Chapter 9
Canonical ensemble
9.1
System in contact with a heat reservoir
We consider a small system A1 characterized by
E1 , V1 and N1 in thermal interaction with a heat
reservoir A2 characterized by E2 , V2 and N1 in
thermal interaction such that A1 � A2 , A1 has
hence fewer degrees of freedom than A2 .
E2 � E 1
N2 � N 1
N1 = const.
N2 = const.
with
E1 + E2 = E = const.
Both systems are in thermal equilibrium at temperature T . The wall between them allows interchange of heat but not of particles. The
system A1 may be any relatively small macroscopic system such as, for instance, a bottle
of water in a lake, while the lake acts as the heat reservoir A2 .
Distribution of energy
states
The question we want to answer is the following:
“Under equilibrium conditions, what is the
probability of finding the small system A1
in any particular microstate α of energy
Eα ? In other words, what is the distribution function ρ = ρ(Eα ) of the system
A1 ?”
We note that the energy E1 is not fixed, only the total energy E = E1 +E2 of the combined
system.
Hamilton function. The Hamilton function of the combined system A is
H(q, p) = H1 (q(1), p(1)) + H2 (q(2), p(2)) ,
103
104
CHAPTER 9. CANONICAL ENSEMBLE
were we have used the notation
q = (q(1), q(2)),
p = (p(1), p(2)) .
Microcanonical ensemble of the combined system. Since the combined system A
is isolated, the distribution function in the combined phase space is given by the microcanonical distribution function ρ(q, p),
�
δ (E − H(q, p)))
ρ(q, p) = �
,
dq dp δ(E − H) = Ω(E) ,
(9.1)
dq dp δ (E − H(q, p))
where Ω(E) is the density of phase space (8.4).
Tracing out A2. It is not the distribution function ρ(q, p) = ρ(q(1), p(1), q(2), p(2)) of
the total system A that we are interested in, but in the distribution function ρ1 (q(1), p(1))
of the small system A1 . One hence needs to trace out A2 :∗
�
ρ1 (q(1), p(1)) ≡
dq(2) dp(2) ρ(q(1), p(1), q(2), p(2))
�
dq(2) dp(2) δ(E − H1 − H2 )
=
Ω(E)
Ω2 (E − H1 )
.
(9.2)
≡
Ω(E)
where Ω2 (E2 ) = Ω(E − H1 ) is the phase space density of A2 .
Small E1 expansion. Now, we make use of the fact that A1 is a much smaller system
than A2 and therefore the energy E1 given by H1 is much smaller than the energy of the
combined system:
E1 � E .
In this case, we can approximate (9.2) by expanding the slowly varying logarithm of
Ω2 (E2 ) = Ω2 (E − H1 ) around the E2 = E as
�
�
∂ ln Ω2
ln Ω2 (E2 ) = ln Ω2 (E − H1 ) � ln Ω2 (E) −
H1 + . . .
(9.3)
∂E2 E2 =E
and neglect the higher-order terms since H1 = E1 � E.
Derivatives of the entropy. Using (8.14), namely that
�
�
�
�
Γ(E, V, N )
Ω(E)Δ
= kB ln
,
S = kB ln
Γ0
Γ0
(9.4)
where Δ is the width of the energy shell, we find that derivatives of the entropy like
∂S
∂ ln Ω(E)
1
=
= kB
T
∂E
∂E
(9.5)
�
A marginal distribution function p(x) = p(x, y)dy in generically obtained by tracing out other
variables from a joint distribution function p(x, y).
∗
9.1. SYSTEM IN CONTACT WITH A HEAT RESERVOIR
105
can be taken with respect to the logarithm of the phase space density Ω(E).
Boltzmann factor. Using (9.5) for the larger system A2 we may rewrite (9.3) as
�
�
�
∂ ln Ω2 (E2 ) ��
H1 + . . .
