268 Grand canonical ensemble
A ̃depends on a single extensive variable,V, it is a homogeneous function of degree 1
inV, i.e.A ̃(μ,λV,T) =λA ̃(μ,V,T). From Euler’s theorem,
A ̃=V∂
A ̃
∂V
, (6.3.5)
which, according to eqn. (6.3.4), becomes
A ̃=−PV. (6.3.6)Thus,−PVis the natural free energy of the grand canonical ensemble. Unlikeother
ensembles,A ̃=−PV is not given a unique symbol. Rather, because it leads directly
to the equation of state, the free energy is simply denoted−PV.
6.4 Grand canonical phase space and the partition function
Since the grand canonical ensemble usesμ,V, andTas its control variables, it is
convenient to think of this ensemble as a canonical ensemble coupledto aparticle
reservoir, which drives the fluctuations in the particle number. As the name implies,
a particle reservoir is a system that can gain or lose particles withoutappreciably
changing its own particle number. Thus, we imagine two systems coupled to a common
thermal reservoir at temperatureT, such that system 1 hasN 1 particles and volume
V 1 and system 2 hasN 2 particles and a volumeV 2. The two systems can exchange
particles, with system 2 acting as a particle reservoir (see Fig. 6.1).Hence,N 2 ≫N 1.
The total particle number and volume are
N , V , E 2 2 2
H 2 ( x 2 )N , V , E 1 1 1
H 1 ( x 1 )Fig. 6.1Two systems in contact with a common thermal reservoir at temperatureT. System
1 hasN 1 particles in a volumeV 1 ; system 2 hasN 2 particles in a volumeV 2. The dashed
lines indicate that systems 1 and 2 can exchange particles.