论文标题
通过中央极限定理和玻尔兹曼的熵公式的麦克斯韦速度分布的随机方法
A Stochastic Approach to Maxwell Velocity Distribution via Central Limit Theorem and Boltzmann's Entropy Formula
论文作者
论文摘要
麦克斯韦的速度分布在系统和阶段之间是普遍有效的。在这里，我们提出了一种使用概率理论的中心极限定理（CLT）的新的通用推导。这本质上使用了这样的想法，即重复分子间碰撞会引入速度矢量各个组件的速度变化中的随机性，从而导致CLT，即高斯分布。为了完成派生，我们接下来表明，均方根速度或标准偏差完全来自Boltzmann的公式与状态密度相关的熵，从而避免了明确使用状态的理想气体方程。我们还证明了谐波振荡器系统的麦克斯韦速度分布。这种推导提供了对麦克斯韦速度分布中玻尔兹曼常数的起源的进一步见解。我们建议这种推导提供了一种解释麦克斯韦高斯分布的普遍性的方法。
Maxwell's velocity distribution is known to be universally valid across systems and phases. Here we present a new and general derivation that uses the central limit theorem (CLT) of the probability theory. This essentially uses the idea that repeated intermolecular collisions introduce randomness in the velocity change in the individual components of the velocity vector, leading to, by the CLT, a Gaussian distribution. To complete the derivation, we next show that the mean-square velocity or the standard deviation follows exactly from Boltzmann's formula relating entropy to the density of states, thus avoiding the use of the ideal gas equation of state explicitly. We furthermore prove the Maxwell velocity distribution for a system of harmonic oscillators. This derivation provides a further insight into the origin of Boltzmann's constant in the Maxwell velocity distribution and also in the equipartition theorem. We propose that this derivation provides an approach that explains the universality of Maxwell Gaussian distribution.
