标题： 波动玻尔兹曼方程的大偏差原理 显示英文标题

标题： Large Deviations Principle for the Fluctuating Boltzmann Equation

摘要： 玻尔兹曼方程是最重要的方程之一，在现代科学中有着广泛的应用。 在当前的研究中，我们考虑了二元碰撞的随机性，并将经典的玻尔兹曼方程推广到随机框架中。 分别推导了离散或连续时间及状态空间下的相应柯尔莫哥洛夫前向方程和刘维尔方程，其特征线作为大数定律的结果给出玻尔兹曼方程。 然后建立了这些方程的大偏差原理，这不仅解释了玻尔兹曼方程中H定理的概率起源，还提供了一种自然的方法将玻尔兹曼方程纳入更广泛的哈密顿结构中。 所谓的哈密顿-玻尔兹曼方程具有许多重要的优点，如时间可逆性、质量、动量和能量的守恒定律、麦克斯韦-玻尔兹曼分布作为平衡解等。 我们同时也给出了扩散极限下的结果。 最后，在BGK近似下，根据我们的哈密顿-玻尔兹曼方程推导出了包括13个矩的宏观流体动力学模型。 我们希望我们的研究能从随机角度或哈密顿观点出发，为经典玻尔兹曼方程带来新的见解。 显示英文摘要

摘要： The Boltzmann equation is one of the most famous equations and has vast applications in modern science. In the current study, we take the randomness of binary collisions into consideration and generalize the classical Boltzmann equation into a stochastic framework. The corresponding Kolmogorov forward equations and Liouville equation in either discrete or continuous time and state space are derived respectively, whose characteristic line gives the Boltzmann equation as a consequence of the law of large numbers. Then the large deviations principle for these equations is established, which not only explains the probabilistic origin of the H-theorem in the Boltzmann equation, but also provides a natural way to incorporate the Boltzmann equation into a broader Hamiltonian structure. The so-called Hamilton-Boltzmann equation enjoys many significant merits, like time reversibility, the conservation laws of mass, momentum and energy, Maxwellian-Boltzmann distribution as the equilibrium solution, etc. We also present results under the diffusive limit in parallel. Finally, the macroscopic hydrodynamic models including 13 moments are derived with respect to our Hamilton-Boltzmann equation under the BGK approximation. We expect our study can inspire new insights into the classical Boltzmann equation from either the stochastic aspect or a Hamiltonian view.