Ω2 (E − H1 ) = exp ln Ω2 (E) −
�
∂E2
E2 =E
�
�
H1
= Ω2 (E) exp −
.
kB T 2
The temperature T2 of the heat reservoir A2 by whatever small amount of energy the
large system A2 gives to the small system A1 . Both systems are thermally coupled, such
that T1 = T2 = T . We hence find with (9.2)
ρ1 (q(1), p(1)) =
H
Ω2 (E) − kH1T
− 1
e B ∝ e kB T
Ω(E)
.
(9.6)
The factor exp[−H1 /(kB T )] is called the Boltzmann factor.
Distribution function of the canonical ensemble. The prefactor Ω2 (E)/Ω(E) in
(9.6) is independent of H1 . We may hence obtain the the normalization of ρ1 alternatively
by integrating over the phase space of A1 :
ρ1 (q(1), p(1)) = �
9.1.1
e−βH1 (q(1),p(1))
dq(1) dp(1) e−βH1 (q(1),p(1))
,
β =
1
.
kB T
(9.7)
Boltzmann factor
The probability Pα of finding the system A1 (which is in thermal equilibrium with the
heat reservoir A2 ) in a microstate α with energy Eα is given by
e−βEα
Pα = � −βEα
αe
(9.8)
Boltzmann distribution
when rewriting (9.7) in terms of Pα .
– The number of states Ω2 (E2 ) = Ω2 (E − H1 ) accessible to the reservoir is a rapidly
increasing function of its energy.
– The number of states Ω2 (E2 ) = Ω2 (E − H1 ) accessible to the reservoir decreases
therefore rapidly with increasing E1 = E − E2 . The probability of finding states
with large E1 is accordingly also rapidly decreasing.
The exponential dependence of Pα on Eα in equation (9.8) expresses this fact in mathematical terms.
106
CHAPTER 9. CANONICAL ENSEMBLE
Example. Suppose a certain number of states accessible to A1 and A2
for various values of their respective
energies, as given in the figure, and
that the total energy of the combined
system is 1007.
– Let A1 be in a state α with energy 6. E2 is then in one of the
3 · 105 states with energy 1001.
– If A1 is in a state γ with energy
7, the reservoir must be in one
of the 1 · 105 states with energy
1000.
The number of realizations of states
with E1 = 6 the ensemble contains is hence much higher than the number of realization
of state with E1 = 7.
Canonical ensemble. An ensemble in contact with a heat reservoir at temperature
T is called a canonical ensemble, with the Boltzmann factor exp(−βEα ) describing the
canonical distribution (9.8).
Energy distribution function. The Boltzmann distribution (9.8) provides the probability
Pα to find an individual microstates α. There
are in general many microstates in a given energy, for which
P (E) =
�
E<Eα <E+Δ
Pα ∝ Ω(E) e−βE ,
(9.9)
is the corresponding energy distribution function. Ω(E) = Ω1 (E) is, as usual, the density of phase space.
– P (E) is rapidly decreasing for increasing energies due to the Boltzmann factor
exp(−βEα ).
– P (E) is rapidly decreasing for decreasing energies due to the decreasing phase space
density Ω(E).
The energy density is therefore sharply peaked. We will discuss the the width of the peak,
viz the energy fluctuations, more in detail in Sect. 9.6.
9.2. CANONICAL PARTITION FUNCTION
9.2
107
Canonical partition function
We rewrite the distribution function (9.7) of the canonical ensemble as
ρ(q, p) = �
e−βH(q,p)
,
d3N q d3N p e−βH(q,p)
where we dropped all the indices ”1” for simplicity, though in fact we are still describing
the properties of a “small” system (which is nevertheless macroscopically big) in thermal
equilibrium with a heat reservoir.
Partition function. The canonical partition function (“kanonische Zustandssumme”)
ZN is defined as
� 3N 3N
d q d p −βH(q,p)
ZN =
.
(9.10)
e
h3N N !
