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IV.G Examples

The two examples of sections (IV.C) and (IV.D) are now reexamined in the . 1. Two level systems: The N impurities are described by a macro-state M (T,N). ≡ Subject to the Hamiltonian = ǫ N n , the canonical of the micro-states H i=1 i µ n , are given by ≡ { i} � 1 N p ( n ) = exp βǫ n . (IV.71) { i} Z − i � i=1 � � From the partition function,

N 1 1 Z(T,N) = exp βǫ n = e− βǫn1 e− βǫnN − i · · · � i=1 � �n1=0 � �n =0 � (IV.72) {�ni} � � �N N = 1 + e−βǫ ,

we obtain the free� �

F (T,N) = k T ln Z = Nk T ln 1 + e− ǫ/(kB T ) . (IV.73) − B − B � � The is now given by

−ǫ/(kB T ) ∂F −ǫ/(kB T ) ǫ e S = = NkB ln 1 + e +NkBT . (IV.74) − ∂T k T 2 1 + e−ǫ/(kB T ) �N � B � � � � � F/T � − � �� � The , Nǫ E = F + T S = , (IV.75) 1 + eǫ/(kB T ) can also be obtained from ∂ ln Z Nǫe−βǫ E = = . (IV.76) − ∂β 1 + e−βǫ Since the joint 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

e−βǫn p(n) = . (IV.77) 1 + e−βǫ This result coincides with eqs.(IV.25), obtained through a more elaborate analysis in the . As expected, in the large N limit, the canonical and microcanon­ ical ensembles describe exactly the same , both at the macroscopic and microscopic levels.

88 2. The Ideal : For the canonical macro-state M (T,V,N), the joint PDF for the ≡ micro-states µ p� , �q , is ≡ { i i}

1 N p 2 1 for �qi box p( p� , �q ) = exp β i { } ∈ . (IV.78) { i i} Z − 2m · � i=1 � � � 0 otherwise Including the modifications to the phase of in eq.(IV.51), the dimensionless partition function is computed as

1 N d3�q d3p� N p 2 Z(T,V,N) = i i exp β i N! h3 − 2m � i=1 � i=1 � � � (IV.79) V N 2πmk T 3N/2 1 V N = B = , N! h2 N! λ(T )3 � � � � where h λ(T ) = , (IV.80) √2πmkBT 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 . The free energy is given by

3N 2πmk T F = k T ln Z = Nk T ln V + Nk T ln N Nk T k T ln B − B − B B − B − 2 B h2 � � V e 3 2πmk T = Nk T ln + ln B . − B N 2 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

∂F V e 3 2πmk T 3 F E S = = Nk ln + ln B Nk T = − , (IV.82) − ∂T − B N 2 h2 − B 2T T �V,N � � �� � � we obtain the� internal energy E = 3NkBT/2. The is obtained from ∂F Nk T P = = B , = P V = Nk T, (IV.83) − ∂V V ⇒ B �T,N � � and the is � given by ∂F F E T S + P V µ = = + k T = − = k T ln nλ3 . (IV.84) ∂N N B N B �T,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 and . The macrostates M (T, J), are specified ≡ in terms of the external and acting on the system; the thermodynamic coordinates x appear as additional random variables. The system is maintained at constant 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 J x. H − · 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(µ , x) = exp [ β (µ ) + βJ x] / (T,N, J) , (IV.85) S − H S · Z

with the Gibbs partition function,

J x (N,T, J) = eβ · −β (µS ) . (IV.86) Z H 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 , which is discussed next.) In this ensemble, the expectation value of the coordinates is obtained from

∂ln x = k T Z , (IV.87) � � B ∂J

which together with the thermodynamic identity x = ∂G/∂J, suggests the identification −

G(N,T, J) = k T ln , (IV.88) − B Z

where G = E T S x J is the . (The same conclusion can be reached − − · by equating in eq.(IV.86) to the term that maximizes the probability with respect to Z x.) The H E x J is easily obtained in this ensemble from ≡ − · ∂ln Z = x J = H. (IV.89) − ∂β �H − · �

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� , �q , with a V occurs with the probability ≡ { i i}

1 N p2 1 for q�i box of volume V p( p� , �q , V ) = exp β i βP V { } ∈ . (IV.90) { i i} − 2m − · � i=1 � � Z � 0 otherwise The normalization factor is now

∞ 1 N d3�q d3p� N p2 (N,T,P ) = dV e− βP V i i exp β i Z N! h3 − 2m 0 i=1 � i=1 � � � � � (IV.91) ∞ 1 1 = dV V N e− βP V = . N!λ(T )3N 3 N �0 (βP λ(T ) ) The Gibbs free energy is given by

5 3 h2 G = k T ln = Nk T ln P ln (k T ) + ln . (IV.92) − B Z B − 2 B 2 2πm � � �� Starting from dG = SdT + V dP + µdN, the volume of the gas is obtained as −

