Chapter

Implications of Langevin's theory

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/9780199664504.003.0009
Implications of Langevin's theory

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This chapter develops the major implications of the Langevin theory of diffusion. The chapter begins by deriving Langevin's formula for the mean-squared displacement of a solute molecule. That formula enables us not only to identify the diffusion coefficient, but also to see how Langevin's theory of diffusion contains Einstein's theory as a special limiting approximation. The chapter shows how these considerations lead naturally to: a fundamental relation between the diffusion coefficient and the drag coefficient called the Einstein formula; a quantity called the characteristic diffusion length; a confirmation of the reasoning the chapter used to derive the stochastic bimolecular chemical reaction rate in Chapters 3 and 4; and new information on the validity conditions for the discrete-stochastic model of diffusion discussed in Chapters 5 and 6. The chapter derives exact formulas for numerically simulating the trajectory of a solute molecule according to Langevin's theory. Using those formulas, the chapter makes a series of side-by-side comparisons of Langevin-simulated trajectories with Einstein-simulated trajectories for the bead-in- water system considered in Chapters 5 and 6. The chapter re-examines the relative motion of two solute molecules in the context of the Langevin theory. The chapter shows how the Langevin theory's more realistic rendering of the velocity of a diffusing molecule not only makes possible an alternative definition of the coefficient of diffusion, but also provides insight into the energetics of diffusion. Finally, the chapter gives a careful critique of what it means to say that a diffusing molecule is “overdamped”.

Keywords: Langevin equation; mean-squared displacement; diffusion coefficient; relaxationtime; Einstein's formula; characteristic diffusion length; simulated diffusion trajectories; relative diffusion; velocity auto-covariance formula; discrete-stochastic model

Chapter.  14440 words.  Illustrated.

Subjects: Physics

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