It is proportional to the canonical distribution function ρ(q, p), but with a different normalization, and analogous to the microcanonical space volume Γ(E) in units of Γ0 :
�
1
Γ(E)
= 3N
d3N q d3N p
Γ0
h N ! E<H(q,p)<E+Δ
� 3N 3N
�
d q d p�
Θ(E + Δ − H) − Θ(E − H) ,
=
3N
h N!
where Θ is the step function.
Free energy. We will show that it is possible to obtain all thermodynamic observables
by differentiating the partition function ZN . We will prove in particular that
F (T, V, N ) = −kB T ln ZN (T ) ,
ZN = e−βF (T,V,N ) ,
(9.11)
where F (T, V, N ) is the Helmholtz free energy.
Proof. In order to proof (9.11) we perform the differentiation
1 ∂ZN
∂
ln ZN =
∂β
ZN ∂β
�
� ��
�
�
∂
dqdp −βH
dqdp −βH �
=
e
e
∂β
h3N N !
h3N N !
�
dqdp (−H) e−βH
�
=
dqdp e−βH
= −�H� = −U .
where we have used the shortcut dqdp = d3N qd3N p and that �H� = E = U is the internal
energy.
108
CHAPTER 9. CANONICAL ENSEMBLE
With (5.13), namely that U = ∂(βF )/∂β, we find that
−
∂
∂
ln ZN = U =
(βF ),
∂β
∂β
ln ZN = −βF,
ZN = e−βF
,
which is what we wanted to prove.
Integration constant. Above derivation allows to identify ln ZN = −βF only up to an
integration constant (or, equivalently, ZN only up to a multiplicative factor). Setting this
constant to zero results in the correct result for the ideal gas, as we will show lateron in
Sect. 9.5.
Thermodynamic properties. Once the partition function ZN and the free energy
F (T, V, N ) = −kB T ln ZN (T, V, N ) are calculated, one obtains the pressure P , the entropy
S and the chemical potential µ as usual via
�
�
�
�
�
�
∂F
∂F
∂F
P = −
,
S = −
,
µ =
.
∂V T,N
∂T V,N
∂N T,V
Specific heat. The specific heat CV is given in particular by
� �
�
∂ 2F
∂2 �
∂S
CV
= − 2 =
=
kB T ln ZN ,
T
∂T V
∂T
∂T 2
(9.12)
where we have used F = −kB T ln ZN .
9.3
Canonical vs. microcanonical ensemble
We have seen that the calculations in the microcanonical and canonical ensembles reduce
to a phase space integration and a calculation of a thermodynamic potential:
Phase space
integration
Thermodynamic
potential
Microcanonical ensemble
Canonical ensemble
Density of states:
�
ΩN (E) = d3N q d3N p δ(E − H)
Partition function:
� 3N 3N
d q d p −βH(q,p)
e
ZN (T ) =
h3N N !
S(E, V, N ) = kB ln
�
ΩN (E)Δ
h3N N !
�
F (T, V, N ) = −kB T ln ZN (T )
Laplace transforms. The relation between the density of states ΩN (E) and the partition
function ZN (T ) can be defined as a Laplace transformation in the following way. We use
the definition (9.1) of the density of states Ω(E),
�
dq dp δ(E − H) = Ω(E),
H = H(q, p) ,
9.4. ADDITIVITY OF F (T, V, N )
109
in order to obtain
�
∞
0
dE e−βE
ΩN (E)
h3N N !
=
=
�
�
d
3N
qd
3N
p
�
∞
0
dE e−βE
δ(E − H)
h3N N !
d3N q d3N p −βH(q,p)
e
=
h3N N !
ZN (T ) .
(9.13)
We have thus shown that ZN (T ) is the Laplace transform† of ΩN (E).
Additive Hamilton functions. In both the microcanonical and in the canonical ensemble we have to perform�an integration which is usually difficult. When the Hamilton
function is additive, H = i Hi , the integration in the canonical ensemble can be factorized, which is not the case for the microcanonical ensemble. Therefore, it is usually easier
to calculate in the canonical ensemble than in the microcanonical ensemble.
