Theory, worked examples and 1 previous-year questions with full solutions — IIT-JAM, GATE, JEST and CSIR NET-JRF, 2020 onwards. Click Show solution under any question.
Theory
9.1 Why a grand ensemble?
In the canonical ensemble the number of particles \(N\) is fixed. Many physical systems do not fix \(N\). A gas in contact with its liquid exchanges molecules by evaporation and condensation; molecules adsorbed on a surface are continually exchanged with the vapour above it; photons in a cavity are created and absorbed by the walls; and any small open region inside a large fluid gains and loses particles all the time. For such systems it is natural to let the system exchange particles as well as energy with a large reservoir. The reservoir then fixes the temperature \(T\) and the chemical potential \(\mu\), and the resulting ensemble is the grand canonical ensemble.
There is also a practical reason for introducing it. For identical quantum particles the constraint \(\sum_in_i=N\) makes the canonical sum awkward to perform. Once the constraint is removed, the sum factorises into independent sums over single levels (§9.7 and Chapter 10). This is the principal use of the grand ensemble in examination problems.
Throughout this chapter \(Z\) denotes the canonical partition function and \(Z_\mu\) the grand partition function. All the thermodynamics will be obtained from a single potential,
\[F'\equiv F-\mu N ,\]
which plays in the grand ensemble exactly the role that the Helmholtz free energy \(F\) plays in the canonical ensemble.
9.2 Derivation of the probability distribution
Let the system \(A\) be in contact with a reservoir \(A'\) much larger than itself (Figure 9.1). The combined system \(A^{(0)}=A+A'\) is isolated, so that
where \((r,s)\) labels a microstate of \(A\) containing \(N_r\) particles and having energy \(E_s\).
Figure 9.1 A statistical system \(A\) immersed in a particle–energy reservoir \(A'\).
Counting reservoir states. By the postulate of equal a priori probabilities, the probability that \(A\) is in the particular state \((r,s)\) is proportional to the number of states then available to the reservoir, \(P_{r,s}\propto\Omega'(E^{(0)}-E_s,\;N^{(0)}-N_r)\). Since \(E_s\ll E^{(0)}\) and \(N_r\ll N^{(0)}\), we expand the logarithm of \(\Omega'\) to first order:
