Chapter

Langevin's theory of diffusion

Daniel T. Gillespie and Effrosyni Seitaridou

in Simple Brownian Diffusion

Published in print October 2012 | ISBN: 9780199664504
Published online January 2013 | e-ISBN: 9780191748516 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199664504.003.0008
Langevin's theory of diffusion

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Although Einstein's theory of diffusion is adequate for many purposes, the previous chapters have shown that it is physically incorrect on small timescales. A theory of diffusion that has a firmer foundation in the physics of molecular motion was proposed in 1908 by Paul Langevin. The point of departure of Langevin's analysis was Newton's Second Law for the solute molecule, but innovatively framed as what mathematicians today would call a stochastic differential equation. The presentation of Langevin's theory in this chapter is updated in the sense that it exploits some mathematical insights that were not fully appreciated in Langevin's time. The chapter begins by describing the key assumption underlying Langevin's theory. The chapter corroborates that assumption by demonstrating its agreement with a rudimentary but plausible physical model of how solute molecules move in a sea of many much smaller solvent molecules. Then the chapter shows how Langevin's theory of diffusion unfolds via the mathematics of the Ornstein- Uhlenbeck process of Chapter 7. Finally, the chapter shows how a basic requirement of statistical thermodynamics fixes the only free parameter in Langevin's theory, thereby making the theory complete. Topics covered include: the fluctuation-dissipation theorem; explicit formulas for the position and velocity of a diffusing molecule as functions of time; the correlation between the position and the velocity as a function of time; and the correlations of the position at two different times and the velocity at two different times.

Keywords: Langevin equation; drag and fluctuating forces; drag coefficient; relaxation time; thermal equilibrium; Gaussian white noise; fluctuation-dissipation theorem; position-velocity correlation; two-time auto-correlations

Chapter.  9275 words.  Illustrated.

Subjects: Physics

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