Chapter 9

Canonical ensemble

9.1 System in contact with a heat reservoir

We consider a small system A1 characterized byE1, V1 and N1 in thermal interaction with a heatreservoir A2 characterized by E2, V2 and N1 inthermal interaction such that A1 � A2, A1 hashence fewer degrees of freedom than A2.

E2 � E1 N1 = const.N2 � N1 N2 = const.

withE1 + E2 = E = const.

Both systems are in thermal equilibrium at tem-perature T . The wall between them allows interchange of heat but not of particles. Thesystem A1 may be any relatively small macroscopic system such as, for instance, a bottleof water in a lake, while the lake acts as the heat reservoir A2.

Distribution of energystates

The question we want to answer is the following:

“Under equilibrium conditions, what is theprobability of finding the small system A1

in any particular microstate α of energyEα? In other words, what is the distri-bution function ρ = ρ(Eα) of the systemA1?”

We note that the energy E1 is not fixed, only the total energy E = E1+E2 of the combinedsystem.

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 Ais isolated, the distribution function in the combined phase space is given by the micro-canonical distribution function ρ(q, p),

ρ(q, p) =δ (E −H(q, p)))�

dq dp δ (E −H(q, p)),

�dq dp δ(E −H) = Ω(E) , (9.1)

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)) ofthe 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)

Ω(E). (9.2)

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 systemthan A2 and therefore the energy E1 given by H1 is much smaller than the energy of thecombined system:

E1 � E .

In this case, we can approximate (9.2) by expanding the slowly varying logarithm ofΩ2(E2) = Ω2(E −H1) around the E2 = E as

lnΩ2(E2) = lnΩ2(E −H1) � lnΩ2(E)−�∂ lnΩ2

∂E2

�

E2=E

H1 + . . . (9.3)

and neglect the higher-order terms since H1 = E1 � E.

Derivatives of the entropy. Using (8.14), namely that

S = kB ln

�Γ(E, V,N)

Γ0

�= kB ln

�Ω(E)Δ

Γ0

�, (9.4)

where Δ is the width of the energy shell, we find that derivatives of the entropy like

1

T=

∂S

∂E= kB

∂ lnΩ(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

Ω2(E −H1) = exp

�lnΩ2(E)− ∂ lnΩ2(E2)

∂E2

����E2=E

H1 + . . .

�

= Ω2(E) exp

�− H1

kBT2

�.

The temperature T2 of the heat reservoir A2 by whatever small amount of energy thelarge system A2 gives to the small system A1. Both systems are thermally coupled, suchthat T1 = T2 = T . We hence find with (9.2)

ρ1(q(1), p(1)) =Ω2(E)

Ω(E)e− H1

kBT ∝ e− H1

kBT . (9.6)

The factor exp[−H1/(kBT )] 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 alternativelyby integrating over the phase space of A1:

ρ1(q(1), p(1)) =e−βH1(q(1),p(1))

�dq(1) dp(1) e−βH1(q(1),p(1))

, β =1

kBT. (9.7)

9.1.1 Boltzmann factor

The probability Pα of finding the system A1 (which is in thermal equilibrium with theheat reservoir A2) in a microstate α with energy Eα is given by

Pα =e−βEα

�α e

−βEα

Boltzmann distribution

(9.8)

when rewriting (9.7) in terms of Pα.

– The number of states Ω2(E2) = Ω2(E −H1) accessible to the reservoir is a rapidlyincreasing function of its energy.

– The number of states Ω2(E2) = Ω2(E − H1) accessible to the reservoir decreasestherefore rapidly with increasing E1 = E − E2. The probability of finding stateswith large E1 is accordingly also rapidly decreasing.

The exponential dependence of Pα on Eα in equation (9.8) expresses this fact in mathe-matical terms.

106 CHAPTER 9. CANONICAL ENSEMBLE

Example. Suppose a certain num-ber of states accessible to A1 and A2

for various values of their respectiveenergies, as given in the figure, andthat the total energy of the combinedsystem is 1007.

– Let A1 be in a state α with en-ergy 6. E2 is then in one of the3 · 105 states with energy 1001.

– If A1 is in a state γ with energy7, the reservoir must be in oneof the 1 · 105 states with energy1000.

The number of realizations of stateswith E1 = 6 the ensemble contains is hence much higher than the number of realizationof state with E1 = 7.

Canonical ensemble. An ensemble in contact with a heat reservoir at temperatureT is called a canonical ensemble, with the Boltzmann factor exp(−βEα) describing thecanonical distribution (9.8).

