PDF Ebook Normal and Anomalous Diffusion: A Tutorial
The art of doing research in physics usually starts with the observation of a natural phenomenon. Then follows a qualitative idea on "How the phenomenon can be interpreted", and one proceeds with the construction of a model equation or a simulation, with the aim that it resembles very well the observed phenomenon. This progression from natural phenomena to models and mathematical prototypes and then back to many similar natural phenomena, is the methodological beauty of our research in physics.
Diffusion belongs to this class of phenomena. All started from the observations of several scientists on the irregular motion of dust, coal or pollen inside the air or a fluid. The roman Lucretius in his poem on the Nature of Things (60 BC) described with amazing details the motion of dust in the air, Jan Ingenhousz described the irregular motion of coal dust on the surface of alcohol in 1785, but Brownian motion is regarded as the discovery of the botanist Robert Brown in 1827, who observed pollen grains executing a jittery motion in a fluid. Brown initially thought that the pollen particles were "alive", but repeating the experiment with dust confirmed the idea that the jittery motion of the pollen grains was due to the irregular motion of the fluid particles.
The mathematics behind "Brownian motion" was first described by Thiele (1880), and then by Louis Bachelier in 1900 in his PhD thesis on "the theory of speculation", in which he presented a stochastic analysis of the stock and option market. Albert Einstein’s independent research in 1905 brought to the attention of the physicists the main mathematical concepts behind Brownian motion and indirectly confirmed the existence of molecules and atoms (at that time the atomic nature of matter was still a controversial idea). As we will see below, the mathematical prototype behind Brownian motion became a very useful tool for the analysis of many natural phenomena.
Several articles and experiments followed Einstein’s and Marian Smoluchowski’s work and confirmed that the molecules of water move randomly, therefore a small particle suspended in the fluid experiences a random number of impacts of random strength and direction in any short time. So, after Brown’s observations of the irregular motion of "pollen grains executing a jittery motion", and the idea of how to interpret it as "the random motion of particles suspended inside the fluid", the next step is to put all this together in a firm mathematical model, "the continuous time stochastic process." The end result is a convenient prototype for many phenomena, and today’s research on "Brownian motion" is used widely for the interpretation of many phenomena.
This tutorial is organized as follows: In Sec. 2, we give an introduction to Brownian motion and classical random walk. Sec. 3 presents different models for classical diffusion, the Langevin equation, the approach through Fick’s law, Einstein’s approach, the Fokker-Planck equation, and the central limit theorem. In Sec. 4, the characteristics of anomalous diffusion are described, and a typical example, the rotating annulus, is presented. Sec. 5 introduces Continuous Time Random Walk, the waiting and the velocity model are explained, methods to solve the equations are discussed, and also the Levy distributions are introduced. In Sec. 6, it is shown how, starting from random walk models, fractional diffusion equations can be constructed. In Sec. 7 we show how a quasi-linear diffusion equation can be derived for Hamiltonian systems. Sec. 8 briefly comments on alternative ways to deal with anomalous diffusion, Sec. 9 contains applications to physics and astrophysics, and Sec. 10 presents the conclusions.
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PDF Ebook Normal and Anomalous Diffusion: A Tutorial
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