Statistical Mechanics - Physics at Oregon State University

1.6. ENTROPY AND TEMPERATURE. 17
are in thermal contact, the configuration with the maximum number of states
available follows from finding the maximal value of
t(UA) = gA(NA, UA)gB(NB, U − UA) (1.53)
where U is the total energy of the combined system. We assume that the
systems are very large, and that the energy is a continuous variable. Therefore,
the condition for equilibrium is
0 = dt
=
dUA
or
1
gA(NA, UA)
∂gA
(NA, UA)gB(NB, U−UA)+gA(NA, UA)
∂U N
∂gA
(NA, UA) =
∂U N
1
gB(NB, U − UA)
This leads us to define the entropy of a system by
∂gB
(NB, U−UA)(−1)
∂U N
(1.54)
∂gB
(NB, U − UA)
∂U N
(1.55)
S(U, N, V ) = kBlogg(U, N, V ) (1.56)
where we have added the variable V. Other extensive variables might also be
included. The symbol kB denotes the Boltzmann factor. Note that the entropy
is always positive. The entropy is a measure of the number of states available
for a system with specified extensive parameters. We have to be careful, though.
Strictly speaking, this definition gives us an entropy analogue. We need to show
that we recover all the laws of thermodynamics using this definition. In this
chapter we will consider changes in temperature. In the next chapter we discuss
changes in volume and mechanical work. In the chapter after that changes in
the particle number and the corresponding work term.
The temperature of a system is defined by
1
T =
∂S
∂U
N,V
(1.57)
where the partial derivative is taken with respect to U, keeping all other extensive
parameters the same. Hence our criterion for thermal equilibrium is
TA = TB
(1.58)
which is not unexpected.
The multiplicity function for the total system gtot is found by summing over
all possible configurations of the combination A plus B, but as shown before the
value of the logarithm of this sum is equal to the logarithm of the largest term.
If the energy of system A in equilibrium is ÛA we have
log(gtot)(U) = log(gA( ÛA)) + log(gB(U − ÛA)) (1.59)