Biomedical Engineering Reference
In-Depth Information
particles, their standard deviation about the mean, the temperature and so on. In statistical
thermodynamics, we do not enquire about the behaviour of the individual particles that
make up a macroscopic sample; we just enquire about their average properties.
If we were to measure the pressure exerted on the walls at time intervals
t
1
,
t
2
, ...,
t
n
then
we might record results
p
(
t
1
),
p
(
t
2
), ...,
p
(
t
n
). We could calculate a sample mean <
p
> and
a sample standard deviation using these results:
n
1
n
p
=
p
(
t
i
)
i
=
1
n
1
n
−
)
2
σ
p
=
(
p
(
t
i
)
p
(8.1)
i
=
1
We might expect that the greater the number of measurements, the closer the sample mean
would be to the true mean, and the smaller the sample deviation would become.
8.1 The Ensemble
When we considered Figure 8.1, I was careful to draw your attention to the differ-
ence between particle properties and bulk properties. I also mentioned that classical
thermodynamics is essentially particle-free; all that really matters to such a thermody-
namicist are bulk properties such as the number of particles
N
, the temperature
T
and the
volume of the container
V
. I have represented this information in the right-hand box in
Figure 8.1.
Rather than worry about the time development of the particles in the left-hand box in
Figure 8.1, what we do is to make a very large number of copies of the system on the
right-hand side. We then calculate average values over this large number of replications
and, according to the
ergodic theorem
, the average value we calculate is exactly the same as
the time average we would calculate by studying the time evolution of the original system.
The two are the same.
I am not suggesting that all the
cells
in the ensemble are exact replicas at the molecu-
lar level: all we do is to ensure that each cell has a certain number of thermodynamic
properties that are the same. There is no mention of molecular properties at this stage of
the game.
Figure 8.2 shows an ensemble of cells all with the same values of
N
,
V
and
T
.
This array of cells is said to form a
canonical ensemble
. There are three other important
ensembles in the theory of statistical thermodynamics, and they are classified according to
what is kept constant in each cell. Apart from the canonical ensemble, where
N
,
V
and
T
are kept constant, we have the following.
•
The
microcanonical ensemble
where
N
, the total energy
E
and
V
are kept constant in
each cell. In fact, this is a very simple ensemble because energy cannot flow from one
cell to another.
•
In an
isothermal-isobaric
ensemble,
N
,
T
and the pressure
p
are kept constant.
