TOPIC 13D The canonical ensemble
➤ Why do you need to know this material?
Whereas Topics 13B and 13C deal with independent molecules, in practice molecules do interact. Therefore, this
material is essential for constructing models of real gases,
liquids, and solids and of any system in which intermolecular interactions cannot be neglected.
➤ What is the key idea?
A system composed of interacting molecules is described
in terms of a canonical partition function, from which its
thermodynamic properties may be deduced.
➤ What do you need to know already?
The calculations here, which are not carried through in
detail, are essentially the same as in Topic 13A. Calculations
of mean energies are also essentially the same as in Topic
13C and are not repeated in detail.
The crucial concept needed in the treatment of systems of
­interacting particles, as in real gases and liquids, is the ‘ensemble’. Like so many scientific terms, the term has basically its normal meaning of ‘collection’, but in statistical thermodynamics it
has been sharpened and refined into a precise ­significance.
13D.1
The concept of ensemble
To set up an ensemble, take a closed system of specified volume,
composition, and temperature, and think of it as replicated N
times (Fig. 13D.1). All the identical closed systems are regarded
as being in thermal contact with one another, so they can ex Because
change energy. The total energy of the ensemble is E.
the members of the ensemble are all in thermal equilibrium
with each other, they have the same temperature, T. The volume
of each member of the ensemble is the same, so the energy levels available to the molecules are the same in each system, and
each member contains the same number of molecules, so there
is a fixed number of molecules to distribute within each system.
This imaginary collection of replications of the actual system
with a common temperature is called the canonical ensemble.1
1
The word ‘canon’ means ‘according to a rule’.
N,
V,
T 1
N,
V,
T
11
2
12
3
4
5
6
7
8
16
17
18
9 10
y
Energ
13 14
15
19
20
Figure 13D.1 A representation of the canonical ensemble, in this
case for N = 20. The individual replications of the actual system
all have the same composition and volume. They are all in mutual
thermal contact, and so all have the same temperature. Energy
may be transferred between them as heat, and so they do not all
have the same energy. The total energy of all 20 replications is a
constant because the ensemble is isolated overall.
There are two other important types of ensembles. In the microcanonical ensemble the condition of constant temperature
is replaced by the requirement that all the systems should have
exactly the same energy: each system is individually ­isolated.
In the grand canonical ensemble the volume and temperature
of each system is the same, but they are open, which means
that matter can be imagined as able to pass between them; the
composition of each one may fluctuate, but now the property
known as the chemical potential (μ, Topic 5A) is the same in
each system. In summary:
Ensemble
Common properties
Microcanonical
V, E, N
Canonical
V, T, N
Grand canonical
V, T, μ
The microcanonical ensemble is the basis of the discussion in
Topic 13A; the grand canonical ensemble will not be considered explicitly.
The important point about an ensemble is that it is a collection of imaginary replications of the system, so the number
of members can be as large as desired; when appropriate, N
can be taken as infinite. The number of members of the ensemble in a state with energy Ei is denoted N i, and it is possible to speak of the configuration of the ensemble (by analogy
with the configuration of the system used in Topic 13A) and
13D The canonical ensemble
N is the number of molecules in the system, the same in
each member of the ensemble.
T is the common temperature of the members.
Ei is the total energy of one member of the ensemble (the
one labelled i).
N i is the number of replicas that have the energy Ei.
E is the total energy of the entire ensemble.
N is the total number of replicas (the number of members
of the ensemble).
W is the weight of the configuration {N 0,N 1, …}.
(a)
Dominating configurations
Just as in Topic 13A, some of the configurations of the canonical ensemble are very much more probable than others. For
instance, it is very unlikely that the whole of the total energy
of the ensemble will accumulate in one system to give the con
figuration {N,0,0,
…}. By analogy with the discussion in Topic
13A, there is a dominating configuration, and thermodynamic
properties can be calculated by taking the average over the ensemble using that single, most probable, configuration. In the
thermodynamic limit of N → ∞, this dominating configuration is overwhelmingly the most probable, and dominates its
properties.
The quantitative discussion follows the argument in Topic
13A (leading to the Boltzmann distribution) with the modification that N and Ni are replaced by N and N i. The weight W of
a configuration {N 0,N 1, …} is
N!
W = N 0 ! N1 !…
Weight
i
Q = ∑e − β E i
Canonical distribution
The canonical distribution in eqn 13D.2 is only apparently an exponentially decreasing function of the energy of the system. Just
as the Boltzmann distribution gives the occupation of a single
state of a molecule, the canonical distribution function gives the
