Particle-based stochastic approximations of the Boltzmann equation are popular tools for simulations of non-equilibrium gas flows, for which the Navier-Stokes-Fourier equations fail to provide accurate description. However, these numerical methods are computationally demanding, especially in the near-continuum regime, where the collisions become overwhelming. On the other hand, the Fokker-Planck kinetic models offer an efficient alternative, as the binary collisions are described by a diffusive process. Despite the intuitive advantage, rigorous and efficient Fokker-Planck approximations of the Boltzmann equation remain an open problem. On one hand, the moment projection of the Fokker-Planck operator should be consistent with that of the Boltzmann operator. On the other hand, the Fokker-Planck model should be constructed in such a way that the H-theorem is satisfied. The central aim of this study is fulfilling these two categorically different constraints, i.e. moment matching and entropy dissipation, within a flexible and tractable Fokker-Planck framework. To this end, we introduce a Fisher information-based entropic constraint and demonstrate that, with a suitable polynomial expansion of the drift term, it is possible to simultaneously achieve weak moment matching while honouring the H-theorem. We support our theoretical result by numerical experiments on the shock problem, validating our Fisher Entropic Fokker-Planck framework.

翻译：基于粒子随机近似的玻尔兹曼方程是模拟非平衡气体流动的常用工具，适用于纳维-斯托克斯-傅里叶方程无法准确描述的情形。然而，这些数值方法计算成本高昂，在近连续流区域尤为明显，因为该区域粒子碰撞极为频繁。相比之下，福克-普朗克动力学模型通过扩散过程描述二元碰撞，提供了高效替代方案。尽管存在直观优势，玻尔兹曼方程的严格高效福克-普朗克近似仍是一个开放性问题。一方面，福克-普朗克算子的矩投影应与玻尔兹曼算子保持一致；另一方面，福克-普朗克模型需满足H定理的构造要求。本研究核心目标是在灵活可处理的福克-普朗克框架内同时满足矩匹配与熵耗散这两类本质不同的约束条件。为此，我们引入基于费希尔信息的熵约束条件，证明通过对漂移项进行适当多项式展开，可在满足H定理的同时实现弱矩匹配。我们通过激波问题的数值实验验证理论结果，证明了费希尔熵福克-普朗克框架的有效性。