Lecture 9: Grand Canonical Ensemble This Describes a System in Contact with a Reservoir with Which It Can Exchange Energy and Particles

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Physics 127a: Class Notes

Lecture 9: Grand Canonical Ensemble This describes a system in contact with a reservoir with which it can exchange energy and particles. The equilibrium is characterized by the reservoir temperature T and chemical potential µ. As for the canonical ensemble, we can derive properties of the grand canonical ensemble by treating system plus reservoir as an isolated system together described by the microcanonical ensemble. The results are:

(N) • Probability for the system to have N particles and be in state j with energy Ej is

−β(E(N)−µN) Pj,N ∝ e j . (1)

(remember this comes from counting the number of states in the reservoir)

• Deﬁne the grand canonical partition function X X −β(E(N)−µN) Q = e j , (2) N j

and the corresponding grand potential

=−kT ln Q. (3)

• For a macroscopic system we can replace the sums by integrals Z Z dE Q ' dN e−β(E−µN−T S(E,N) 1 (4)

where eS/k is the number of states in the energy shell 1. For large N the integrand is dominated by E ' E¯ and N ' N¯ which minimize E − µN − TSgiven by differentiation ∂S µ = T , ∂N (5) E ∂S 1 = T . (6) ∂E N

These two equations implicitly give E,¯ N¯ , and tell us the physical result that the temperature and chemical potential of the system are given by the results of the isolated system at the most probable E and N. Furthermore in evaluating ln Q only the value of the integrand at its maximum contributes at O(N) ' U − TS− µN (7) (strictly we should use the most probable values E,¯ N¯ , but can replace them by the means U,N since the distribution is so narrow).

• Differentiating and using dU = TdS− PdV + µdN gives the thermodynamic identity in the form

d =−SdT − Ndµ− PdV. (8)

• Making a system at T,µ,V by adding little volumes dV at the same T,P gives us (since then dT = dµ = 0) =−PV. (9)

1 Grand Potential for the Classical Ideal Gas We can write X N Q = z QN (10) N βµ with z = e known as the fugacity and QN the canonical partition function for N particles. Also, for the classical ideal gas N (Q1) QN = , (11) N! so that X N (zQ1) zQ Q = = e 1 . N (12) N ! Thus PV − = = Q = zQ kT kT ln 1. (13) For a monatomic gas we found / 2πmkT 3 2 Q = V . (14) 1 h2 This is the ideal gas law—but in terms of µ rather than N. Normally we would eliminate µ in terms of the mean N using ∂ N =− . (15) ∂µ T,V From Eq. (13) we see that the only µ dependence of is in the fugacity z = eβµ, and differentiating gives

N = zQ1 (16) which with Eq. (14) gives the same expression for µ(N, T, V ) as we found before, and with Eq. (13)gives the ideal gas law in the familiar form.

Number ﬂuctuations In the grand canonical ensemble, the probability of ﬁnding N particles in the system is

P −β(E(N)−µN) e j zN Q P(N)= j = N . P P (N) (17) −β(Ej −µN) Q N j e

For the ideal gas N (Q1) QN = (18) N! and N N −zQ P(N)= z Q e 1 . (19) 1 P As we have seen hNi = zQ1 (this can of course be derived as N NP (N) from equation (19)), so that for the ideal gas the number ﬂuctuations are described by the Poisson distribution hNiN e−hNi P(N)= . (20) N!