Energy distribution function. The Boltz-mann distribution (9.8) provides the probabilityPα to find an individual microstates α. Thereare in general many microstates in a given en-ergy, for which

P (E) =�

E<Eα<E+Δ

Pα ∝ Ω(E) e−βE , (9.9)

is the corresponding energy distribution func-tion. Ω(E) = Ω1(E) is, as usual, the density of phase space.

– P (E) is rapidly decreasing for increasing energies due to the Boltzmann factorexp(−βEα).

– P (E) is rapidly decreasing for decreasing energies due to the decreasing phase spacedensity 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 107

9.2 Canonical partition function

We rewrite the distribution function (9.7) of the canonical ensemble as

ρ(q, p) =e−βH(q,p)

�d3Nq d3Np e−βH(q,p)

,

where we dropped all the indices ”1” for simplicity, though in fact we are still describingthe properties of a “small” system (which is nevertheless macroscopically big) in thermalequilibrium with a heat reservoir.

Partition function. The canonical partition function (“kanonische Zustandssumme”)ZN is defined as

ZN =

�d3Nq d3Np

h3NN !e−βH(q,p) . (9.10)

It is proportional to the canonical distribution function ρ(q, p), but with a different nor-malization, and analogous to the microcanonical space volume Γ(E) in units of Γ0:

Γ(E)

Γ0

=1

h3NN !

�

E<H(q,p)<E+Δ

d3Nq d3Np

=

�d3Nq d3Np

h3NN !

�Θ(E +Δ−H)−Θ(E −H)

�,

where Θ is the step function.

Free energy. We will show that it is possible to obtain all thermodynamic observablesby differentiating the partition function ZN . We will prove in particular that

F (T, V,N) = −kBT lnZN(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

∂

∂βlnZN =

1

ZN

∂ZN

∂β

=

�∂

∂β

�dqdp

h3NN !e−βH

���� dqdp

h3NN !e−βH

�

=

�dqdp (−H) e−βH

�dqdp e−βH

= −�H� = −U .

where we have used the shortcut dqdp = d3Nqd3Np and that �H� = E = U is the internalenergy.

108 CHAPTER 9. CANONICAL ENSEMBLE

With (5.13), namely that U = ∂(βF )/∂β, we find that

− ∂

∂βlnZN = U =

∂

∂β(βF ), lnZN = −βF, ZN = e−βF ,

which is what we wanted to prove.

Integration constant. Above derivation allows to identify lnZN = −βF only up to anintegration constant (or, equivalently, ZN only up to a multiplicative factor). Setting thisconstant to zero results in the correct result for the ideal gas, as we will show lateron inSect. 9.5.

Thermodynamic properties. Once the partition function ZN and the free energyF (T, V,N) = −kBT lnZN(T, V,N) are calculated, one obtains the pressure P , the entropyS and the chemical potential µ as usual via

P = −�∂F

∂V

�

T,N

, S = −�∂F

∂T

�

V,N

, µ =

�∂F

∂N

�

T,V

.

Specific heat. The specific heat CV is given in particular by

CV

T=

�∂S

∂T

�

V

= −∂2F

∂T 2=

∂2

∂T 2

�kBT lnZN

�, (9.12)

where we have used F = −kBT lnZN .

9.3 Canonical vs. microcanonical ensemble

We have seen that the calculations in the microcanonical and canonical ensembles reduceto a phase space integration and a calculation of a thermodynamic potential:

Microcanonical ensemble Canonical ensemble

Phase space Density of states: Partition function:

integration ΩN(E) =�d3Nq d3Np δ(E −H) ZN(T ) =

�d3Nq d3Np

h3NN !e−βH(q,p)

Thermodynamic

potential S(E, V,N) = kB ln

�ΩN(E)Δ

h3NN !

�F (T, V,N) = −kBT lnZN(T )

Laplace transforms. The relation between the density of states ΩN(E) and the partitionfunction ZN(T ) can be defined as a Laplace transformation in the following way. We usethe 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

h3NN !ΩN(E) =

�d3Nq d3Np

� ∞

0

dE e−βE

h3NN !δ(E −H)

=

�d3Nq d3Np

h3NN !e−βH(q,p) = 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 en-semble we have to perform an integration which is usually difficult. When the Hamiltonfunction is additive, H =

�i Hi, the integration in the canonical ensemble can be factor-

ized, which is not the case for the microcanonical ensemble. Therefore, it is usually easierto 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 considertwo systems in thermal equilibrium. Neglect-ing the interaction among the systems, the totalHamilton function can be written as a sum ofthe Hamiltonians of the individual systems,

H = H1 +H2, N = N1 +N2 .

