The dichotomous history of diffusion

© 2009 American Institute of Physics, S-0031-9228-0907-030-6 Diffusion implies spreading, either observable (physical), or abstract and probabilistic (stochastic). At the beginning of the 19th century, the mathematics of both were established by Joseph Fourier and Pierre Simon Laplace. In 1807 Fourier submitted a monograph, Théorie de la propagation de la chaleur dans les solides (Theory of the propagation of heat in solids), to the French Academy of Sciences and introduced the partial differential equation describing heat flow.1,2 The academy appointed Joseph Lagrange and Laplace as two of the four reviewers of the monograph; they questioned Fourier’s use of trigonometric series to solve the heat equation. Hence, the work was never approved. At the time, Laplace was working on his theory of probability—in particular, estimating the probability that the sum of n random variables may be equal to or less than a certain value when n is very large. In 1809 he formulated a partial differential equation with the same form as Fourier’s heat equation, whose solution would provide an estimate of the probability.3 Soon thereafter, Laplace published Théorie analytique des probabilités (Analytical theory of probability)4 and Fourier published Théorie analytique de la chaleur (Analytical theory of heat).5 Fourier’s theory had an immediate impact on mathematical physics by opening up new avenues for the study of heat. It also inspired scientists of the day to apply the diffusion model to electricity, molecules, and fluid flow in resistive media. However, Laplace’s stochastic diffusion, based on his probability studies, went largely unnoticed for decades. Toward the close of the 19th century, Lord Rayleigh6,7 and economist and statistician Francis Edgeworth8 drew on Laplace’s treatise on probability theory to formulate the stochastic diffusion equation with probability density as the dependent variable. Shortly thereafter, Louis Bachelier9 formulated a diffusion equation to model random price fluctuations in the stock market. At the turn of the 20th century, mathematics of diffusion had established two separate identities—one as a physically observable process, the other as a probabilistic description. In 1905 Albert Einstein recognized the equivalence between two diffusion coefficients—one describing random events, the other describing a physical process—and used that equivalence to experimentally establish a kinetic theory of heat based on molecular motion and estimate Avogadro’s number, a quantity of fundamental importance in chemistry.10 The diffusion equation is widely used in physical, biological, geological, and social sciences in physical or stochastic form. Commonly, mathematicians and scientists focus attention on the similarity of mathematical patterns in applying the diffusion model to one field or another and ignore the inherent differences in the nature of the processes. Nevertheless, the differences between physical and stochastic diffusion are as notable as their similarities. In his investigations, Fourier was inspired by curiosity to comprehend the role of heat in Earth’s lithosphere, oceans, and atmosphere. He thus found it surprising that mathematical results bearing on physical diffusion were also relevant to problems in the abstract theory of probability.5 To provide insight into diverse phenomena and the evolution of the mathematical ideas, this paper traces the twofold history of diffusion.