Two Mathematical Approaches to Random Fluctuations: A Historical Perspective

Abstract

Since Einstein’s work on Brownian motions in 1905, physicists and mathematicians in the first half of the twentieth century had established a research program on various continuous-time fluctuations: agitation of a pollen powder, wobbling of a meter’s reading, and background “noise” in an electric circuit. The researchers understood such fluctuations as an outcome of the molecular or electronic random movement, and developed a mathematical theory of stochastic processes to grapple with their properties. Historical reviews have portrayed the development of the stochastic theory of fluctuations as a linear progress toward a unified mathematical and conceptual framework. In this paper, I argue that two distinct approaches were actually at work. One approach operated in the time domain, as it aimed to formulate the diffusion-type equation in time-space (the Fokker-Planck equation) for the probability density function of a random fluctuation and solve the equation to obtain relevant statistical quantities. The other approach operated in the frequency domain, as it focused on the spectral analysis of the fluctuation and the properties associated with its power spectral density. These approaches reflected two important research traditions in early twentieth-century physical sciences. The time-domain analysis was marshaled by statistical physicists (Einstein, Smoluchowski, Orstein, Uhlenbeck, etc.), who viewed fluctuations as instantiations of equipartition, ensemble average, and Boltzmann distribution. The frequency-domain analysis was promoted by physicists and mathematicians working on engineering problems (Schottky, Nyquist, Wiener, Rice, etc.). They treated fluctuations as random signals, and preferred the canonical method in electrical engineering: Fourier analysis.?