Statistical Mechanics: Classical & Quantum Statistical Mechanics

Statistical Mechanics

Partition functions, quantum statistics, and ideal and interacting gases.

Classical Statistical Mechanics

Phase Space

6N-dimensional space of positions and momenta for N particles

Phase Space Trajectory Position q Momentum p q1 q2 q3 p1 p2 p3 t = 0 t = T dGamma Phase space cell: dGamma = dq dp

Phase space volume element:

$$d\Gamma = \prod_{i=1}^{N} d^3\mathbf{r}_i d^3\mathbf{p}_i$$

Liouville's Theorem

Phase space density is conserved along trajectories:

\(\frac{d\rho}{dt} = \frac{\partial \rho}{\partial t} + \{\rho, H\} = 0\)

Phase Space Volume Conservation t = 0 Initial volume V Hamiltonian evolution t = tau Same volume V V = V

Classical Partition Function

\(Z = \frac{1}{N!h^{3N}} \int e^{-\beta H(\mathbf{r},\mathbf{p})} d\Gamma\)

The factor $1/N!$ accounts for indistinguishability (Gibbs correction)

Equipartition Theorem

Each quadratic term in the energy contributes $\frac{1}{2}k_B T$ to the average energy. For example:

  • Harmonic oscillator: $\langle E \rangle = k_B T$ (kinetic + potential)
  • Ideal gas molecule: $\langle E_{\text{trans}} \rangle = \frac{3}{2}k_B T$ (3 translational DOF)

Quantum Statistical Mechanics

The classical formalism above rests on a single object — the phase-space distribution $\rho(\mathbf{r}, \mathbf{p})$ — from which every observable follows by integrating against $d\Gamma$. Quantum mechanics forces us to generalize on two fronts. First, a quantum system in thermal contact with a reservoir is not in a definite pure state $ \psi\rangle$ but in a statistical mixture of energy eigenstates, so the distribution is promoted from a function on phase space to an operator on Hilbert space — the density operator $\rho$. Second, because position and momentum no longer commute, there is no joint $(\mathbf{r},\mathbf{p})$ distribution to integrate over; the phase-space integral $\frac{1}{h^{3N}}\int (\cdots)\, d\Gamma$ is replaced by the basis-independent trace $\text{Tr}(\cdots)$, which sums the diagonal matrix elements over any complete set of states. The dictionary is direct:
\[\rho(\mathbf{r},\mathbf{p}) \;\longrightarrow\; \hat{\rho}, \qquad \frac{1}{h^{3N}}\int d\Gamma \;\longrightarrow\; \text{Tr}, \qquad \langle A \rangle = \frac{1}{h^{3N}}\int A\,\rho\, d\Gamma \;\longrightarrow\; \langle A \rangle = \text{Tr}(\hat{\rho}\hat{A}).\]

The Gibbs $1/N!$ that we inserted by hand classically now appears automatically, encoded in the (anti)symmetry of the many-body Hilbert space. The two formalisms meet in the classical limit: when the thermal de Broglie wavelength is small compared to the inter-particle spacing, the trace over states reduces to the phase-space integral and $\hat{\rho}$ becomes diagonal in the classical sense, recovering the Boltzmann weight $e^{-\beta H}$ as an ordinary function.

Density Matrix

For a mixed state:

\(\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|\)

Canonical density matrix: $\rho = \frac{e^{-\beta H}}{Z}$

Density Matrix Structure rho rho_11 rho_12 rho_13 rho_21 rho_22 rho_23 rho_31 rho_32 rho_33 Legend Diagonal: populations Off-diag: coherences

Quantum Partition Function

\(Z = \text{Tr}(e^{-\beta H}) = \sum_n e^{-\beta E_n}\)

Fermi-Dirac Statistics

For fermions (half-integer spin)

Average occupation number:

\(\langle n_i \rangle = \frac{1}{e^{\beta(\epsilon_i - \mu)} + 1}\)

Fermi-Dirac Distribution Function Energy epsilon Occupation <n> 1 0.5 0 mu (Fermi level) T = 0 K T > 0 K Thermal broadening

At T = 0, becomes a step function at the Fermi energy

Bose-Einstein Statistics

For bosons (integer spin)

Average occupation number:

\(\langle n_i \rangle = \frac{1}{e^{\beta(\epsilon_i - \mu)} - 1}\)