∂G Nk T V = = B , = P V = Nk T. (IV.93) ∂P P ⇒ B �T,N � � The enthalpy H = E + PV � is easily calculated from � � ∂ln 5 H =Z = Nk T, − ∂β 2 B

from which we get CP = dH/dT = 5/2NkB. 2. Spins in an external magnetic field B� , provide a common example for usage of the Gibbs canonical ensemble. Adding the work done against the magnetic field to the internal Hamiltonian , results in the Gibbs partition function H

(N,T,B) = tr exp β + βB� M� , Z − H · � � �� where M� is the net magnetization. The symbol tr is used to indicate the sum over all 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 σ = 1 . The corresponding magnetization along the field direction is given { i ± } 91 N by M = µ0 i=1 σi, where µ0 is a microscopic magnetic . Assuming that there are no interactions between spins ( = 0), the probability of a microstate is � H

N 1 p ( σ ) = exp βBµ σ . (IV.94) { i} 0 i � i=1 � Z � 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,T,B) = [2 cosh(βµ B)]N , (IV.95) Z 0

and the Gibbs free energy

G = k T ln = Nk T ln [2 cosh(βµ B)] . (IV.96) − B Z − B 0

The average magnetization is given by

∂G M = = Nµ tanh(βµ B). (IV.97) −∂B 0 0

Expanding eq.(IV.97) for small B results in the well-known Curie law for magnetic sus­ ceptibility of non-interacting spins,

∂M Nµ2 χ(T ) = = 0 . (IV.98) ∂B k T �B=0 B � � The enthalpy is simply H = BM = �BM, and C = B∂M/∂T . �H − � − B −

IV.I The Grand Canonical Ensemble

The previous sections demonstrate that while the canonical and microcanonical en­ sembles are completely equivalent in the , 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(µ ) = exp [βµN(µ ) β (µ )] / . (IV.99) S S − H S Q The normalization factor is the grand partition function,

(T, µ, x) = eβµN(µS )−β (µS ) . (IV.100) Q H µS � We can reorganize the above summation by grouping together all micro-states with a given number of particles, i.e.

∞ (T, µ, x) = eβµN e− β N (µS) . (IV.101) Q H N�=0 ( µ�S |N) The restricted sums in eq.(IV.101) are just the N-particle partition functions. As each term in is the total weight of all micro-states of N particles, the unconditional probability of Q finding N particles in the system is

eβµN Z(T,N, x) p(N) = . (IV.102) (T, µ, x) Q The average number of particles in the system is 1 ∂ ∂ N = = ln , (IV.103) � � ∂(βµ)Q ∂(βµ) Q Q while the number fluctuations are related to the variance

1 ∂2 ∂ 2 ∂2 ∂ N N 2 = N 2 N 2 = ln ln = ln = � � . � �C � � − � � ∂(βµ)2 Q − ∂(βµ) Q ∂(βµ)2 Q ∂(βµ) � � Q (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. ≈ ∞ ∗ ∗ (T, µ, x) = lim eβµN Z(T,N, x) = eβµN Z(T, N ∗ , x) = eβµN −βF Q N→∞ N=0 (IV.105) � ∗ = e− β(−µN +E−TS) = e−βG ,

93 where (T, µ, x) = E T S µN = k T ln , (IV.106) G − − − B Q is the . Thermodynamic is obtained by using d = SdT G − − Ndµ + J dx, as · ∂ ∂ ∂ S = G , N = G , J = G . (IV.107) − ∂T − ∂µ i ∂x �µ,x �T,x i �T,µ � � � � � � 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� , �q , �p , �q , have indefinite particle number. The grand { 1 1 2 2 · · ·} partition function is given by

∞ 1 N d3�q d3p� p 2 (T, µ, V ) = e βµN i i exp β i Q N! h3 − 2m N=0 � �i=1 � � i � � � � ∞ βµN N e V h (IV.108) = 3 (with λ = ) N! λ √2πmkBT N�=0 � � V = exp e βµ , λ3 � � and the grand potential is

V (T, µ, V ) = k T ln = k T eβµ . (IV.109) G − B Q − B λ3 But, since = E T S µN = PV , the gas can be obtained directly as G − − − ∂ e βµ P = G = G = k T . (IV.110) −V − ∂V B λ3 �µ,T � � The particle number and the chemical potential� are related by

∂ e βµV N = G = . (IV.111) − ∂µ λ3 �T,V � � The equation of state is obtained by comparing� eqs.(IV.110) and (IV.111), as P =

kBT N/V . Finally, the chemical potential is given by

N P λ3 µ = k T ln λ3 = k T ln . (IV.112) B V B k T � � � B �

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8.333 Statistical I: of Particles Fall 2013

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