probability of occurrence of members in a single state i of energy
Ei. There may in fact be numerous states with almost identical
energies. For example, in a gas the identities of the molecules
moving slowly or quickly can change without necessarily affecting the total energy. The energy density of states, the number
of states in an energy range divided by the width of the range
(Fig. 13D.2), is a very sharply increasing function of energy. It
follows that the probability of a member of an ensemble having
a specified energy (as distinct from being in a specified state) is
given by eqn 13D.2, a sharply decreasing function, multiplied by
a sharply increasing function (Fig. 13D.3). Therefore, the overall
distribution is a sharply peaked function. That is, most members
of the ensemble have an energy very close to the mean value.
Width of
range
Number of
states
Figure 13D.2 The energy density of states is the number of states
in an energy range divided by the width of the range.
(13D.1)
The configuration of greatest weight, subject to the constraints
that the total energy of the ensemble is constant at E and that
is given by the
the total number of members is fixed at N,
­canonical distribution:
N i e − β E
=
Q
N
Fluctuations from the most probable
distribution
(b)
Energy
its weight, W , the number of ways of achieving the configuration {N 0,N 1, …}. Note that N is unrelated to N, the number of
molecules in the actual system; N is the number of imaginary
replications of that system. In summary:
555
Probability
of state
Number of
states
Probability
of energy
(13D.2)
i
where the sum is over all members of the ensemble. The quantity Q , which is a function of the temperature, is called the
­canonical partition function, and β = 1/kT. Like the molecular partition function, the canonical partition function contains all the thermodynamic information about a system but
allows for the possibility of interactions between the constituent molecules.
Energy
Figure 13D.3 To construct the form of the distribution of
members of the canonical ensemble in terms of their energies, the
probability that any one is in a state of given energy (eqn 13D.2) is
multiplied by the number of states corresponding to that energy
(a steeply rising function). The product is a sharply peaked function
at the mean energy (here considerably magnified), which shows
that almost all the members of the ensemble have that energy.
556 13
Statistical thermodynamics
Brief illustration 13D.1
A function that increases rapidly is xn, with n large. A function that decreases rapidly is e − nx, once again, with n large.
The product of these two functions, normalized so that the
maxima for different values of n all coincide,
f ( x ) = en x ne − nx
is plotted for three values of n in Fig. 13D.4. Note that the
width of the product does indeed decrease as n increases.
1
By the same argument that led to eqn 13C.4a (⟨ε ⟩= −(1/q )
(∂ q / ∂β )V , when ε gs ≡ 0),
⟨E⟩ = −
1  ∂Q 
 ∂lnQ  Q  ∂β  V = −  ∂β  V
Mean energy
of a system
(13D.5)
As in the case of the mean molecular energy, the ground-state
energy of the entire system must be added to this expression if
it is not zero.
Independent molecules
revisited
13D.3
0.8
10
0.6
When the molecules are in fact independent of each other, Q
can be shown to be related to the molecular partition ­function q.
20
f (x)
50
0.4
How is that done? 13D.1 Establishing the relation
0.2
0
between Q and q
0
0.5
1
1.5 x
2
2.5
3
Figure 13D.4 The product of the two functions discussed in
Brief illustration 13D.1, for three different values of n.
13D.2
The mean energy of a system
Just as the molecular partition function can be used to calculate the mean value of a molecular property, so the canonical
partition function can be used to calculate the mean energy of
an entire system composed of molecules which might or might
not be interacting with one another. Thus, Q is more general
than q because it does not assume that the molecules are independent. Therefore Q can be used to discuss the properties of
condensed phases and real gases where molecular interactions
are important.
and there
Because the total energy of the ensemble is E,
N.
are N members, the mean energy of a member is ⟨E⟩ = E/
Because the fraction, pi, of members of the ensemble in a
state i with energy Ei is given by the analogue of eqn 13A.10b
( pi = e − βε /q with pi = Ni/N) as
i
− β Ei
e
pi = Q (13D.3)
i
1
Q
∑E e
i
i
− β Ei
Step 1 Consider a system of independent, distinguishable
­molecules
The total energy of a collection of N independent molecules is
the sum of the energies of the molecules. Therefore, the total
energy of a state i of the system is written as
Ei = εi(1) + εi(2) + … + εi(N)
In this expression, εi(1) is the energy of molecule 1 when the
system is in the state i, εi(2) the energy of molecule 2 when the
system is in the same state i, and so on. The canonical partition function is then
Q = ∑e − βε (1) −βε (2) − − βε ( N )
i
i
i
i
Provided the molecules are distinguishable (in a sense
described below), the sum over the states of the system can
be reproduced by letting each molecule enter all its own individual states. Therefore, instead of summing over the states
i of the system, sum over all the individual states j of molecule
1, all the states j of molecule 2, and so on. This rewriting of the
original expression leads to
q
q
q






