Chapter

The discrete‐stochastic approach

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/9780199664504.003.0005
The discrete‐stochastic approach

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This chapter introduces a model of diffusion in which the position of the solute molecule is described by a discrete stochastic variable, in contrast to the continuous stochastic variable of the Einstein model. The aim of this new model is to provide a description of diffusion which is for most practical purposes equivalent to the Einstein model, but which affords new ways of describing it mathematically. The system volume is imagined to be subdivided into equal-size cells, and the current state of the system is given by the current numbers of solute molecules in the cells. The diffusion process is modeled by supposing that the solute molecules independently jump to adjacent cells according to a specific probability rule. The chapter establishes consistency between that cell-jumping rule and both the Fickian and the Einstein models of diffusion, and the chapter deduces a generalization of the cell-jumping rule for unequal cell sizes. The chapter shows that there must be some lower bound on the cell size, and it derives a generalization of the standard cell-jumping rule that relaxes that lower bound. The chapter examines in detail the discrete-stochastic model's version of Fick's Law. Finally, the chapter illustrates the model in the context of a laboratory experiment in which the solute molecules are tiny polystyrene beads diffusing in water inside a specially fabricated microfluidic chip.

Keywords: discrete-stochastic; cell jumping; jump probability rate; microfluidic chip; Fick's Law; ballistic refinement; cell-size limitations; time-increment limitations

Chapter.  14871 words.  Illustrated.

Subjects: Physics

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