Phase Space Distributions and Microcanonical Averages

In the previous chapters, we considered the approximation of Hamiltonian trajectories. In this chapter we study the paths emanating from the collection of all initial conditions within a given set. This is the starting point for statistical mechanics which allows the calculation of averages. 

To discuss the changes in the relative concentrations of phase points in one or another region, we introduce the concept of a probability measure for the distribution of phase space points. Normally this is represented by a density, a nonnegative function of the phase space variables that can be used to characterize sets 

Leimkuhler, C. Matthews, Molecular Dynamics, Interdisciplinary Applied 

Our treatment of statistical mechanics is abbreviated and specialized as our focus is ultimately on certain types of systems Ifequently arising in molecular dynamics and their numerical treatment. We refer the interested reader to more detailed treatments including [57,70,168,197,264,297,372]. 

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