9.4
Additivity of F (T, V, N )
An important property of the free energy is that it has to be additive.
Non-interacting systems. Let us consider
two systems in thermal equilibrium. Neglecting the interaction among the systems, the total
Hamilton function can be written as a sum of
the Hamiltonians of the individual systems,
H = H 1 + H2 ,
N = N1 + N2 .
Multiplication of partition functions. The partition function of the total system is
�
1
d3N q d3N p e−β(H1 +H2 ) ,
ZN (T, V ) = 3N
h N1 !N2 !
where have made use of the fact that there is not exchange of particles between the two
systems. The factor in the denominator is therefore proportional to N1 !N2 ! and NOT to
N !. It then follows that the partition function factorizes,
�
1
ZN (T, V ) =
d3N1 q d3N1 p e−βH1 (q1 ,p1 )
3N
1
h N1 !
�
1
× 3N2
d3N2 q d3N2 p e−βH2 (q2 ,p2 )
h N2 !
= ZN1 (T, V1 ) ZN2 (T, V2 ) ,
and that the free energy F = −kB T ln ZN is additive:
F (T, V, N ) = F1 (T, V1 , N1 ) + F2 (T, V2 , N2 ) .
†
The Laplace transform F (s) of a function f (t) is defined as F (s) =
�∞
0
f (t) exp(−st)dt.
110
CHAPTER 9. CANONICAL ENSEMBLE
Convolution of densities of states. That the overall partition function factorizes
follows also from the fact that the density of states Ω(E) of the combined system,
�
Ω(E) =
d3N q d3N p δ(E − H1 − H2 )
�
�
3N1
3N1
3N2
3N2
=
d qd pd qd p
dE2 δ(E − H1 − E2 ) δ(E2 − H2 )
�
=
dE2 Ω1 (E − E2 ) Ω2 (E2 ) ,
is given by the ( convolution) of the density of states Ωi (Ei ) of the individual systems.
Using the representation (9.13) for the partition function we obtain‡
�
dE e−βE
ZN =
Ω(E)
h3N N1 !N2
�
�
dE e−β(E1 +E2 )
dE2 Ω1 (E − E2 ) Ω2 (E2 ) .
=
�
��
�
h3N N1 !N2 !
Ω1 (E1 )
A change of the integration variable from dE to dE1 then leads again to
ZN (T, V ) = ZN1 (T, V1 ) ZN2 (T, V2 ) .
(9.14)
Note that this relation is only valid if H = H1 + H2 and H12 = 0.
9.5
Ideal gas in the canonical ensemble
We consider now the ideal gas in the canonical ensemble, for which the Hamilton function,
H =
N
�
p�i 2
i=1
2m
,
ZN (T, V ) =
�
d3N q d3N p −β �Ni=1 p�2i /(2m)
e
,
h3N N !
(9.15)
contains just the kinetic energy.
Factorization. The integral leading to ZN factorizes in (9.15):
�3N
�� +∞
VN
dp −β p2
ZN (T, V ) =
e 2m
N!
−∞ h
�� +∞ √
�3N
VN
2kB T m −x2
e dx
=
,
N!
h
−∞
(9.16)
where we have used the variable substitution
p2
x =
,
2kB T m
2
�
‡
dp
dx = √
,
2kB T m
�
+∞
2
dx e−x =
√
π.
−∞
Note that a variable transfomation (E, E2 ) → (E1 , E2 ) with a Jacobian determinant,
dE1 dE2 |J|, where J is the respective Jacobian.
(9.17)
�
dE dE2 =
9.5. IDEAL GAS IN THE CANONICAL ENSEMBLE
111
Thermal wavelength. Evaluating (9.16) explicitly with the help of (9.17) we get
VN
ZN (T, V ) =
N!
�√
2πmkB T
h
�3N
1
≡
N!
�
V
λ3T
�N
,
(9.18)
where we have defined the thermal wavelength λT as
λT = √
h
2πmkB T
.
