On the diffusion coefficient: The Einstein relation and beyond

Peskir, Goran (2003) On the diffusion coefficient: The Einstein relation and beyond. Stochastic Models, 19 (3). pp. 383-405. ISSN 1532-4214

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We present a detailed derivation of the closed-form expression for the diffusion coefficient that was initially obtained by Einstein.[4] The present derivation does not make use of a fictitious force as did the original Einstein derivation, but instead concentrates directly on establishing a dynamic equilibrium between the forces of pressure and friction acting on a Brownian particle. This approach makes it easier to understand the true essence of the argument, and thus makes it simpler to apply the argument in a more general case or setting. We demonstrate this by deriving the equation of motion of a Brownian particle that is under the influence of an external force in the fluid with a non-constant temperature. This equation extends the well-known Smoluchowski approximation[24] to the case of non-constant temperature, and offers new insights into the Ludwig–Soret and Enskog–Chapman effects (providing also a scholar example explaining the need for a stochastic integral). The key point in the derivation is reached by applying the Einstein dynamic equilibrium argument together with the conservation of the number of particles law. We show that this approach leads directly to the Kolmogorov forward equation whenever the setting is Markovian. The same method can also be applied in the case of interacting Brownian particles satisfying the van der Waals equation. In this setting we first demonstrate that the presence of short-range repulsive forces between Brownian particles tends to increase the diffusion coefficient, and the presence of long-range attractive forces between Brownian particles tends to decrease it. The method of derivation then leads to a nonlinear partial differential equation which in the case of weak interaction reduces to the Fokker–Planck equation. One of the main aims of the present article is to demonstrate that the Einstein argument leads to a truly dynamical theory of diffusion.

Item Type: Article
Uncontrolled Keywords: Diffusion coefficient, The Einstein relation, Brownian motion, Diffusion, The Enskog–Chapman effect, The Ludwig–Soret effect, Interacting Brownian particles, Fick's law, The general-gas law, The diffusion equation, The Kolmogorov forward equation (the Fokker–Planck equation), The heat equation, The van der Waals equation, The Smoluchowski approximation, Markov process, Itô integral, The long-range attractive forces, The short-range repulsive forces, Count data, Primary 60J60, 60J65, 35K05, 35Q99, Secondary 82C22, 35K55, 82B40, 82C31
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 35 Partial differential equations
MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
MSC 2010, the AMS's Mathematics Subject Classification > 82 Statistical mechanics, structure of matter
Depositing User: Ms Lucy van Russelt
Date Deposited: 21 Jul 2006
Last Modified: 20 Oct 2017 14:12
URI: https://eprints.maths.manchester.ac.uk/id/eprint/406

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