Examples, the Gibbs Canonical Ensemble, the Grand Canonical Ensemble

IV.G Examples The two examples of sections (IV.C) and (IV.D) are now reexamined in the canonical ensemble. 1. Two level systems: The N impurities are described by a macro-state M ≡ (T, N ). �N Subject to the Hamiltonian H = ǫ i=1 ni , the canonical probabilities of the micro-states µ ≡ {ni }, are given by � � N � 1 ni . p ({ni }) = exp −βǫ Z i=1 (IV.71) From the partition function, � � � 1 � � 1 � N � � � � Z(T, N ) = exp −βǫ ni = e−βǫn1 · · · e−βǫnN n1 =0 i=1 {ni } nN =0 (IV.72) � �N = 1 + e−βǫ , we obtain the free energy � � F (T, N ) = −kB T ln Z = −N kB T ln 1 + e−ǫ/(kB T ) . The entropy is now given by � � � � � ∂F �� e−ǫ/(kB T ) ǫ −ǫ/(kB T ) S=− . T = N k ln 1 + e +N k B B ∂T �N � kB T 2 1 + e−ǫ/(kB T ) �� � −F/T The internal energy, E = F + TS = can also be obtained from E=− Nǫ , 1 + eǫ/(kB T ) ∂ ln Z N ǫe−βǫ = . ∂β 1 + e−βǫ (IV.73) (IV.74) (IV.75) (IV.76) Since the joint probability in eq.(IV.71) is in the form of a product, the excitations of different impurities are independent of each other, with the unconditional distribution p(n) = e−βǫn . 1 + e−βǫ (IV.77) This result coincides with eqs.(IV.25), obtained through a more elaborate analysis in the microcanonical ensemble. As expected, in the large N limit, the canonical and microcanon­ ical ensembles describe exactly the same physics, both at the macroscopic and microscopic levels. 88 2. The Ideal Gas: For the canonical macro-state M ≡ (T, V, N ), the joint PDF for the micro-states µ ≡ {p�i , �qi }, is � � � N 1 for {�qi } ∈ box 2 � 1 pi p({p�i , �qi }) = exp −β · . Z 2m i=1 0 otherwise (IV.78) Including the modifications to the phase space of identical particles in eq.(IV.51), the dimensionless partition function is computed as � � N N � 1 � d3 �qi d3 p�i pi2 Z(T, V, N ) = exp −β 2m N ! i=1 h3 i=1 � �3N/2 �N � V N 2πmkB T 1 V = = , N! h2 N ! λ(T )3 � where λ(T ) = √ (IV.79) h , 2πmkB T (IV.80) is a characteristic length associated with the action h. It shall be demonstrated later on that this length scale controls the onset of quantum mechanical effects in an ideal gas. The free energy is given by 3N kB T ln F = −kB T ln Z = −N kB T ln V + N kB T ln N − N kB T − 2 �� � � � � Ve 3 2πmkB T = −N kB T ln . + ln N 2 h2 � 2πmkB T h2 � (IV.81) Various thermodynamic properties of the ideal gas can now be obtained from dF = −SdT − P dV + µdN . For example, from the entropy � � �� � 3 F −E ∂F �� Ve 3 2πmkB T −S = = −N kB ln + ln = , − N kB T � 2 ∂T V,N h N 2 2T T (IV.82) and the chemical potential is given by � � � E − TS + PV ∂F �� F = kB T ln nλ3 . µ= = + kB T = � ∂N T,V N N (IV.84) we obtain the internal energy E = 3N kB T /2. The equation of state is obtained from � ∂F �� N kB T = , =⇒ P V = N kB T, (IV.83) P =− ∂V �T,N V Also, according to eq.(IV.78), the momenta of the N particles are taken from independent Maxwell–Boltzmann distributions, consistent with eq.(IV.39). 89 IV.H The Gibbs Canonical Ensemble We can also define a generalized canonical ensemble in which the internal energy changes by the addition of both heat and work. The macrostates M ≡ (T, J), are specified in terms of the external temperature and forces acting on the system; the thermodynamic coordinates x appear as additional random variables. The system is maintained at constant force through external elements (e.g. pistons or magnets). Including the work done against the forces, the energy of the combined system that includes these elements is H − J · x. Note that while the work done on the system