发表机构
Technische Universität München（慕尼黑工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出统一研究非线性Lévy型涨落修正的纲领，建立新非线性涨落-耗散关系，推导主涨落动理学方程及α-稳定Dean-Kawasaki方程，并揭示动理学涨落与流体动力学涨落的标度关系。
AI 中文摘要
本文提出了一个研究涨落动理学理论的纲领，旨在以一种统一的方式研究非线性Lévy型涨落对动理学偏微分方程的修正。Vlasov--Fokker--Planck (VFP)、Landau和Boltzmann方程均可分解为一个动理学输运项和一个由非线性对偶耗散势对生成的广义梯度结构。超越线性Onsager结构和经典涨落--耗散关系，我们提出了一种新的非线性Lévy型涨落--耗散关系，该关系识别了与这一广义梯度结构相关的热涨落。在正则倾斜类上，广义梯度结构的典范作用与由所得SPDE预测的Freidlin--Wentzell型作用一致。这种一致性为新的非线性涨落--耗散关系提供了热力学支持，并导出了一个主涨落动理学方程。我们进一步建立了非线性对偶耗散势的一个充分条件，在该条件下其中心累积量允许Lévy--Khintchine表示，并证明了由此产生的Lévy流可以通过高斯白噪声和补偿泊松随机测度实现。作为该框架的一个应用，推导了一个α-稳定的Lévy型Dean-Kawasaki方程。此外，通过在噪声强度与流体动力学标度参数之间的一个特定关系下重新标度涨落Boltzmann方程，我们证明了其哈密顿量收敛到Landau--Lifshitz--Navier--Stokes方程的哈密顿量。这揭示了动理学涨落与涨落流体动力学之间的标度关系。
英文摘要
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.
Comments60 pages
