Chapter

Diffusion in an external force field

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/9780199664504.003.0010
Diffusion in an external force field

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This chapter considers what happens when diffusing solute molecules are subjected to an externally applied conservative force. The chapter starts by presenting the traditional derivation of the Smoluchowski equation, which is the standard generalization of the classical diffusion equation to accommodate such a force. For a rudimentary model of gradient-sensing bacterial chemotaxis, the chapter shows how the Smoluchowski equation reduces to the Keller-Segel equation. After taking note of some weaknesses in the traditional derivation of the Smoluchowski equation, the chapter presents the more rigorous derivation that is afforded by the Kramers equation in Langevin's theory of diffusion. The chapter shows that the solution of the time-stationary Kramers equation is just what we should expect on the basis of classical statistical mechanics. The chapter gives a careful critique of what are commonly called the “low friction” and “high friction” limits of the Kramers equation, and it shows that in the latter limit we obtain the Smoluchowski equation as an approximation, although with some subtle caveats. The chapter concludes by examining in detail the case of a constant external force field.

Keywords: Langevin equation; external force; Smoluchowski equation; gradient- sensing chemotaxis; coefficient of chemotaxis; Keller-Segel equation; Kramers equation; energetics of diffusion; low-friction and high-friction limits

Chapter.  12705 words.  Illustrated.

Subjects: Physics

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