Statistics of the various ensembles

1. What is this chapter doing?

It's explaining how to describe thermal equilibrium of quantum systems using density matrices for three main ensembles:

Microcanonical – isolated system (E, V, N fixed)
Canonical – can exchange energy with a heat bath (T fixed, E fluctuates, V, N fixed)
Grand canonical – can exchange both energy and particles (T, μ fixed, E and N fluctuate)

In each case we ask: what density matrix ρ̂ describes the system in equilibrium?

2. Microcanonical ensemble (isolated system)

Physical situation

Number of particles N is fixed.
Volume V is fixed.
Energy is fixed within a narrow window: \(E - \tfrac{\Delta}{2} < E < E + \tfrac{\Delta}{2}\).
The system is isolated: no energy or particles go in or out.

Let Γ be the number of quantum states whose energies lie in that small window. These are the accessible microstates.

Key postulate: equal a priori probabilities

Every accessible microstate is equally likely.

So the probability of being in any one of those Γ states is:

\[ \rho_n = \begin{cases} 1/\Gamma & \text{if state } n \text{ is accessible} \\ 0 & \text{otherwise.} \end{cases} \]

In the energy basis, this means the density matrix is diagonal:

\[ \rho_{mn} = \rho_n \delta_{mn}. \]

Entropy is then

\[ S = k \ln \Gamma. \]

That's the familiar statistical–mechanics formula: entropy is proportional to the log of the number of accessible states.

2.1 Pure vs mixed case

If Γ = 1: only one accessible state.
That's a pure state: the system is definitely in that one state.
The density matrix of a pure state satisfies \(\rho^2 = \rho\).
If Γ > 1: there is a mixture of states, each with probability 1/Γ.
This is a mixed state.

They also check that if you change basis (use a different set of states instead of the energy eigenstates), the "pure-state" density matrix still satisfies \(\rho^2 = \rho\).

2.2 Why is the microcanonical ρ̂ proportional to the identity?

In energy basis, within the allowed energy window, all diagonal elements are equal (1/Γ) and off-diagonal ones are zero.

The author then argues that, due to:

equal a priori probabilities, and
random phases (the phases of the probability amplitudes are uncorrelated),

the density matrix in any reasonable basis must become

\[ \rho_{mn} = c \delta_{mn}, \]

i.e. proportional to the identity (within the allowed subspace). For the microcanonical ensemble the constant is \(c = 1/\Gamma\).

So:

Microcanonical ensemble: the system is equally likely to be in any of Γ allowed states → density operator is (1/Γ) times the identity on that energy shell.

3. Canonical ensemble (contact with a heat bath)

Now we relax the condition of fixed energy.

Physical situation

System is in contact with a big heat bath at temperature T.
T, V, N are fixed.
The system's energy E is not fixed; it can fluctuate around some average.

The probability that the system has energy eigenvalue \(E_n\) is:

\[ \rho_n = C e^{-\beta E_n},\quad \beta = 1/(kT). \]

Here C is a normalization constant chosen so that all probabilities add up to 1:

\[ C = \frac{1}{\sum_n e^{-\beta E_n}} = \frac{1}{Q_N(\beta)}, \]

where

\[ Q_N(\beta) = \sum_n e^{-\beta E_n} \]

is the canonical partition function.

So the density matrix (in the energy basis) is

\[ \rho_{mn} = \rho_n \delta_{mn}. \]

In operator form:

\[ \hat{\rho} = \frac{1}{Q_N(\beta)} e^{-\beta \hat{H}} = \frac{e^{-\beta \hat{H}}}{\mathrm{Tr}\left(e^{-\beta \hat{H}}\right)}. \]

Expectation value of an observable \(\hat{G}\)

\[ \langle G \rangle_N = \mathrm{Tr}(\hat{\rho} \hat{G}) = \frac{\mathrm{Tr}\left(\hat{G} e^{-\beta \hat{H}}\right)} {\mathrm{Tr}\left(e^{-\beta \hat{H}}\right)}. \]

That's the standard formula for thermal averages in quantum statistical mechanics.

4. Grand canonical ensemble (can exchange particles too)

Now we let the system exchange both energy and particles with a reservoir.

Physical situation

Temperature T, volume V, and chemical potential μ are fixed.
Energy E and number of particles N can both fluctuate.
There is a Hamiltonian operator \(\hat{H}\) and a number operator \(\hat{n}\) (whose eigenvalues are 0, 1, 2, …).

The density operator that generalizes the canonical one is

\[ \hat{\rho} = \frac{1}{\mathcal{Q}(\mu, V, T)} e^{-\beta \hat{H} - \mu \hat{n}}, \]

where

\[ \mathcal{Q}(\mu, V, T) = \sum_{r, s} e^{-\beta(E_{r,s} - \mu N_s)} = \mathrm{Tr}\left(e^{-\beta \hat{H} - \mu \hat{n}}\right) \]

is the grand partition function ("grand" because both E and N fluctuate).

Often one writes \(z = e^{\beta \mu}\) and expands things as sums over N with factors \(z^N\); that's the fugacity.

Expectation value of an observable \(\hat{G}\)

\[ \langle G \rangle = \frac{1}{\mathcal{Q}(\mu, V, T)} \mathrm{Tr}\left(\hat{G} e^{-\beta \hat{H} - \mu \hat{n}}\right). \]

They also show you can rewrite this as a weighted average of the canonical-ensemble averages at fixed N, using the fugacity expansion.

5. Big picture summary

Microcanonical: completely isolated; fixed (E, V, N).
ρ̂ is uniform over all states with that energy: \(\hat{\rho} = \frac{1}{\Gamma} \mathbf{1}\) on that energy shell.
Entropy \(S = k \ln \Gamma\).
Canonical: in contact with a heat bath; fixed (T, V, N).
Probabilities weighted by Boltzmann factor \(e^{-\beta E_n}\).
\(\hat{\rho} = e^{-\beta \hat{H}} / \mathrm{Tr}(e^{-\beta \hat{H}})\).
Grand canonical: in contact with heat and particle reservoir; fixed (T, V, μ).
Probabilities weighted by \(e^{-\beta(E - \mu N)}\).
\(\hat{\rho} = e^{-\beta \hat{H} - \mu \hat{n}} / \mathrm{Tr}(e^{-\beta \hat{H} - \mu \hat{n}})\).

In all cases, once you know ρ̂, the average of any physical quantity G is always

\[ \langle G \rangle = \mathrm{Tr}(\hat{\rho} \hat{G}). \]