Chapter

Master equations and simulation algorithms for 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/9780199664504.003.0006
Master equations and simulation algorithms for the discrete‐stochastic approach

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The discrete-stochastic model of diffusion introduced in the preceding chapter implies that the system state evolves in time as what is known as a jump Markov process. The time evolution of such a process is generally governed by what is called a master equation. In this case, the assumption that the solute molecules move about independently of each other gives rise to two master equations, one for a single solute molecule and one for the entire collection of solute molecules. In this chapter the chapter derives both of those master equations. The chapter obtains the exact time-dependent solution of the single-molecule master equation, and the exact time-independent (equilibrium) solution of the many-molecule master equation. The chapter also derives companion stochastic simulation algorithms for the two master equations, and the chapter applies those algorithms to the microfluidic diffusion experiment of Chapter 5. The chapter uses simulation to test the validity of the discrete-stochastic version of Fick's Law.

Keywords: discrete-stochastic; cell jumping; diffusion master equations; stochastic simulation algorithms; Fick's Law; Monte Carlo averaging; moment equations

Chapter.  19998 words.  Illustrated.

Subjects: Physics

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