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Home , Boltzmann distribution, Canonical ensemble, Microcanonical ensemble

Chapter 9

Canonical ensemble

9.1 System in contact with a heat reservoir

We consider a small systemA 1 characterized by E1,V 1 andN 1 in thermal interaction with a heat reservoirA 2 characterized byE 2,V 2 andN 1 in thermal interaction such thatA A ,A has 1 � 2 1 hence fewer degrees of freedom thanA 2.

E2 E 1 N1 = const. N �N N = const. 2 � 1 2 with E1 +E 2 =E = const. Both systems are in at tem- peratureT . The wall between them allows interchange of heat but not of particles. The systemA 1 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 reservoirA 2. Distribution of energy The question we want to answer is the following: states “Under equilibrium conditions, what is the offinding the small systemA 1 in any particular microstateα of energy Eα? In other words, what is the distri- bution functionρ=ρ(E α) of the system A1?”

We note that the energyE 1 is notfixed, only the total energyE=E 1 +E2 of the combined system. Hamilton function. The Hamilton function of the combined systemA is

H(q, p) =H 1(q(1), p(1)) +H 2(q(2), p(2)),

103 104 CHAPTER 9.

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 systemA is isolated, the distribution function in the combined is given by the micro- canonical distribution functionρ(q, p), δ(E H(q, p))) ρ(q, p) = − , dq dpδ(E H) =Ω(E), (9.1) dq dpδ(E H(q, p)) − − � whereΩ(E) is the density� of phase space ( 8.4). Tracing outA2. It is not the distribution functionρ(q, p) =ρ(q(1),p(1), q(2), p(2)) of the total systemA that we are interested in, but in the distribution functionρ 1(q(1), p(1)) of the small systemA 1. One hence needs to trace outA 2:∗

ρ (q(1), p(1)) dq(2) dp(2)ρ(q(1), p(1), q(2), p(2)) 1 ≡ � dq(2) dp(2)δ(E H H ) = − 1 − 2 Ω(E) � Ω (E H ) 2 − 1 . (9.2) ≡ Ω(E) whereΩ (E ) =Ω(E H ) is the phase space density ofA . 2 2 − 1 2 SmallE 1 expansion. Now, we make use of the fact thatA 1 is a much smaller system thanA 2 and therefore the energyE 1 given byH 1 is much smaller than the energy of the combined system: E E. 1 � In this case, we can approximate (9.2) by expanding the slowly varying logarithm of Ω (E ) =Ω (E H ) around theE =E as 2 2 2 − 1 2

∂ lnΩ 2 lnΩ 2(E2) = lnΩ 2(E H 1) lnΩ 2(E) H1 +... (9.3) − � − ∂E2 � �E2=E and neglect the higher-order terms sinceH =E E. 1 1 � Derivatives of the . Using (8.14), namely that Γ(E,V,N) Ω(E)Δ S=k ln =k ln , (9.4) B Γ B Γ � 0 � � 0 � whereΔ is the width of the energy shell, wefind that derivatives of the entropy like 1 ∂S ∂ lnΩ(E) = =k (9.5) T ∂E B ∂E

∗ A marginal distribution functionp(x) = p(x, y)dy in generically obtained by tracing out other variables from a joint distribution functionp(x, y). � 9.1. SYSTEM IN CONTACT WITH A HEAT RESERVOIR 105

can be taken with respect to the logarithm of the phase space densityΩ(E).

