Chapter 9 Canonical 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 thermal equilibrium 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 probability offinding the small systemA 1 in any particular microstateα of energy Eα? In other words, what is the distribution 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. 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 systemA is isolated, the distribution function in the combined phase space is given by the microcanonical 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 entropy. 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− Boltzmann distribution � 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 number 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 energy 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 temperature 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 energy, for which βE P(E) = P Ω(E)e − , (9.9) α ∝ E<E <E+Δ �α is the corresponding energy distribution function.Ω(E)=Ω 1(E) is, as usual, the density of phase space. –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 normalization, and analogous to the microcanonical space volumeΓ(E) in units ofΓ 0: Γ(E) 1 = d3N q d3N p Γ h3N N! 0 �E<H(q,p)<E+Δ d3N q d3N p = Θ(E+Δ H) Θ(E H) , h3N N! − − − � � � whereΘ is the step function. Free energy. We will show that it is possible to obtain all thermodynamic observables by differentiating the partition functionZ N . We will prove in particular that 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 Helmholtz free energy. 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 internal energy. � � 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 gas, 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 .

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