluctuation--dissipation relation, we propose a new nonlinear Lévy-type fluctuation--dissipation relation that identifies the thermal fluctuations associated with this generalized gradient structure. On a regular tilted class, the canonical action of the generalized gradient structure agrees with the Freidlin--Wentzell-type action predicted by the resulting SPDE. This consistency gives thermodynamic support for the new nonlinear fluctuation--dissipation relation and leads to a master fluctuating kinetic equation. We further establish a sufficient condition on the nonlinear dual dissipation potential under which its centered cumulant admits a Lévy--Khintchine representation, and show that the resulting Lévy current can be realized through Gaussian white noise and a compensated Poisson random measure. As an application of this framework, an α-stable Lévy-type Dean-Kawasaki equation is derived. Furthermore, by rescaling the fluctuating Boltzmann equation under a distinguished relation between the noise intensity and the hydrodynamic scaling parameter, we prove that its Hamiltonian converges to the Hamiltonian of the Landau--Lifshitz--Navier--Stokes equations. This reveals a scaling relation between kinetic fluctuations and fluctuating hydrodynamics.
ER -