AI 中文总结
本文利用任意分布等待时间的碰撞更新过程推导Boltzmann方程，证明指数等待时间对应经典Navier-Stokes方程，幂律等待时间对应分数阶方程，从而提供随机求解方法。
AI 中文摘要
Navier-Stokes方程是流体力学中的关键方程，至今尚未解决。发现Navier-Stokes方程的解是千禧年难题之一。1900年，Hilbert提出了一种潜在的方法来解决这个问题，即通过建立微观动力学与宏观连续介质方程之间的关系。关键的桥梁是Boltzmann方程的推导和概率论。在本文中，我们将使用具有任意分布等待时间的碰撞更新过程来推导粒子速度和位移概率随时间演化的Boltzmann方程，在此基础上我们证明，具有指数碰撞等待时间的更新过程等价于经典Navier-Stokes方程，而具有幂律等待时间的更新过程等价于分数阶Navier-Stokes方程。由于具有任意分布等待时间的碰撞更新过程是一个随机过程，并且易于对轨迹进行随机模拟以获得相应的解，我们实际上找到了一种随机方法来求解经典和分数阶Navier-Stokes方程。
英文摘要
The Navier-Stokes equations, which remain unsolved, are crucial equations in fluid mechanics. Discovering the solutions to the Navier-Stokes equations is one of the challenging Millennium problems. In 1900, Hilbert proposed a potential approach to tackle this problem by establishing the relationship between microscopic dynamics and the macroscopic continuum equations. The key bridge is the derivation of Boltzmann equation and the theory of probability. In this paper, we shall use the collision renewal process with arbitrarily distributed waiting times to derive the Boltzmann equation for the time evolution of the probability of the velocity and the displacement of the particle, based on which we prove that the renewal process with exponential collision waiting time is equivalent to the classical Navier-Stokes equations, and that with power-law waiting time is equivalent to the fractional Navier-Stokes equations. Since the collision renewal process with arbitrarily distributed waiting times is a random process and is easy to perform the stochastic simulations of trajectories to obtain the corresponding solution, we actually find a stochastic approach to solve the classical and fractional Navier-Stokes equations.
