Statistical Mechanics - Physics at Oregon State University

36 CHAPTER 2. THE CANONICAL ENSEMBLE
Statistics is easy.
In order to get the root-mean-square fluctuations ∆ɛ in the energy, we have
to calculate < ɛ 2 s >. These fluctuations are related to the heat capacity at
constant volume, CV , as can be shown in the following manner. By definition:
CV =
∂U
=
∂T V,N
1 1
Z kBT 2
U =< ɛs >= 1
Z
CV = 1
kBT 2 < ɛ2 s > − 1
Z
s
s
ɛ 2 ɛs − 1
kB T
se −
Z2
s
ɛs
− k ɛse B T (2.53)
ɛs − k ɛse B T
s
ɛs − k ɛse B T
∂ log(Z)
∂T
CV = 1
kBT 2 < ɛ2 s > − < ɛs > U
kBT 2
which gives with the relation U =< ɛs > that
kBT 2 CV =< ɛ 2 s > − < ɛs > 2 = (∆ɛ) 2
∂Z
∂T V,N
V,N
(2.54)
(2.55)
(2.56)
(2.57)
Two important conclusions are based on this formula. First of all, the right
hand side is positive, and hence the heat capacity is positive. Hence the internal
energy U is an increasing function of the temperature! Second, the heat capacity
is an extensive quantity and hence ∆ɛ ∝ √ N. This tells us that the fluctuations
in the energy increase with increasing system size. A more important quantity
is the energy per particle, U
N . The fluctuations in the energy per particle are
therefore proportional to 1 √ . The energy per particle is therefore well defined
N
in the thermodynamic limit, fluctuations are very small. Also, this formula
for the energy fluctuations is very useful in numerical simulations. In these
calculations one often follows the state of a system as a function of time at
a given temperature, and the fluctuations in the energy are a direct result of
the calculations. By varying the temperature, one also obtains U(T,V). Hence
there are two independent ways of calculating the heat capacity, and this gives
a very good test of the computer programs! Finally, the formula relating the
heat capacity and the fluctuations in energy is an example of a general class of
formulas. Response functions are always related to fluctuations in a system in
equilibrium. In our case, if the fluctuations in energy of a given system at a
given temperature are small, it is apparently difficult to change the energy of
this system. Hence we expect also that we need a large change in temperature
to make a small change in energy, and this implies that the heat capacity is
small.