Boltzmann factor. Using (9.5) for the larger systemA 2 we may rewrite (9.3) as

∂ lnΩ 2(E2) Ω2(E H 1) = exp lnΩ 2(E) H1 +... − − ∂E2 � �E2=E � � H � =Ω (E) exp 1 . � 2 −k T � B 2 �

The temperatureT 2 of the heat reservoirA 2 by whatever small amount of energy the large systemA 2 gives to the small systemA 1. Both systems are thermally coupled, such thatT 1 =T 2 =T . We hencefind with ( 9.2)

Ω2(E) H1 H1 ρ (q(1), p(1)) = e− kB T e − kB T . (9.6) 1 Ω(E) ∝

The factor exp[ H /(k T )] is called the Boltzmann factor. − 1 B Distribution function of the canonical ensemble. The prefactorΩ 2(E)/Ω(E) in (9.6) is independent ofH 1. We may hence obtain the the normalization ofρ 1 alternatively by integrating over the phase space ofA 1:

βH1(q(1),p(1)) e− 1 ρ1(q(1), p(1)) = ,β= . (9.7) βH1(q(1),p(1)) dq(1) dp(1)e − kBT � 9.1.1 Boltzmann factor

The probabilityP α offinding the systemA 1 (which is in thermal equilibrium with the heat reservoirA 2) in a microstateα with energyE α is given by

βEα e− Pα = (9.8) βEα α e− � when rewriting (9.7) in terms ofP α.

– The number of statesΩ 2(E2) =Ω 2(E H 1) accessible to the reservoir is a rapidly increasing function of its energy. −

– The number of statesΩ 2(E2) =Ω 2(E H 1) accessible to the reservoir decreases therefore rapidly with increasingE =−E E . The probability offinding states 1 − 2 with largeE 1 is accordingly also rapidly decreasing.

The exponential dependence ofP α onE α 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 toA 1 andA 2 for various values of their respective energies, as given in thefigure, and that the total energy of the combined system is 1007.

– LetA 1 be in a stateα with en- ergy 6.E 2 is then in one of the 3 10 5 states with energy 1001. ·

– IfA 1 is in a stateγ with energy 7, the reservoir must be in one of the 1 10 5 states with energy 1000. ·

The number of realizations of states withE 1 = 6 the ensemble contains is hence much higher than the number of realization of state withE 1 = 7. Canonical ensemble. An ensemble in contact with a heat reservoir at T is called a canonical ensemble, with the Boltzmann factor exp( βEα) describing the canonical distribution (9.8). − Energy distribution function. The Boltz- mann distribution (9.8) provides the probability Pα tofind an individual microstatesα. There are in general many microstates in a given en- ergy, for which

βE P(E) = P Ω(E)e − , (9.9) α ∝ E

–P(E) is rapidly decreasing for increasing energies due to the Boltzmann factor exp( βE ). − α –P(E) is rapidly decreasing for decreasing energies due to the decreasing phase space densityΩ(E).

The energy density is therefore sharply peaked. We will discuss the the width of the peak, viz the energyfluctuations, 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

βH(q,p) e− ρ(q, p) = 3N 3N βH(q,p) , d q d p e−

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) Z = e− . (9.10) N h3N N! � It is proportional to the canonical distribution functionρ(q, p), but with a different nor- malization, and analogous to the microcanonical space volumeΓ(E) in units ofΓ 0:

Γ(E) 1 = d3N q d3N p Γ h3N N! 0 �E

βF(T,V,N) F(T,V,N)= k T lnZ (T) ,Z =e − , (9.11) − B N N

whereF(T,V,N) is the . Proof. In order to proof (9.11) we perform the differentiation

∂ 1 ∂ZN lnZ N = ∂β ZN ∂β

∂ dqdp βH dqdp βH = e− e− ∂β h3N N! h3N N! � � � �� � βH � dqdp( H)e − = − dqdp e βH � − = H = U. −� �� − where we have used the shortcut dqdp=d 3N qd3N p and that H =E=U is the . � � 108 CHAPTER 9. CANONICAL ENSEMBLE

With (5.13), namely thatU=∂(βF)/∂β, wefind that

∂ ∂ βF lnZ =U= (βF), lnZ = βF, Z =e − , −∂β N ∂β N − N which is what we wanted to prove. Integration constant. Above derivation allows to identify lnZ = βF only up to an N − integration constant (or, equivalently,Z N only up to a multiplicative factor). Setting this constant to zero results in the correct result for the ideal , as we will show lateron in Sect. 9.5.

