Diffusion as a random walk

There are many different derivations of the diffusion equation. The one that I used here follows (Hughes 1995). In biology, of course, we trace the notion of diffusion to the botanist Robert Brown, who reported the irregular movement of pollen particles observed under the microscopic (Brown 1828). Hence, diffusion is often called - even by mathematicians - Brownian motion. In his miraculous year of 1905, Einstein published papers on the photoelectric effect (for which he was awarded the Nobel prize), special relativity and Brownian motion (a very nice reprint of this paper is found in Stachel (1998); there is also a Dover edition containing it). The paper on Brownian motion is particularly interesting because at the time he wrote it, there was still discussion about whether the atomic theory of matter was correct. Einstein wondered what the atomic theory of matter would mean for a large particle surrounded by a large number of randomly moving small ones. In answering this question, he derived the solution in Eq. (2.51) and connected the diffusion coefficient to temperature and Boltzmann's constant. Einstein had evidently heard about Brownian motion, but had not read the paper because he wrote ''It is possible that the motions to be discussed here are identical with so-called Brownian molecular motion; however, the data available to me on the latter are so imprecise that I could not form a judgment on the question'' (p. 85 in Stachel (1998)). The history of diffusion itself is quite interesting. As starting points, I suggest that you look at Wheatley and Augutter (1996) and Narasimhan (1999), which give two very interesting perspectives. In his interesting and provocative essay, Simberloff (1980) notes that 1859 was the year of publication of both Origin of Species and of Maxwell's work on the statistical distribution of velocities of particles in a gas; thus beginning the revolutions against determinism in both biology and physics coincide. We will have much more to say about diffusion and the random walk in Chapter 7 and 8.

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