The Microcanonical Probability Measure

In this case we say that the operator Cf is skew-adjoint. The equations 

An important consequence of the skew symmetry of Cf for a divergence-free 

In this section, we continue the discussion of the Hamiltonian case by introducing the microcanonical probability measure, the natural measure associated to a surface of fixed energy. The framework we have discussed so far is useful when the density is known to be a C°° function of phase space, but this is not always the case. 

A simple example which we would wish to include in our framework is the point distribution which concentrates all measure at a single point. This type of object cannot be directly represented by a smooth function (or even by one in L ), but it is nonetheless possible to extend our framework in a way that makes sense in such cases. 

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From a mathematical perspective, we may prefer to have a concept of the microcanonical probability measure that does not involve delta functions. It is possible to understand computations of averages with respect to a Dirac-type generalized function like po as integrals on a set in a space of dimension one less than that of the ambient phase space (which is typically an even dimensional Euclidean space). We assume the level sets He = (q[Pg.191]

The Microcanonical Probability Measure The limit of this as AE 0 defines a quantity... 

In the case of an equilibrium system the Hamiltonian is the same as that of an ensemble of conservative systems in statistical equilibrium. If the energy of the system is measured to lie between Ek and EK + AE, then the representative ensemble is also restricted to the energy shell [AE K. From the hypotheses of equal a priori probabilities and random a priori phases it then follows that the diagonal elements of the density matrix lying inside [AE]k are all equal and that all others vanish. The density matrix of the quantum statistical microcanonical ensemble is thereby determined as... 

Here and My are the microcanonical measures of the interiors of basins a and y at energy E. Equation (31) ensures that total probability is conserved at all times. 

The microcanonical ensemble describes an "isolated" system. The dynamical approach says all states for a given energy are equally likely and the energy is fixed in an isolated system. From conservation of probability, = 1/Q( ) if = E and is zero otherwise. In the variational approach, one maximizes S[P.y] subject to the constraints that = 1 and that E = E. The result is the same from both approaches. There is a microcanonical temperature, which is a response function and is an intensive quantity. The inverse temperature P = f Iv,n measures how the entropy changes as the energy is varied. 

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