Thermodynamic properties. Once the partition functionZ N and the free energy F(T,V,N)= k BT lnZ N (T,V,N) are calculated, one obtains the pressureP , the entropy S and the chemical− potentialµ as usual via

∂F ∂F ∂F P= ,S= , µ= . − ∂V − ∂T ∂N � �T,N � �V,N � �T,V

Specific heat. The specific heatC V is given in particular by

2 2 CV ∂S ∂ F ∂ = = 2 = 2 kBT lnZ N , (9.12) T ∂T V − ∂T ∂T � � � � where we have usedF= k T lnZ . − B N 9.3 Canonical vs.

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:

Microcanonical ensemble Canonical ensemble

Phase space Density of states: Partition function: 3N 3N 3N 3N d q d p βH(q,p) integration Ω (E)= d q d pδ(E H) Z (T)= e− N − N h3N N! � Thermodynamic � Ω (E)Δ potential S(E,V,N)=k ln N F(T,V,N)= k T lnZ (T) B h3N N! − B N � �

Laplace transforms. The relation between the density of statesΩ N (E) and the partition functionZ N (T ) can be defined as a Laplace transformation in the following way. We use the definition (9.1) of the density of statesΩ(E),

dq dpδ(E H) =Ω(E),H=H(q, p), − � 9.4. ADDITIVITY OFF(T,V,N) 109

in order to obtain

βE βE ∞ dE e ∞ dE e − Ω (E) = d3N q d3N p − δ(E H) h3N N! N h3N N! − �0 � �0 3N 3N d q d p βH(q,p) = e− = Z (T) . (9.13) h3N N! N �

We have thus shown thatZ N (T ) is the Laplace transform† ofΩ N (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 Hamilton

function 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 easier to calculate in the canonical� ensemble than in the microcanonical ensemble.

9.4 Additivity ofF(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. Neglect- ing 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 +H 2,N=N 1 +N 2 .

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

1 3N 3N β(H1+H2) Z (T,V)= d q d p e− , N h3N N !N ! 1 2 � 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 toN 1!N2! and NOT to N!. It then follows that the partition function factorizes,

1 3N1 3N1 βH1(q1,p1) ZN (T,V) = d q d p e− h3N1 N ! 1 � 1 3N2 3N2 βH2(q2,p2) d q d p e− × h3N2 N ! 2 � =Z N1 (T,V1)Z N2 (T,V2), and that the free energyF= k T lnZ is additive: − B N

F(T,V,N)=F 1(T,V1,N1) +F 2(T,V2,N2).

† The Laplace transformF(s) of a functionf(t) is defined asF(s) = ∞ f(t) exp( st)dt. 0 − � 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 statesΩ(E) of the combined system,

Ω(E) = d3N q d3N pδ(E H H ) − 1 − 2 � = d3N1 q d3N1 p d3N2 q d3N2 p dE δ(E H E )δ(E H ) 2 − 1 − 2 2 − 2 � � = dE Ω (E E )Ω (E ), 2 1 − 2 2 2 � is given by the( convolution) of the density of statesΩ i(Ei) of the individual systems. Using the representation (9.13) for the partition function we obtain‡ dE e βE Z = − Ω(E) N h3N N !N � 1 2 β(E1+E2) dE e− = 3N dE2 Ω1(E E 2) Ω2(E2). h N1!N2! − � � Ω1(E1)

A change of the integration variable from dE to dE�1 then�� leads� again to

ZN (T,V)=Z N1 (T,V1)Z N2 (T,V2). (9.14)

Note that this relation is only valid ifH=H 1 +H 2 andH 12 = 0.

