# The Grand Canonical Ensemble

a) The Grand Canonical Ensemble applies for when the total energy and the number of particles of a system can fluctuate. For a system at thermal equilibrium at temperature T and chemical potential $\mu$ the grand canonical partition function is

$Z_{GC} =\sum_{\alpha} \exp[\mu*N(\alpha)/(KT)-U(\alpha)/(KT)]$

where the summation is over all microstates $\alpha$ and the volume $V$ is fixed.

What is the probability to be in a specific microstate $\alpha$.

b) Consider the probability to be in a microstate with energy $U_1$ and $N_1$ and also the probability to be in a microstate with energy $U_2$ and $N_2$. What is their ratio $R$?

c) Consider the general definition of entropy $S=-K*\sum_{\alpha}P(\alpha)*log(P(\alpha))$. Show that

$P=(KT/V)*log(Z)$

a)

For Grand canonical ensemble the partition function is

$Z=∑_i exp((μN_i-U_i)/kT)$

which means the probability for state j is

$P_j= (exp((μN_j-U_j)/kT))/Z=(exp((μN_j-U_j)/kT))/(∑_i exp((μN_i-U_i)/kT) )$

b)

Defining the grand potential as

$Ω=-kT*lnZ=<U>-TS-μ<N>$

One has

$1/Z=exp(Ω/kT)=exp((<U> -TS-μ<N>)/kT)$

And therefore

$R=P_1/P_2 =\frac{exp(-S(1)/k+(μN_1)/kT-U_1/kT)}{exp(-S(2)/k+(μN_2)/kT-U_2/kT)}$

because $<U>(=E)$ and $<N>$ are average values.

c)

If we start from the entropy definition:

$S=-k*∑_i P_i*ln(P_i )=-k∑_iP_i*(ln 1/Z+(μN_i-U_i)/kT)$

$S=(-k*ln 1/Z ∑_i P_i)-k*(μ<N>-<U>)/kT$

$TS+μ<N>-<U> =+kT*ln(Z)$

Euler theorem is (here p lowercase is pressure)

$E=μN-pV+TS$

Which means

$pV=kT*ln(Z)$