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Brownian Motion, 2nd Edition by Björn Böttcher, Lothar Partzsch, René L. Schilling

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21 On diffusions

Diffusion is a physical phenomenon which describes the tendency of two (or more) substances, e. g. gases or liquids, to reach an equilibrium: Particles of one type, say B, move or ‘diffuse’ in(to) another substance, say S, leading to a more uniform distribution of the type B particles within S. Since the particles are physical objects, it is reasonable to assume that their trajectories are continuous (in fact, differentiable) and that the movement is influenced by various temporal and spatial (in-)homogeneities which depend on the nature of the substance S. Diffusion phenomena are governed by Fick’s law. Denote by p = p(t, x) the concentration of the B particles at a space-time point (t, x), and by J = J(t, x) the flow of particles. ...

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