Bose-Einstein Distribution Function Energy epsilon Occupation <n> 1 2 0 mu = 0 High T Medium T Low T diverges! BEC Region As T approaches Tc, occupation diverges at ground state (condensation)

Allows for Bose-Einstein condensation when $\mu \to 0^-$

Ideal Gases

The ideal gas is the hydrogen atom of statistical mechanics — the one model simple enough to solve completely, yet rich enough to expose the deep difference between classical and quantum statistics. The single control parameter is how the thermal de Broglie wavelength $\lambda$ compares to the inter-particle spacing $n^{-1/3}$:

  • When $\lambda \ll n^{-1/3}$ (hot or dilute), wavepackets don’t overlap and the gas behaves classically.
  • When $\lambda \gtrsim n^{-1/3}$ (cold or dense), wavepackets overlap and quantum statistics — Fermi or Bose — take over.

Classical Ideal Gas

In the classical regime the partition function factorizes over particles, with the $1/N!$ Gibbs factor for indistinguishability:

\[Z = \frac{V^N}{N!\lambda^{3N}}, \qquad \lambda = \sqrt{\frac{2\pi\hbar^2}{mk_BT}}.\]

Differentiating $\ln Z$ recovers the familiar equation of state $PV = Nk_BT$ — a reassuring check that microstate-counting reproduces 19th-century gas laws.

Quantum Ideal Gases

Once $\lambda \gtrsim n^{-1/3}$, the spin-statistics of the particles dominates, and fermions and bosons could hardly behave more differently.

Fermi Gas

The Pauli exclusion principle forbids two fermions from sharing a state, so even at $T = 0$ the particles stack up to the Fermi energy, filling a sphere in momentum space:

\[E_F = \frac{\hbar^2}{2m}(3\pi^2 n)^{2/3}.\]

Only the thin shell within $\sim k_B T$ of $E_F$ can be excited, giving the characteristic linear low-temperature heat capacity $C_V \propto T$. This degeneracy pressure is what holds up white dwarfs and neutron stars against gravity.

Bose Gas

Bosons have the opposite tendency — they favor sharing a state. Below a critical temperature a macroscopic fraction of them collapses into the single ground state, forming a Bose-Einstein condensate:

\[T_c = \frac{2\pi\hbar^2}{mk_B}\left(\frac{n}{2.612}\right)^{2/3}.\]

This is not ordinary condensation in real space but condensation in momentum space, first realized experimentally in dilute atomic gases in 1995.

Interacting Systems

Interactions are where statistical mechanics gets hard — and interesting. Once particles influence each other, the partition function no longer factorizes, and exact solutions become rare. Two complementary strategies dominate: expand systematically in the strength of interactions (the virial expansion, good for dilute gases), or replace the many-body environment of each particle with a single average field (mean-field theory, good for capturing collective ordering).

Virial Expansion

For a weakly interacting gas, corrections to the ideal-gas law come as a power series in density:

\[\frac{PV}{Nk_BT} = 1 + B_2(T)n + B_3(T)n^2 + \dots\]

The second virial coefficient $B_2$ measures the net effect of pairwise interactions — repulsive cores push it positive, attractive tails pull it negative:

\[B_2(T) = -\frac{1}{2V}\int \left(e^{-\beta u(r)} - 1\right)d^3r.\]

Mean Field Theory

Rather than track every pairwise interaction, mean-field theory lets each particle feel the average effect of all the others — a single self-consistent field. For the Ising model, each spin sees an effective field set by the average magnetization $m$ of its $z$ neighbors, giving the self-consistency equation

\[m = \tanh\!\left(\frac{m z J}{k_B T}\right).\]

This has only the trivial solution $m = 0$ at high temperature, but a nonzero (spontaneously magnetized) solution appears below the critical temperature $T_c = zJ/k_B$ — mean-field theory’s prediction of a phase transition. It gets the existence of the transition right but the critical exponents wrong, because it ignores the fluctuations that dominate near $T_c$ — exactly what the renormalization group was invented to handle.

Correlation Functions

Two-point correlation: \(G(r) = \langle s_i s_j \rangle - \langle s_i \rangle\langle s_j \rangle\)

Near critical point: $G(r) \sim \frac{e^{-r/\xi}}{r^{d-2+\eta}}$

See Also

Next: Phase Transitions & Graduate Formalism →