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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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