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Properties of entropy
Entropy, as we have defined it, has some dependence on the resolution
to which the energy of macrostates is measured. Recall that
is the
number of accessible microstates with energy in the range to .
Suppose that we choose a new resolution and define a new
density of states
which is the number of states with energy
in the range to
. It can easily be seen that

(227) 
It follows that the new entropy
is related to the
previous entropy
via

(228) 
Now, our usual estimate that
gives , where
is the number of degrees of freedom. It follows that even if
were to differ from by of order (i.e., twenty four orders of
magnitude), which is virtually inconceivable, the second term on the righthand
side of the above equation is still only of order , which is utterly
negligible compared to . It follows that

(229) 
to an excellent approximation, so our definition of entropy is completely
insensitive to the resolution to which we measure energy (or any other
macroscopic parameter).
Note that, like the
temperature, the entropy of a macrostate is only welldefined if
the macrostate is in equilibrium. The crucial point is that
it only makes sense to talk about the
number of accessible states if the systems
in the ensemble are given sufficient time to thoroughly explore all of the possible
microstates consistent with the known
macroscopic constraints. In other words, we can only
be sure that a given microstate is inaccessible when the systems in the ensemble have
had ample opportunity to move into it, and yet have not done so. Note that for an
equilibrium state, the entropy is just as welldefined as more familiar quantities
such as the temperature and the mean pressure.
Consider, again, two systems and which are in thermal contact but can do
no work on one another (see Sect. 5.2).
Let and be the energies of the two systems,
and
and
the respective densities of states.
Furthermore, let
be the conserved energy of the
system as a whole and
the corresponding density of states.
We have from Eq. (158) that

(230) 
where
. In other words, the number of states accessible to the
whole system is the product of the numbers of states accessible to each subsystem,
since every microstate of can be combined with every microstate of
to form a distinct microstate of the whole system. We know, from Sect. 5.2,
that in equilibrium the mean energy of takes the value
for which
is maximum, and the
temperatures of and are equal. The distribution of around the
mean value is of order
, where is the number of
degrees of freedom. It follows that the total number of accessible microstates is
approximately the number of states which lie within
of .
Thus,

(231) 
The entropy of the whole system is given by

(232) 
According to our usual estimate,
, the first term on the
righthand
side is of order whereas the second term is of order
. Any reasonable choice for the energy subdivision should be
greater than , otherwise there would
be less than one microstate per subdivision. It follows that the second term
is less than or of order , which is utterly negligible compared to
. Thus,

(233) 
to an excellent approximation,
giving

(234) 
It can be seen that the probability distribution for
is so strongly peaked
around its maximum value that, for the purpose of calculating the entropy, the
total number of states is equal to the maximum number of states [i.e.,
].
One consequence
of this is that the entropy has the simple additive property shown in
Eq. (234). Thus, the total entropy of two thermally interacting systems
in equilibrium is the sum of the entropies of each system in isolation.
Next: Uses of entropy
Up: Statistical thermodynamics
Previous: Entropy
Richard Fitzpatrick
20060202