Multiplication of partition functions. The partition function of the total system is

ZN(T, V ) =1

h3NN1!N2!

�d3Nq d3Np e−β(H1+H2) ,

where have made use of the fact that there is not exchange of particles between the twosystems. The factor in the denominator is therefore proportional to N1!N2! and NOT toN !. It then follows that the partition function factorizes,

ZN(T, V ) =1

h3N1N1!

�d3N1q d3N1p e−βH1(q1,p1)

× 1

h3N2N2!

�d3N2q d3N2p e−βH2(q2,p2)

= ZN1(T, V1)ZN2(T, V2) ,

and that the free energy F = −kBT lnZN 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 factorizesfollows also from the fact that the density of states Ω(E) of the combined system,

Ω(E) =

�d3Nq d3Np δ(E −H1 −H2)

=

�d3N1q d3N1p d3N2q d3N2p

�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‡

ZN =

�dE e−βE

h3NN1!N2

Ω(E)

=

�dE e−β(E1+E2)

h3NN1!N2!

�dE2 Ω1(E − E2)� �� �

Ω1(E1)

Ω2(E2) .

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�

i=1

�pi2

2m, ZN(T, V ) =

�d3Nq d3Np

h3NN !e−β

�Ni=1 �p

2i /(2m) , (9.15)

contains just the kinetic energy.

Factorization. The integral leading to ZN factorizes in (9.15):

ZN(T, V ) =V N

N !

�� +∞

−∞

dp

he−β p2

2m

�3N

=V N

N !

�� +∞

−∞

√2kBTm

he−x2

dx

�3N

, (9.16)

where we have used the variable substitution

x2 =p2

2kBTm, dx =

dp√2kBTm

,

� +∞

−∞dx e−x2

=√π . (9.17)

‡ Note that a variable transfomation (E,E2) → (E1, E2) with a Jacobian determinant,�dE dE2 =�

dE1 dE2 |J |, where J is the respective Jacobian.

9.5. IDEAL GAS IN THE CANONICAL ENSEMBLE 111

Thermal wavelength. Evaluating (9.16) explicitly with the help of (9.17) we get

ZN(T, V ) =V N

N !

�√2πmkBT

h

�3N

≡ 1

N !

�V

λ3T

�N

, (9.18)

where we have defined the thermal wavelength λT as

λT =h√

2πmkBT.

For air (actually nitrogen, N2, with m = 4.65 · 10−26 kg) at T = 298K, the thermal wave-length is 0.19A◦, which is actually smaller than the Bohr radius. Quantum mechanicaleffects start to play a role only once λT becomes larger than the typical interparticleseparation.

Thermal momentum. Heisenberg’s uncertainty principle Δx ·Δp ∼ h allows to definea thermal momentum pT as

pT =h

λT

=�

2πmkBT ,p2T2m

= πkBT =2π

3Ekin, Ekin =

3

2kBT ,

where we have used (3.5) for the average energy Ekin per particle. The thermal momentumpT is hence of the same order of magnitude as the average momentum p̄ of the gas, asdefined by Ekin = p̄2/(2m), but not identical.

Free energy. From (9.18) we obtain (with logN ! ≈ N logN −N)

F (T, V,N) = −kBT ln

�1

N !

�V

λ3T

�N�

= −kBT

�ln

1

N !+ ln

�V

λ3T

�N�

= −kBT

�−N lnN +N +N ln

V

λ3T

�(9.19)

and hence

F (T, V,N) = −NkBT

�ln

�V

Nλ3T

�+ 1

�

for the free energy of the ideal gas.

Entropy. Using

∂λT

∂T=

−λT

2T,

∂

∂Tln

�V

Nλ3T

�= −3

∂ lnλT

∂T=

3

2T

we then have

S = −�∂F

∂T

�

V,N

= NkB

�ln

�V

Nλ3T

�+ 1

�+NkBT

�3

2T

�,

112 CHAPTER 9. CANONICAL ENSEMBLE

which results in the Sackur-Tetrode equation

S = NkB

�ln

V

Nλ3T

+5

2

�. (9.20)

Comparing (9.20) with (8.25), namely with the microcanonical Sackur-Tetrode equation

S = kBN

�ln

��4πmE

3h2N

�3/2V

N

�+

5

2

�,

one finds that they coincide when E/N = 3kBT/2.

