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
本文提出泊松型涨落玻尔兹曼方程，通过构造有限维粗粒化跳跃过程保证非负性，并证明其路径大偏差极限与硬球气体截断玻尔兹曼率函数一致。
AI 中文摘要
我们引入一个非线性泊松型涨落玻尔兹曼方程，其噪声同时编码了底层硬球气体的涨落和路径大偏差率函数。与涨落流体力学中通常研究的高斯噪声方程不同，本方程提出了一个新的困难：泊松随机测度产生的跳跃可能破坏解的非负性。为解决此问题，我们构造了一个有限维粗粒化跳跃过程，并证明其适定性和非负性。对每个固定网格，该过程满足一个良好的路径大偏差原理。然后我们固定一个正则化速度截断，并研究当网格细化时离散率函数的渐近行为。在一类有偏正则路径上，其极限是对应的硬球气体截断玻尔兹曼大偏差率函数。这建立了粗粒化涨落玻尔兹曼模型与底层粒子系统在路径大偏差层面的一致性。
英文摘要
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.
Comments71 pages
