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18 - Stochastic simulations of chemical reaction kinetics

Published online by Cambridge University Press:  05 December 2011

Yiannis N. Kaznessis
Yiannis N. Kaznessis
Affiliation:
University of Minnesota
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Summary

An abundance of systems of scientific and technological importance are not at the thermodynamic limit. Stochasticity naturally emerges then as an intrinsic feature of system dynamics. In Chapter 13 we presented the essential elements of a theory for stochastic processes. The master equation was developed as the salient tool for capturing the change of the system's probability distribution in time. Its conceptual strength notwithstanding, the master equation is impossible to solve analytically for any but the simplest of systems. A need emerges then for numerical simulations either to solve the master equation or, alternatively, to sample numerically the probability distribution and its changes in time. This need resembles the need to sample equilibrium probability distributions of equilibrium ensembles.

Attempts to develop a numerical solution of the master equation invariably face insurmountable difficulties. In principle, the probability distribution can be expressed in terms of its moments, and the master equation be equivalently written as a set of ordinary differential equations involving these probability moments. In this scheme, unless severe approximations are made, higher moments are present in the equation for each moment. A closure scheme then is not feasible and one is left with an infinite number of coupled equations. Numerous, approximate closure schemes have been proposed in the literature, but none has proven adequate.

Type
Chapter
Information
Statistical Thermodynamics and Stochastic Kinetics
An Introduction for Engineers
 , pp. 295 - 307
Publisher: Cambridge University Press
Print publication year: 2011

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