The mathematics of Brownian motion and Johnson noise
The mathematics of Brownian motion and Johnson noise
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DOI：
10.1119/1.18210
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发表时间：
1996-03-01
影响因子：
0.9
通讯作者：
Gillespie, DT
中科院分区：
文献类型：
--
作者：
Gillespie, DT
布朗运动和约翰逊噪声之所以难教，原因之一是它们的数学要求超出了普通微分学的能力。这里提出的是一个必要的概括微积分，即连续马尔可夫过程理论的阐述，在一种形式，应该是先进的物理本科生访问。它显示了这个数学框架如何使一个人能够给出清晰，简洁的推导布朗运动和约翰逊噪声的所有主要结果，包括波动耗散公式，自协方差输运公式，谱密度公式，奈奎斯特公式，白色和1/f（2）噪声的概念，和一个精确的数值模拟算法。一个额外的好处，这一论述是一个更清楚的数学之间的联系，两个非常不同的方法布朗运动所采取的爱因斯坦和朗之万在其开创性的论文1905年和1908年。(C)1996年美国物理教师协会。
One reason why Brownian motion and Johnson noise are difficult subjects to teach is that their mathematical requirements transcend the capabilities of ordinary differential calculus. Presented here is an exposition of the needed generalization of calculus, namely continuous Markov process theory, in a form that should be accessible to advanced physics undergraduates. It is shown how this mathematical framework enables one to give clear, concise derivations of all the principal results of Brownian motion and Johnson noise, including fluctuation-dissipation formulas, auto-covariance transport formulas, spectral density formulas, Nyquist's formula, the notions of white and 1/f(2) noise, and an accurate numerical simulation algorithm. An added benefit of this exposition is a clearer view of the mathematical connection between the two very different approaches to Brownian motion taken by Einstein and Langevin in their pioneering papers of 1905 and 1908. (C) 1996 American Association of Physics Teachers.