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Chapter 3 showed how Einstein's analysis of Brownian motion shifted the focus of the classical diffusion equation from the average behavior of many solute molecules to the probabilistic behavior of an individual solute molecule. This chapter deduces some further implications of the Einstein theory of diffusion, using not only analytical reasoning but also numerical simulation. This chapter shows how simulation can be used to construct plots of a single diffusing molecule's position probability density function for some simple boundary conditions. The chapter also shows how simulation can be used to construct “snapshots” of an unrestricted solute molecule's position at successive instants of time. But the chapter will discovers that such snapshots expose a serious physical limitation of the Einstein theory. The chapter rationalizes a quick fix that allows us to complete the derivation which has begun in Chapter 3 of a formula for the stochastic rate of a diffusion-controlled bimolecular chemical reaction. But from a broader view, it will become apparent that, while the Einstein theory of diffusion has a wide range of practical utility, a physically more accurate theory is needed.

*Keywords: *
stochastic simulation;
rt-point diffusion trajectories;
infinite speeds;
path lengths;
ballistic motion;
Maxwell-Boltzmann distribution;
stochastic bimolecular;
reaction rate;
diffusion-controlled reactions;
Collins-Kimball theory;
dilute gas limit;
Smoluchowski limit

*Chapter.*
*12461 words.*
*Illustrated.*

*Subjects: *
Physics

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