The Canonical Density and Thermodynamic Averages

The distributional approach described above and in the previous chapter and used to define microcanonical averages on a bounded energy surface must be modified to make it applicable to the canonical ensemble. Recall that in the microcanonical setting we assumed p was a smooth density defined on the bounded energy surface 

more generally, a distribution. For a canonical system there is no longer a constraint of the form // = E and thus we must allow the phase space to include essentially all points for which the position coordinates lie in the defined configurational domain (a 3-torus, if periodic boundary conditions are used). In general the momenta are unbounded. It thus becomes necessary to restrict the set of functions on which the functional -, p) acts. In the case of periodic boundary conditions we will assume that, for any observable of interest. 

If the position domain Q is unbounded (for example choosing Q = then some other restriction is needed on the potential energy function U (and on the observables) in order that averages are well defined. Typical assumptions for potentials on the unbounded domain are that (i) U is bounded below, and (ii) U grows sufficiently rapidly as cxd. This ensures that 

Under assumptions on the potential energy as discussed above, and with j) continuous and allowed to grow at most polynomially fast at infinity, we may proceed to define averages with respect to the canonical density p, namely 

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