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Chemistry LibreTexts

11. The Microcanonical Ensemble

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  • The goal of equilibrium statistical mechanics is to calculate the diagonal elements of \(\hat{\rho}_{en}\) so we can evaluate average observables \(<A> =Tr\{\hat{A}\hat{\rho}_{en}\}=A\) that give us fundamental relations or equations of state. Just as thermodynamics has its potentials \(U\), \(A\), \(H\), \(G\) etc., so statistical mechanics has its ensembles, which are useful depending on what macroscopic variables are specified. We first consider the microcanonical ensemble because it is the one directly defined in postulate II of statistical mechanics.

    In the microcanonical ensemble \(U\) is fixed (Postulate I), and other constraints that are fixed are the volume \(V\) and mole number \(n\) (for a simple system), or other extensive parameters (for more complicated systems).