Microcanonical average

In this interpretation Q is the number of linearly independent eigenfunctions of the unperturbed Hamiltonian in the interval AE. From (36) the microcanonical average of an observable represented by the operator A in an arbitrary basis, is... 

As we mentioned in motivating the microcanonical distribution, we could alternatively view microcanonical averages as integrals with respect to the singular measure on the ambient space 2) defined by... 

A result in [338] provides a formula to calculate the perturbation of the microcanonical average induced by the Hamiltonian perturbation. This result has been recast in the more convenient form in [14] ... 

Suppose that w is an arbitrary smooth vector field such that w VH Oon He-Then the perturbation of the microcanonical average of a smooth function g is given... 

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... 

In this section, we use square brackets ([]) to indicate quantities which have been microcanonically summed, braces ( ) to indicate quantities which have been microcanonically averaged, and angle brackets (<>) to indicate quantities which have been thermally averaged. The appropriate 3D microcanonically averaged rotational cross section with which we wish to compare is ... 

In a TST picture, the partial widths can be identified as the states of the transition state. Thus, equating as the microcanonically averaged width ... 

Monte Carlo methods can also be applied to systems in the microcanonical ensemble,and the techniques just described can be extended to this ensemble as well. For a cluster of N atoms with the same mass m, microcanonical averages of observables are obtained by calculating statistical expectations using the density function... 

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