D.61 Checks on the expression for entropy

According to the microscopic definition, the differential of the entropy
should be

where the sum is over all system energy eigenfunctions and is their probability. The differential can be simplified to

the latter equality since the sum of the probabilities is always one, so 0.

This is to be compared with the macroscopic differential for the
entropy. Since the macroscopic expression requires thermal
equilibrium, in the microscopic expression above can be equated
to the canonical value where is the
energy of system eigenfunction . It simplifies the
microscopic differential of the entropy to

The macroscopic expression for the differential of entropy is given by
(11.18),

Substituting in the differential first law (11.11),

and plugging into that the definitions of and ,

and differentiating out the product in the first term, one part drops out versus the second term and what is left is the differential for according to the microscopic definition (D.38). So, the macroscopic and microscopic definitions agree to within a constant on the entropy. That means that they agree completely, because the macroscopic definition has no clue about the constant.

Now consider the case of a system with zero indeterminacy in energy.
According to the fundamental assumption, all the eigenfunctions with
the correct energy should have the same probability in thermal
equilibrium. From the entropy’s point of view, thermal
equilibrium should be the stable most messy state, having the maximum
entropy. For the two views to agree, the maximum of the microscopic
expression for the entropy should occur when all eigenfunctions of the
given energy have the same probability. Restricting attention to only
the energy eigenfunctions with the correct energy, the
maximum entropy occurs when the derivatives of

with respect to the are zero. Note that the constraint that the sum of the probabilities must be one has been added as a penalty term with a Lagrangian multiplier, {D.48}. Taking derivatives produces

showing that, yes, all the have the same value at the maximum entropy. (Note that the minima in entropy, all zero except one, do not show up in the derivation; is zero when 0, but its derivative does not exist there. In fact, the infinite derivative can be used to verify that no maxima exist with any of the equal to zero if you are worried about that.)

If the energy is uncertain, and only the expectation energy is known,
the penalized function becomes

and the derivatives become

which can be solved to show that

with and constants. The requirement to conform with the given definition of temperature identifies as and the fact that the probabilities must sum to one identifies as 1/.

For two systems and in thermal contact, the probabilities of the combined system energy eigenfunctions are found as the products of the probabilities of those of the individual systems. The maximum of the combined entropy, constrained by the given total energy , is then found by differentiating

can be simplified by taking apart the logarithm and noting that the probabilities and sum to one to give

Differentiation now produces

which produces and and the common constant then implies that the two systems have the same temperature.