Grand Canonical Ensemble

We have both energy E and the number of particles N vary!

The same considerations as for canonical ensemble (substitute N for E).

Probability of a state:

$$\Prob(A_{i,N}) = \frac{1}{\Xi}\exp(-E_i/kT + \mu N/kT)$$

Here

$$\Xi = \sum_{i,N} \exp(-E_i/kT+\mu N/kT)$$

is called grand partition function , and

$$\mu = -\frac{1}{T}\left(\frac{\partial S}{\partial N}\right)_{E,V}$$

is called chemical potential . It is an intensive variable (why?)

Another formula:

$$dS = \frac{1}{T}dE - \frac{\mu}{T}dN +\frac{p}{T}\,dV$$

so

$$dE = \mu\,dN + T\,dS -p\,dV,\quad \mu =\left(\frac{\partial E}{\partial N}\right)_{S,V}$$

Yet another formula: since A=E-TS,

$$dA = \mu\,dN - S\,dT -p\,dV, \quad \mu =\left(\frac{\partial A}{\partial N}\right)_{T,V}$$

Averages:

$$\left\langle f\right\rangle = \frac{1}{\Xi}\sum_{i,N} f_{i,N} \exp(-E_i/kT+\mu N/kT)$$

$\Omega$-potential (compare to free energy A!)

$$\Omega = -kT\ln\Xi$$

Average entropy:
$$\begin{multline*} \left\langle S\right\rangle = -k\sum_{E,N} \Prob(E,N)\ln\bigl(... ...gle E\right\rangle}{T} - \frac{\left\langle N\right\rangle\mu}{T}\end{multline*}$$
Or:

$$\Omega = E - TS - N\mu$$

$\Omega$-potential is a Legendre transform of E!  

$$d\Omega = -S\,dT-P\,dV-N\,d\mu$$ (1)

So $\Omega=\Omega(V,T,\mu)$

Consequence:

$\Omega$-potential is just the product of volume and pressure:

$$\Omega = -PV$$

Proof:

In thermodynamic limit

$$T\to\mathit{const},\quad \mu\to\mathit{const},\quad \Omega\sim V\to\infty$$

Therefore

$$\Omega = \omega(T,\mu)V$$

Then

$$\omega = \left(\frac{\partial\Omega}{\partial V}\right)_{T,\mu}$$

and from (1)

$$\omega = -P$$