Statistical Mechanics - Physics at Oregon State University

26 CHAPTER 2. THE CANONICAL ENSEMBLE
At this point we seem to have a contradiction. The basic assumption of statistical
mechanics tells us that all accessible quantum states are equally probably.
The Boltzmann distribution on the other hand gives the states unequal
probabilities! This will always occur when we specify some intensive parameter
like T. In such a case one always needs a reservoir, and the basic assumption
applies to the states of the reservoir plus system. If we specify the state of the
system, this corresponds to a configuration, i.e. a set of states, of the combination
reservoir plus system. The basic assumption only applies to states, not
to configurations. For a configuration one always has to include a multiplicity
function counting the number of states available for that configuration.
The Boltzmann factor is not correct in the limit ɛs → ∞, since that implies
taking a large fraction of the energy of the reservoir and putting it in the system.
The upper limit of ɛs for which the Boltzmann factor is valid can be derived
from the considerations in the beginning of this section. This is not a practical
problem, though, since we can make the reservoir very large. The probability
of finding states with an energy which is too large for our formalism to be valid
then becomes exponentially close to zero!
2.2 Energy and entropy and temperature.
The expectation value of the energy, or average energy, of the system is given
by
U =
ɛsP (s) (2.11)
s
where we have dropped the subscript S for system in the probability function.
This can be related to the partition function by
∂ log(Z)
=
∂T
1
∂Z
=
Z ∂T
1 ɛs
− ɛs
k e B T
Z kBT 2 (2.12)
or
V,N
V,N
U(T, V, N) = kBT 2
∂ log Z)
∂T
V,N
The entropy can be obtained from U(T) through
∂S
∂T
∂S
=
∂U
∂U
∂T
which leads to
V,N
V,N
T
S(T, V, N) = Sref (V, N) +
Tref
V,N
1
T
s
= 1
T
(T, V, N) (2.13)
∂U
∂T V,N
(2.14)
∂U
dT (2.15)
∂T V,N
since we assume that all other variables remain constant. It is useful to have an
explicit form for the entropy, however. Based on