Chemical potential. The chemical potential µ is

µ =

�∂F

∂N

�

T,V

= −kBT

�ln

V

Nλ3T

+ 1

�+NkBT

�V

Nλ3T

· λ3T

V

�

= −kBT ln

�V

Nλ3T

�.

The previous expressions were much simpler obtained than when calculated in the micro-canonical ensemble.

Equivalence of ensembles. In the thermodynamic limit the average value of an ob-servable is in general independent of the ensemble (microcanonical or canonical).

N → ∞, V → ∞,N

V= const.

is taken. One therefore usually chooses the ensemble that is easier to work with.

Fluctuations of observables. Fluctuations of observables, �A2� − �A�2, may howeverbe ensemble dependent! An example for an observable for which this is the case is theenergy, which is constant, by definition, in the microcanonical ensemble, but distributedaccording 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 2

�kBT lnZn

�

=∂

∂T

�kB lnZn +

kBT

ZN

∂Zn

∂β

∂β

∂T

�∂β

∂T=

−1

kBT 2

=∂

∂T

�kB lnZn −

1

TZN

∂Zn

∂β

�.

9.6. ENERGY FLUCTUATIONS 113

Second derivatives. The remaining derivative with respect to the temperature T are

∂

∂TkB lnZn =

−1

T 2ZN

∂Zn

∂β

∂

∂T

−1

TZN

∂Zn

∂β=

1

T 2ZN

∂Zn

∂β+

�1

TZ2N

�∂Zn

∂β

�2

− 1

TZN

∂2Zn

∂β2

�−1

kBT 2.

With the first two terms canceling each other We find

CV =1

kbT 2

�1

ZN

∂2Zn

∂β2−�

1

ZN

∂Zn

∂β

�2�

(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 functioncorresponds to

1

ZN

∂Zn

∂β= −

�d3N q d3Nph3NN !

H e−βH(q,p)

�d3N q d3Nph3NN !

e−βH(q,p), ZN =

�d3Nq d3Np

h3NN !e−βH(q,p) ,

viz to1

ZN

∂Zn

∂β= −�E�,

1

ZN

∂2Zn

∂β2=�E2�. (9.22)

Specific heat. Our results (9.21) and (9.22) lead to the fundamental relation

CV =1

kbT 2

��E2�−�E�2�

(9.23)

between the specific heat CV and the fluctuations �E2�−�E�2

of the energy.

– Both the specific heat CV ∼ N and the right-hand side of (9.23) are extensive. Thelater as a result of the central limit theorem discussed in Sect. 8.6, which states thatthe variance of independent processes are additive.

– The specific heat describes the energy exchange between the system and an heatreservoir. It hence makes that sense that CV is proportional to the size of the energyfluctuations.

Relative energy fluctuations. The relative energy fluctuations,��

E2�−�E�2

�E� ∼ 1√

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 bothCV and the internal energy U = �E� are extensive.

– Eq. (9.24) is consistent with the demand that the canonical the microcanonicalensembles are equivalent in the thermodynamic limit N → ∞. Energy fluctuationsare absent in the microcanonical ensemble.

114 CHAPTER 9. CANONICAL ENSEMBLE

9.7 Paramagnetism

We consider a system with N magnetic atoms per unit volume placed in an externalmagnetic field H. Each atom has an intrinsic magnetic moment µ = 2µ0s with spins = 1/2.

Energy states. In a quantum-mechanical description, the magnetic moments of theatoms 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 asmall 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� peratom as a function of the temperature T :

�µH� =µ eβµH − µ e−βµH

eβµH + e−βµH , �µH� = µ tanhµHkBT

,

where we used that

tanh y =ey − e−y

ey + e−y, y = βµH =

µHkBT

.

Magnetization. We define the magnetization, i.e. the mean magnetic moment per unitvolume, 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 + . . .)

2≈ y ,

so that

�µH� =µ2HkBT

.

9.7. PARAMAGNETISM 115

Curie Law. For the magnetic susceptibility χ, defined as �M� = χH, we then have

χ =Nµ2

kBT.

At temperatures high compared to the magnetic energies, χ ∝ T−1 which is known as theCurie law.

Low-temperature expansion. Low tempera-tures correspond to y � 1,

ey � e−y, tanh y ≈ 1 ,

and hence

�µH� = µ, �M� = Nµ .

The magnetization saturates at the maximalvalue at low temperatures independent of H.

116 CHAPTER 9. CANONICAL ENSEMBLE