9.5 Ideal gas in the canonical ensemble

We consider now the ideal gas in the canonical ensemble, for which the Hamilton function,

N 2 3N 3N �ip d q d p β N �2p/(2m) H= ,Z (T,V)= e− i=1 i , (9.15) 2m N h3N N! i=1 � � � contains just the kinetic energy.

Factorization. The integral leading toZ N factorizes in (9.15):

3N N + 2 V ∞ dp β p Z (T,V) = e− 2m N N! h ��−∞ � N + 3N V ∞ √2kBT m x2 = e− dx , (9.16) N! h ��−∞ � where we have used the variable substitution 2 + 2 p dp ∞ x2 x = , dx= , dx e− = √π. (9.17) 2kBT m √2kBT m �−∞

‡ Note that a variable transfomation (E,E2) (E 1,E2) with a Jacobian determinant, dE dE2 = dE dE J , whereJ is the respective Jacobian.→ 1 2 | | � � 9.5. IDEAL GAS IN THE CANONICAL ENSEMBLE 111

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

3N V N √2πmk T 1 V N Z (T,V)= B , (9.18) N N! h ≡ N! λ3 � � � T �

where we have defined the thermal wavelengthλ T as

h λT = . √2πmkBT

26 For air (actually nitrogen,N , withm=4.65 10 − kg) atT = 298 K, the thermal wave- 2 · length is 0.19 A◦, which is actually smaller than the Bohr radius. Quantum mechanical effects start to play a role only onceλ T becomes larger than the typical interparticle separation. Thermal momentum. Heisenberg’s uncertainty principleΔx Δp h allows to define · ∼ a thermal momentump T as 2 h pT 2π 3 pT = = 2πmkBT, =πk BT= Ekin,Ekin = kBT, λT 2m 3 2 � where we have used (3.5) for the average energyE kin per particle. The thermal momentum pT is hence of the same order of magnitude as the average momentum ¯p of the gas, as 2 defined byE kin = ¯p/(2m), but not identical. Free energy. From (9.18) we obtain (with logN! N logN N) ≈ − 1 V N F(T,V,N) = k T ln − B N! λ3 � � T � � 1 V N = k T ln + ln − B N! λ3 � � T � � V = k T N lnN+N+N ln (9.19) − B − λ3 � T � and hence V F(T,V,N)= Nk T ln + 1 − B Nλ 3 � � T � � for the free energy of the ideal gas. Entropy. Using ∂λ λ ∂ V ∂ lnλ 3 T = − T , ln = 3 T = ∂T 2T ∂T Nλ 3 − ∂T 2T � T � we then have ∂F V 3 S= =Nk ln + 1 +Nk T , − ∂T B Nλ 3 B 2T � �V,N � � T � � � � 112 CHAPTER 9. CANONICAL ENSEMBLE

which results in the Sackur-Tetrode equation

V 5 S= Nk ln + . (9.20) B Nλ 3 2 � T �

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

4πmE 3/2 V 5 S=k N ln + , B 3h2N N 2 � �� � � �

onefinds that they coincide when E/N=3k BT/2. . The chemical potentialµ is

∂F V V λ3 µ= = k T ln + 1 +Nk T T ∂N − B Nλ 3 B Nλ 3 · V � �T,V � T � � T � V = k T ln . − B Nλ 3 � T �

The previous expressions were much simpler obtained than when calculated in the micro- canonical ensemble. Equivalence of ensembles. In the the average value of an ob- servable is in general independent of the ensemble (microcanonical or canonical).

