MPS-RR 2000-25
June 2000

On the Diffusion Coefficient: The Einstein Relation and Beyond

Goran Peskir

Abstract

We present a detailed proof of the closed-form expression for the diffusion coefficient that is initially derived by Einstein [2]. The proof 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 easier to apply the argument in a more general case or setting. We demonstrate this fact 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 recaptures the well-known Smoluchowski approximation [12] in the case of non-constant temperature, and provides a scholar example explaining the need for a stochastic integral. The key point in the proof is to apply 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. We 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 proof leads to a nonlinear partial differential equation which in the case of weak interaction reduces to the Fokker-Planck equation. One of the main points of this article is to demonstrate as much as possible that the Einstein argument leads to a truly dynamical theory of diffusion.

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