For air (actually nitrogen, N2 , with m = 4.65 · 10−26 kg) at T = 298 K, the thermal wavelength is 0.19 A◦ , which is actually smaller than the Bohr radius. Quantum mechanical
effects start to play a role only once λT becomes larger than the typical interparticle
separation.
Thermal momentum. Heisenberg’s uncertainty principle Δx · Δp ∼ h allows to define
a thermal momentum pT as
pT =
�
h
= 2πmkB T ,
λT
p2T
2π
= πkB T =
Ekin ,
2m
3
Ekin =
3
kB T ,
2
where we have used (3.5) for the average energy Ekin per particle. The thermal momentum
pT is hence of the same order of magnitude as the average momentum p̄ of the gas, as
defined by Ekin = p̄2 /(2m), but not identical.
Free energy. From (9.18) we obtain (with log N ! ≈ N log N − N )
�
� �N �
1
V
F (T, V, N ) = −kB T ln
N ! λ3T
�
� �N �
V
1
+ ln
= −kB T ln
N!
λ3T
�
�
V
= −kB T −N ln N + N + N ln 3
λT
and hence
�
�
� �
V
+1
F (T, V, N ) = −N kB T ln
N λ3T
for the free energy of the ideal gas.
Entropy. Using
−λT
∂λT
=
,
∂T
2T
we then have
S = −
�
∂F
∂T
�
V,N
∂
ln
∂T
�
V
N λ3T
�
= −3
3
∂ ln λT
=
∂T
2T
� �
�
�
� �
V
3
+ 1 + N kB T
= N kB ln
,
3
N λT
2T
(9.19)
112
CHAPTER 9. CANONICAL ENSEMBLE
which results in the Sackur-Tetrode equation
S =
N kB
�
V
5
ln
+
3
N λT
2
�
.
(9.20)
Comparing (9.20) with (8.25), namely with the microcanonical Sackur-Tetrode equation
�
� ��
�3/2 �
5
4πmE
V
+
,
S = kB N ln
3h2 N
N
2
one finds that they coincide when E/N = 3kB T /2.
Chemical potential. The chemical potential µ is
�
�
�
�
�
�
∂F
V
V
λ3T
µ=
= −kB T ln
+ 1 + N kB T
·
∂N T,V
N λ3T
N λ3T V
=
−kB T ln
�
V
N λ3T
�
.
The previous expressions were much simpler obtained than when calculated in the microcanonical ensemble.
Equivalence of ensembles. In the thermodynamic limit the average value of an observable is in general independent of the ensemble (microcanonical or canonical).
N → ∞,
V → ∞,
N
= const.
V
is taken. One therefore usually chooses the ensemble that is easier to work with.
Fluctuations of observables. Fluctuations of observables, �A2 � − �A�2 , may however
be ensemble dependent! An example for an observable for which this is the case is the
energy, which is constant, by definition, in the microcanonical ensemble, but distributed
according to (9.9) in the canonical ensemble.
9.6
Energy fluctuations
We evaluated the representation (9.12) for the specific heat in a first step:
CV
T
�
∂2 �
T
ln
Z
k
B
n
∂T 2
�
�
kB T ∂Zn ∂β
∂
kB ln Zn +
=
∂T
ZN ∂β ∂T
�
�
∂
1 ∂Zn
=
kB ln Zn −
.
∂T
T ZN ∂β
=
∂β
−1
=
∂T
kB T 2
9.6. ENERGY FLUCTUATIONS
113
Second derivatives. The remaining derivative with respect to the temperature T are
−1 ∂Zn
∂
kB ln Zn =
∂T
T 2 ZN ∂β
�
�
�
�2
∂Zn
1 ∂ 2 Zn −1
1 ∂Zn
1
∂ −1 ∂Zn
−
.
=
+
∂T T ZN ∂β
T 2 ZN ∂β
T ZN2
∂β
T ZN ∂β 2 kB T 2
With the first two terms canceling each other We find
�
�2 �
�
1 ∂ 2 Zn
1
1 ∂Zn
CV =
−
kb T 2 ZN ∂β 2
ZN ∂β
(9.21)
for the specific heat Cv as a functions of derivatives of the partition function ZN .