Q =  ∑e − βε j   ∑e − βε j   ∑e − βε j 

  j
 j
 j
it follows that
⟨ E ⟩ = ∑ pi Ei =
There are two cases you need to consider because it turns out to
be important to consider initially a system in which the particles are distinguishable (such as when they are at fixed locations
in a solid) and then one in which they are indistinguishable
(such as when they are in a gas and able to exchange places).
(13D.4)
and therefore to Q = q N.
13D The canonical ensemble
Step 2 Consider a system of independent, indistinguishable
molecules
If all the molecules are identical and free to move through
space, it is not possible to distinguish them and the relation
Q = q N is not valid. Suppose that molecule 1 is in some state
a, molecule 2 is in b, and molecule 3 is in c, then one member
of the ensemble has an energy E = εa + εb + εc. This member,
however, is indistinguishable from one formed by putting
molecule 1 in state b, molecule 2 in state c, and molecule
3 in state a, or some other permutation. There are six such
permutations in all, and N! in general. In the case of indistinguishable molecules, it follows that too many states have
been counted in going from the sum over system states to the
sum over molecular states, so writing Q = q N overestimates the
value of Q. The detailed argument is quite involved, but at all
except very low temperatures it turns out that the correction
factor is 1/N!, so Q = q N/N!.
Step 3 Summarize the results
It follows that:
For distinguishable independent molecules:
Q = qN
For indistinguishable independent ­molecules:
N
Q = q /N!
(13D.6a)
(13D.6b)
For molecules to be indistinguishable, they must be of
the same kind: an Ar atom is never indistinguishable from
a Ne atom. Their identity, however, is not the only criterion.
Each identical molecule in a crystal lattice, for instance, can
be ‘named’ with a set of coordinates. Identical molecules in
a lattice can therefore be treated as distinguishable because
their sites are distinguishable, and eqn 13D.6a can be used.
On the other hand, identical molecules in a gas are free to
move to different locations, and there is no way of keeping track of the identity of a given molecule; therefore eqn
13D.6b must be used.
Brief illustration 13D.2
For a gas of N indistinguishable molecules, Q = q N/N!. The
energy of the system is therefore
 ∂ln( q / N !) 
 ∂(ln q − ln N !) 
 ∂( N ln q ) 
⟨E ⟩ = − 
= −
= −


β
β
∂
∂
 ∂β  V

V

V
N
N
 ∂ln q 
= −N 
= N ⟨ε ⟩
 ∂β  V
That is, the mean energy of the gas is N times the mean energy
of a single molecule.
557
The variation of the energy with
volume
13D.4
When there are interactions between molecules, the energy of
a collection depends on the average distance between them,
and therefore on the volume that a fixed number occupy. This
dependence on volume is particularly important for the discussion of real gases (Topic 1C).
To discuss the dependence of the energy on the volume at
constant temperature it is necessary to evaluate (∂⟨E⟩/∂V)T. (In
Topics 2D and 3E, this quantity is identified as the ‘internal
pressure’ of a gas and denoted πT.) To proceed, substitute eqn
13D.5 and obtain
 ∂  ∂lnQ  
 ∂⟨ E ⟩ 
 ∂V  = −  ∂V  ∂β   
T
V T
(13D.7)
If the gas were perfect, Q = q N/N!, with q the partition function for the translational and any internal (such as rotational)
modes. If the gas is monatomic, only the translational mode is
present and q = V/Λ3 with Λ = h/(2πmkT)1/2 and the canonical
partition function would be V N/Λ3NN!. The presence of interactions is taken into account by replacing V N/N! by a factor
called the configuration integral, Z , which depends on the
intermolecular potentials (don’t confuse this Z with the compression factor Z in Topic 1C), and writing
Z
Λ3 N
It then follows that
Q=
(13D.8)
 ∂  ∂ln(Z /Λ3 N )  
 ∂⟨ E ⟩ 
 ∂V  = −  ∂V 
 
