The Canonical Distribution and Stochastic Differential Equations

Until now, we have considered molecular models for isolated collections of atoms. We have seen how to derive equations of motion, and studied the properties of the relevant dynamical systems. We have also observed that the chaotic nature of a typical system imparts a random aspect which suggests that the dynamics defines, at least in some approximate sense, a limiting probability distribution. 

Leimkuhler, C. Matthews, Molecular Dynamics, Interdisciplinary Applied 

From the point of view of defining molecular models, we would like to be able to specify the macroscopic properties, in particular the temperature and pressure which are easily measured in the laboratory setting. The energy (and, in the case of pressure control, also the volume) may then fluctuate over time.  

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Hairer, M. Convergence of Markov processes. The canonical distribution and stochastic differential equations (2010). Unpublished... 

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