Derivatives of the partition function. The definition (9.10) for the partition function
corresponds to
� d3N q d3N p
� 3N 3N
H e−βH(q,p)
1 ∂Zn
d q d p −βH(q,p)
h3N N !
= − � d3N q d3N p
e
,
ZN =
,
ZN ∂β
h3N N !
e−βH(q,p)
3N
h
viz to
N!
� �
1 ∂Zn
= − E ,
ZN ∂β
� 2�
1 ∂ 2 Zn
=
E .
ZN ∂β 2
(9.22)
Specific heat. Our results (9.21) and (9.22) lead to the fundamental relation
1 �� 2 � � � 2 �
E − E
kb T 2
� � �2
between the specific heat CV and the fluctuations �E 2 − E of the energy.
CV =
(9.23)
– Both the specific heat CV ∼ N and the right-hand side of (9.23) are extensive. The
later as a result of the central limit theorem discussed in Sect. 8.6, which states that
the variance of independent processes are additive.
– The specific heat describes the energy exchange between the system and an heat
reservoir. It hence makes that sense that CV is proportional to the size of the energy
fluctuations.
Relative energy fluctuations. The relative energy fluctuations,
�� � � �
2
E2 − E
1
� �
∼ √
E
N
(9.24)
vanish in the thermodynamic limit N → ∞.
– The scaling relation (9.24) if a direct consequence of (9.23) and of the fact that both
CV and the internal energy U = �E� are extensive.
– Eq. (9.24) is consistent with the demand that the canonical the microcanonical
ensembles are equivalent in the thermodynamic limit N → ∞. Energy fluctuations
are absent in the microcanonical ensemble.
114
9.7
CHAPTER 9. CANONICAL ENSEMBLE
Paramagnetism
We consider a system with N magnetic atoms per unit volume placed in an external
magnetic field H. Each atom has an intrinsic magnetic moment µ = 2µ0 s with spin
s = 1/2.
Energy states. In a quantum-mechanical description, the magnetic moments of the
atoms can point either parallel or anti-parallel to the magnetic field.
state
alignment
moment
energy
probability
(+)
parallel to H
+µ
−µH
P+ = c e−βε+ = c e+βµH
(−)
anti-parallel to H
−µ
+µH
P− = c e−βε− = c e−βµH
We assume here that the atoms interact weakly. One can therefore a single atom as a
small system and the rest of the atoms as a reservoir in the terms of a canonical ensemble.
Mean magnetic moment. We want to analyze the mean magnetic moment �µH � per
atom as a function of the temperature T :
�µH � =
µ eβµH − µ e−βµH
,
eβµH + e−βµH
�µH � = µ tanh
µH
kB T
,
where we used that
tanh y =
ey − e−y
,
ey + e−y
y = βµH =
µH
.
kB T
Magnetization. We define the magnetization, i.e. the mean magnetic moment per unit
volume, as
�M � = N �µH �
and analyze its behavior in the limit of high- and of low temperatures.
High-temperature expansion. Large temperatures correspond to y � 1 and hence to
ey = 1 + y + . . . ,
e−y = 1 − y + . . . .
Then,
tanh y =
(1 + y + . . .) − (1 − y + . . .)
≈ y,
2
so that
�µH � =
µ2 H
kB T
.
9.7. PARAMAGNETISM
115
Curie Law. For the magnetic susceptibility χ, defined as �M � = χH, we then have
χ =
N µ2
kB T
.
At temperatures high compared to the magnetic energies, χ ∝ T −1 which is known as the
Curie law.
Low-temperature expansion. Low temperatures correspond to y � 1,
ey � e−y ,
tanh y ≈ 1 ,
and hence
�µH � = µ,
�M � = N µ .
The magnetization saturates at the maximal
value at low temperatures independent of H.
116
CHAPTER 9. CANONICAL ENSEMBLE