is +J · x, the energy change associated with the external elements with coordinates x has the opposite sign. The microstates of this combined system occur with the (canonical) probabilities p(µS , x) = exp [−βH(µS ) + βJ · x] /Z(T, N, J) , (IV.85) with the Gibbs partition function, Z(N, T, J) = � eβJ·x−β H(µS ) . (IV.86) µS ,x (Note that we have explicitly included the particle number N to indicate that there is no chemical work. Chemical work is considered in the Grand Canonical Ensemble, which is discussed next.) In this ensemble, the expectation value of the coordinates is obtained from �x� = kB T ∂ln Z , ∂J (IV.87) which together with the thermodynamic identity x = −∂G/∂J, suggests the identification G(N, T, J) = −kB T ln Z, (IV.88) where G = E − T S − x · J is the Gibbs free energy. (The same conclusion can be reached by equating Z in eq.(IV.86) to the term that maximizes the probability with respect to x.) The enthalpy H ≡ E − x · J is easily obtained in this ensemble from − ∂ln Z = �H − x · J� = H. ∂β (IV.89) Note that heat capacities at constant force (which include work done against the external forces), are obtained from the enthalpy as CJ = ∂H/∂T . 90 The following examples illustrate the use of the Gibbs canonical ensemble: 1. The Ideal Gas in the isobaric ensemble is described by the macrostate M ≡ (N, T, P ). A micro-state µ ≡ {p�i , �qi }, with a volume V occurs with the probability �� � N 1 for {q�i } ∈ box of volume V � 1 p2i p({p�i , �qi }, V ) = exp −β − βP V · 2m Z i=1 0 otherwise . (IV.90) The normalization factor is now Z(N, T, P ) = = � ∞ dV e−βP V 0 � ∞ � � N N � 1 � d3 �qi d3 p�i p2i exp −β 2m N ! i=1 h3 i=1 � N −βP V dV V e 0 1 1 = . N 3N N !λ(T ) (βP λ(T )3 ) The Gibbs free energy is given by � � 2 �� 5 3 h G = −kB T ln Z = N kB T ln P − ln (kB T ) + ln . 2 2 2πm Starting from dG = −SdT + V dP + µdN , the volume of the gas is obtained as � N kB T ∂G �� V = = , =⇒ P V = N kB T. � P ∂P T,N (IV.91) (IV.92) (IV.93) The enthalpy H = �E + P V � is easily calculated from H =− ∂ln Z 5 = N kB T, ∂β 2 from which we get CP = dH/dT = 5/2N kB . � , provide a common example for usage of the 2. Spins in an external magnetic field B Gibbs canonical ensemble. Adding the work done against the magnetic field to the internal Hamiltonian H, results in the Gibbs partition function � � �� � ·M � Z(N, T, B) = tr exp −βH + βB , � is the net magnetization. The symbol tr is used to indicate the sum over all where M spin degrees of freedom, which in a quantum mechanical formulation are restricted to discrete values. The simplest case is spin of 1/2, with two possible projections of the spin along the magnetic field. A microstate of N spins is now described by the set of Ising variables {σi = ±1}. The corresponding magnetization along the field direction is given 91 by M = µ0 �N i=1 σi , where µ0 is a microscopic magnetic moment. Assuming that there are no interactions between spins (H = 0), the probability of a microstate is � � N � 1 p ({σi }) = exp βBµ0 σi . Z i=1 (IV.94) Clearly, this is closely related to the example of two level systems discussed in the canonical ensemble, and we can easily obtain the Gibbs partition function N Z(N, T, B) = [2 cosh(βµ0 B)] , (IV.95) G = −kB T ln Z = −N kB T ln [2 cosh(βµ0 B)] . (IV.96) and the Gibbs free energy The average magnetization is given by M =− ∂G = N µ0 tanh(βµ0 B). ∂B (IV.97) Expanding eq.