N N ,V , = const. →∞ →∞ V is taken. One therefore usually chooses the ensemble that is easier to work with. Fluctuations of observables. Fluctuations of observables, A 2 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 Energyfluctuations

We evaluated the representation (9.12) for the specific heat in afirst step:

C ∂2 V = k T lnZ T ∂T 2 B n ∂ � �k T ∂Z ∂β ∂β 1 = k lnZ + B n = − ∂T B n Z ∂β ∂T ∂T k T 2 � N � B ∂ 1 ∂Z = k lnZ n . ∂T B n − TZ ∂β � N � 9.6. ENERGY FLUCTUATIONS 113

Second derivatives. The remaining derivative with respect to the temperatureT are

∂ 1 ∂Zn kB lnZ n = −2 ∂T T ZN ∂β ∂ 1 ∂Z 1 ∂Z 1 ∂Z 2 1 ∂2Z 1 − n = n + n n − . ∂T TZ ∂β T 2Z ∂β TZ2 ∂β − TZ ∂β2 k T 2 N N � N � � N � B With thefirst two terms canceling each other Wefind

1 1 ∂2Z 1 ∂Z 2 C = n n (9.21) V k T 2 Z ∂β2 − Z ∂β b � N � N � �

for the specific heatC v as a functions of derivatives of the partition functionZ N . Derivatives of the partition function. The definition (9.10) for the partition function corresponds to

d3N q d3N p βH(q,p) 3N 3N 1 ∂Zn h3N N! H e− d q d p βH(q,p) = ,Z = e− , d3N q d3N p N 3N ZN ∂β − e βH(q,p) h N! � h3N N! − � viz to � 2 1 ∂Zn 1 ∂ Zn 2 = E , 2 = E . (9.22) ZN ∂β − ZN ∂β Specific heat. Our results (9.21) and� � (9.22) lead to the fundamental� � relation

1 2 2 CV = 2 E E (9.23) kbT − �� � � � � 2 between the specific heatC and thefluctuations E 2 E of the energy. V � − – Both the specific heatC V 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 thatC V is proportional to the size of the energy fluctuations. Relative energyfluctuations. The relative energyfluctuations,

2 E2 E 1 − (9.24) �� �E � � ∼ √N vanish in the thermodynamic limitN . →∞� � – The scaling relation (9.24) if a direct consequence of (9.23) and of the fact that both C and the internal energyU= E are extensive. V � � – Eq. (9.24) is consistent with the demand that the canonical the microcanonical ensembles are equivalent in the thermodynamic limitN . Energyfluctuations are absent in the microcanonical ensemble. →∞ 114 CHAPTER 9. CANONICAL ENSEMBLE 9.7

We consider a system withN magnetic atoms per unit volume placed in an external magneticfield . Each atom has an intrinsic magnetic momentµ=2µ 0s with s=1/2. H Energy states. In a quantum-mechanical description, the magnetic moments of the atoms can point either parallel or anti-parallel to the magneticfield.

state alignment moment energy probability

βε+ +βµ (+) parallel to +µ µ P =ce − =ce H H − H +

βε βµ ( ) anti-parallel to µ +µ P =ce − − =ce − H − H − H − 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 µ per H atom as a function of the temperatureT: � �

βµ βµ µ e H µe − H µ µ = βµ − βµ , µ =µ tanh H , � H� e H +e − H � H� kBT

where we used that

y y e e − µ tanhy= y − y , y=βµ = H . e +e − H kBT

Magnetization. We define the magnetization, i.e. the mean magnetic moment per unit volume, as M =N µ � � � H� and analyze its behavior in the limit of high- and of low . High-temperature expansion. Large temperatures correspond toy 1 and hence to � y y e = 1 +y+..., e − = 1 y+.... − Then, (1 +y+...) (1 y+...) tanhy= − − y, 2 ≈ so that µ2 µ = H . � H� kBT 9.7. PARAMAGNETISM 115

Curie Law. For the magnetic susceptibilityχ, defined as M =χ , we then have � � H Nµ2 χ= . kBT

1 At temperatures high compared to the magnetic energies,χ T − which is known as the Curie law. ∝ Low-temperature expansion. Low tempera- tures correspond toy 1, � y y e e − , tanhy 1, � ≈ and hence

µ = µ, M =Nµ. � H� � � The magnetization saturates at the maximal value at low temperatures independent of . H 116 CHAPTER 9. CANONICAL ENSEMBLE