物理
布朗运动
数学物理
分布(数学)
连接(主束)
BETA(编程语言)
均方位移
量子力学
分布函数
数学分析
数学
几何学
计算机科学
分子动力学
程序设计语言
作者
G. E. Uhlenbeck,L. S. Ornstein
出处
期刊:Physical Review
[American Physical Society]
日期:1930-09-01
卷期号:36 (5): 823-841
被引量:3527
标识
DOI:10.1103/physrev.36.823
摘要
With a method first indicated by Ornstein the mean values of all the powers of the velocity $u$ and the displacement $s$ of a free particle in Brownian motion are calculated. It is shown that $u\ensuremath{-}{u}_{0}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}t)$ and $s\ensuremath{-}\frac{{u}_{0}}{\ensuremath{\beta}[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}t)]}$ where ${u}_{0}$ is the initial velocity and $\ensuremath{\beta}$ the friction coefficient divided by the mass of the particle, follow the normal Gaussian distribution law. For $s$ this gives the exact frequency distribution corresponding to the exact formula for ${s}^{2}$ of Ornstein and F\"urth. Discussion is given of the connection with the Fokker-Planck partial differential equation. By the same method exact expressions are obtained for the square of the deviation of a harmonically bound particle in Brownian motion as a function of the time and the initial deviation. Here the periodic, aperiodic and overdamped cases have to be treated separately. In the last case, when $\ensuremath{\beta}$ is much larger than the frequency and for values of $t\ensuremath{\gg}{\ensuremath{\beta}}^{\ensuremath{-}1}$, the formula takes the form of that previously given by Smoluchowski.
科研通智能强力驱动
Strongly Powered by AbleSci AI