∂β


T
V
T
 ∂  ∂lnZ    ∂  ∂ln(1/Λ3 N )  
−
= −


 
∂β
 ∂V  ∂β  V  T  ∂V 
V T
∂2f/∂x∂y = ∂2f/∂y∂x for the second (blue) term

 ∂  ∂lnZ    ∂
−
= −


 ∂V  ∂β  V  T  ∂β

0

 ∂ln(1/Λ3 N )  
 

∂V
T
V
∂ ln y/∂x = (1/y)∂y/∂x
 ∂ 1  ∂Z  
 ∂  ∂ln Z  
= −
 ∂β   = −  ∂V Z  ∂β   ∂
V


V T
V T
(13D.9)
In the third line, Λ is independent of volume, so its derivative
with respect to volume is zero.
For a real gas of atoms (for which the intermolecular interactions are isotropic), Z is related to the total potential energy
EP of interaction of all the particles, which depends on all their
relative locations, by
558 13
Statistical thermodynamics
1
Z = N ! ∫ e − β E dτ 1 dτ 2 … dτ N p
(13D.10)
Configuration integral
where dτi is the volume element for atom i and the integration
is over all the variables. The physical origin of this term is that
the probability of occurrence of each arrangement of molecules possible in the sample is given by a Boltzmann distribution in which the exponent is given by the potential energy
corresponding to that arrangement.
Brief illustration 13D.3
where n is the amount of molecules present in the volume V and
a is a constant that is proportional to the area under the attractive part of the potential. In Example 3E.2 of Topic 3E exactly the
same expression (in the form πT = an2/V 2) was derived from the
van der Waals equation of state. The conclusion at this point is
that if there are attractive interactions between molecules in a
gas, then its energy increases as it expands isothermally (because
(∂⟨E⟩/∂V)T > 0, and the slope of ⟨E⟩ with respect to V is positive).
The energy rises because, at greater average separations, the molecules spend less time in regions where they interact favourably.
1
1
( )
Z = N ! ∫ dτ 1 dτ 2 … dτ N = N ! ∫ dτ
N
=
VN
N!
just as it should be for a perfect gas.
Intermolecular separation
0
If the potential energy has the form of a central hard sphere
surrounded by a shallow attractive well (Fig. 13D.5), then detailed calculation, which is too involved to reproduce here (see
A deeper look 7 on the website of this text), leads to
2
 ∂⟨ E ⟩  an
 ∂V  = V 2 T
Determines b
Potential energy, V
Equation 13D.10 is very difficult to manipulate in practice,
even for quite simple intermolecular potentials, except for
a perfect gas for which EP = 0. In that case, the exponential
function becomes 1 and
(13D.11)
Attractive potential
Determines a
Figure 13D.5 The intermolecular potential energy of molecules
in a real gas can be modelled by a central hard sphere that
determines the van der Waals parameter b surrounded by a
shallow attractive well that determines the parameter a. As
mentioned in the text, calculations of the canonical partition
function based on this are consistent with the van der Waals
equation of state (Topic 1C).
Checklist of concepts
☐ 1. The canonical ensemble is a collection of imaginary
replications of the actual system with a common temperature and number of particles.
☐ 3. The mean energy of the members of the ensemble can
be calculated from the canonical partition function.
☐ 2. The canonical distribution gives the most probable
number of members of the ensemble with a specified
total energy.
Checklist of equations
Property
Equation
Comment
Equation number
= ∑e
Definition
13D.2
− β Ei
Canonical partition function
Q
Canonical distribution
Ni /N = e − β Ei /Q
13D.2
Mean energy
⟨ E ⟩ = − (1/Q )(∂Q / ∂β )V = − (∂lnQ / ∂β )V
13D.5
Canonical partition function
Q = Z /Λ
13D.8
Configuration integral
Z = (1/N !)∫ e
Variation of mean energy with volume
i
3N
− β Ep
dτ 1 dτ 2 …dτ N
2
(∂⟨ E ⟩ / ∂V )T = an /V
2
Isotropic interaction
13D.10
Potential energy as specified in Fig. 13D.5
13D.11