(IV.97) for small B results in the well-known Curie law for magnetic sus­ ceptibility of non-interacting spins, � N µ20 ∂M �� χ(T ) = = . ∂B �B=0 kB T (IV.98) The enthalpy is simply H = �H − BM � = −BM , and CB = −B∂M/∂T . IV.I The Grand Canonical Ensemble The previous sections demonstrate that while the canonical and microcanonical en­ sembles are completely equivalent in the thermodynamic limit, it is frequently much easier to perform statistical mechanical computations in the canonical framework. Sometimes it is more convenient to allow chemical work (by fixing the chemical potential µ, rather than at a fixed number of particles), but no mechanical work. The resulting macro-states M ≡ (T, µ, x), are governed by the grand canonical ensemble. The corresponding micro­ states µS , contain an indefinite number of particles N (µS ). As in the case of the canonical ensemble, the system S, can be maintained at a constant chemical potential through contact 92 with a reservoir R, at temperature T and chemical potential µ. The probability distribu­ tion for the micro-states of S is obtained by summing over all states of the reservoir, as in eq.(IV.53), and is given by p(µS ) = exp [βµN (µS ) − βH(µS )] /Q . (IV.99) The normalization factor is the grand partition function, Q(T, µ, x) = � eβµN(µS )−β H(µS ) . (IV.100) µS We can reorganize the above summation by grouping together all micro-states with a given number of particles, i.e. Q(T, µ, x) = ∞ � eβµN N=0 � e−β HN (µS ) . (IV.101) ( µS |N) The restricted sums in eq.(IV.101) are just the N -particle partition functions. As each term in Q is the total weight of all micro-states of N particles, the unconditional probability of finding N particles in the system is p(N ) = eβµN Z(T, N, x) . Q(T, µ, x) (IV.102) The average number of particles in the system is �N � = 1 ∂ ∂ Q= ln Q, Q ∂(βµ) ∂(βµ) (IV.103) while the number fluctuations are related to the variance � �2 1 ∂2 ∂2 ∂ �N � ∂ 2 2 2 ln Q − ln Q = . �N �C = �N � − �N � = ln Q = 2 2 Q ∂(βµ) ∂(βµ) ∂(βµ) ∂(βµ) (IV.104) The variance is thus proportional to N , and the relative number fluctuations vanish in the thermodynamic limit, establishing the equivalence of this ensemble to the previous ones. Because of the sharpness of the distribution for N , the sum in eq.(IV.101) can be approximated by its largest term at N = N ∗ ≈< N >, i.e. Q(T, µ, x) = lim N→∞ ∞ � N=0 −β(−µN ∗ +E−T S) =e ∗ eβµN Z(T, N, x) = eβµN Z(T, N ∗ , x) = eβµN = e−βG , 93 ∗ −βF (IV.105) where G(T, µ, x) = E − T S − µN = −kB T ln Q, (IV.106) is the grand potential. Thermodynamic information is obtained by using dG = −SdT − N dµ + J · dx, as � ∂G �� , −S = ∂T �µ,x � ∂G �� N =− , ∂µ �T,x � ∂G �� Ji = . ∂xi �T,µ (IV.107) As a final example, we compute the properties of the ideal gas of non-interacting particles in the grand canonical ensemble. The macro-state is M ≡ (T, µ, V ), and the corresponding micro-states {p�1 , �q1 , �p2 , �q2 , · · ·} have indefinite particle number. The grand partition function is given by � � � � �� N � p2 1 d3 �qi d3 p�i i Q(T, µ, V ) = e exp −β N! h3 2m i=1 i N=0 � �N ∞ � h eβµN V ) (with λ = √ = 3 N! λ 2πmkB T N=0 � � βµ V = exp e , λ3 ∞ � βµN (IV.108) and the grand potential is G(T, µ, V ) = −kB T ln Q = −kB T eβµ V . λ3 (IV.109) But, since G = E − T S − µN = −P V , the gas pressure can be obtained directly as � eβµ G ∂G �� P =− =− k = T . (IV.110) B ∂V �µ,T V λ3 The particle number and the chemical potential are related by � eβµ V ∂G �� = . N =− λ3 ∂µ �T,V (IV.111) The equation of state is obtained by comparing eqs.(IV.110) and (IV.111), as P = kB T N/V . Finally, the chemical potential is given by � 3N µ = kB T ln λ V � = kB T ln 94 � P λ3 kB T � . (IV.112)