Grand Canonical Ensemble & Chemical Potential
Grand Canonical Ensemble & Chemical Potential
When both energy and particles can exchange with a reservoir, the grand canonical ensemble governs the system. The chemical potential \(\mu\) controls particle number just as temperature controls energy.
Definition
Grand partition function: \(\mathcal{Z} = \sum_{N,i} e^{-\beta(E_i - \mu N)}\). Grand potential: \(\Omega = -k_BT\ln\mathcal{Z}\). Average particle number: \(\langle N\rangle = -\partial\Omega/\partial\mu\).
Key Result
Occupation numbers: Fermi-Dirac \(\langle n\rangle = 1/(e^{\beta(\epsilon-\mu)}+1)\); Bose-Einstein \(\langle n\rangle = 1/(e^{\beta(\epsilon-\mu)}-1)\). Both reduce to Boltzmann \(e^{-\beta(\epsilon-\mu)}\) when \(\epsilon-\mu\gg k_BT\).
Example 1
At \(T=0\), the Fermi-Dirac distribution is a step function: all states below \(\mu = E_F\) (Fermi energy) are filled, all above are empty. The Fermi energy of copper: \(E_F \approx 7\) eV.
Example 2
For photons (\(\mu=0\)), Planck's radiation law emerges: \(u(\omega) = (\hbar\omega^3/\pi^2c^3)/(e^{\beta\hbar\omega}-1)\), explaining blackbody radiation.
Loading grand-canonical-viz...
Practice
- Why must photons have zero chemical potential?
- Derive the Sommerfeld expansion for a Fermi gas at low temperature.
- What is Bose-Einstein condensation, and at what temperature does it occur for rubidium-87?
- Explain the physical meaning of the chemical potential.
Show Answer Key
1. Photons are bosons in thermal equilibrium with the cavity walls. Their number is not conserved (absorbed/emitted freely). The grand potential $\Omega = -k_BT\ln\mathcal{Z}$ is minimized at $\mu = 0$: the constraint $\sum N = \text{fixed}$ does not apply. Equivalently, $\partial F/\partial N = \mu = 0$ because adding a photon costs zero free energy.
2. For a Fermi gas at low $T$: $\langle E \rangle \approx \frac{3}{5}N\epsilon_F[1 + \frac{5\pi^2}{12}(k_BT/\epsilon_F)^2 + \ldots]$. More generally, for any smooth $\phi(\epsilon)$: $\int_0^\infty \phi(\epsilon)f(\epsilon)d\epsilon \approx \int_0^{\mu}\phi(\epsilon)d\epsilon + \frac{\pi^2}{6}(k_BT)^2\phi'(\mu)+\ldots$ The key step is expanding around $\mu$ using $f(\epsilon) = [e^{(\epsilon-\mu)/k_BT}+1]^{-1}$.
3. BEC occurs when $T < T_c = \frac{2\pi\hbar^2}{mk_B}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}$. A macroscopic fraction of bosons occupies the ground state. For $^{87}$Rb ($m = 1.44\times10^{-25}$ kg, $n \sim 10^{19}$ m$^{-3}$): $T_c \sim 100$ nK (achieved experimentally in 1995 by Cornell, Wieman, and Ketterle).
4. Chemical potential $\mu = (\partial F/\partial N)_{T,V}$: the free energy cost of adding one particle. In the grand canonical ensemble, $\mu$ controls the average particle number. For fermions: $\mu \approx \epsilon_F$ at $T=0$ (Fermi energy). For bosons near BEC: $\mu \to 0^-$. For classical ideal gas: $\mu = k_BT\ln(n\lambda^3) < 0$.