Partition Function

Definition

In the canonical ensemble (system at fixed T, V, N in contact with a heat bath), the probability of microstate i with energy Eᵢ is:

P(i) = (1/Z) · exp(−βEᵢ),     β ≡ 1/(k·T)

where the partition function Z is the normalization:

Z(T, V, N) = Σᵢ exp(−βEᵢ)

The sum runs over every accessible microstate. Equivalently, group states by energy and sum over energy levels using the density of states g(E):

Z = Σ_E g(E) · exp(−βE) = ∫ g(E) · exp(−βE) dE  (continuous case)

That's it. The whole machine. Everything else is calculus on ln Z.

From Z to thermodynamics

The Helmholtz free energy is:

F(T, V, N) = −kT · ln Z

Every thermodynamic relation flows from F. A few highlights:

Quantity	From ln Z	From F
Mean energy ⟨E⟩	−∂ln Z/∂β	F − T·∂F/∂T
Entropy S	k·(ln Z + β·⟨E⟩)	−(∂F/∂T)_V
Pressure P	(1/β)·(∂ln Z/∂V)_T	−(∂F/∂V)_T
Heat capacity Cv	k·β²·∂²ln Z/∂β²	−T·(∂²F/∂T²)_V
Variance of energy	∂²ln Z/∂β²	kT²·Cv
Magnetization M (B field)	(1/β)·∂ln Z/∂B	−(∂F/∂B)_T

Worked example — two-level system

Consider a single spin-½ in magnetic field B. Two states: aligned (E = −μB) and anti-aligned (E = +μB).

Z = exp(+βμB) + exp(−βμB) = 2·cosh(βμB)

Take the log and differentiate:

ln Z = ln 2 + ln cosh(βμB)
⟨E⟩ = −∂ln Z/∂β = −μB · tanh(βμB)
S/k = ln Z + β·⟨E⟩ = ln 2cosh(βμB) − βμB·tanh(βμB)

Limits:

• High T (kT ≫ μB): tanh ≈ βμB → ⟨E⟩ ≈ −μ²B²/(kT). Curie's law follows: M ∝ 1/T.
• Low T (kT ≪ μB): tanh ≈ 1 → ⟨E⟩ ≈ −μB. System frozen in ground state. S → 0.

The heat capacity Cv = k·(βμB)²·sech²(βμB) peaks at kT ≈ μB — the famous "Schottky anomaly" of two-level systems.

Factorization for independent subsystems

If the total energy is a sum of independent contributions E = E₁ + E₂ + … + E_N, then:

Z_total = Z₁ · Z₂ · … · Z_N
ln Z_total = ln Z₁ + ln Z₂ + … + ln Z_N

This is why ln Z is extensive (proportional to system size). For an ideal gas of N indistinguishable particles:

Z(N, V, T) = (1/N!) · Z_1(V, T)^N
F = −kT·ln Z = −NkT[ln(V/(N·λ³)) + 1]
   where λ = h/√(2π·m·kT) is the thermal de Broglie wavelength

From this single expression you get the ideal-gas law (PV = NkT), Sackur-Tetrode entropy, equipartition (⟨E⟩ = (3/2)NkT), and the heat capacity Cv = (3/2)Nk. Half of intro thermodynamics drops out of one logarithm.

Numerical examples

System	Energy spectrum	Z (one mode)	Heat capacity
Two-level (spin in B)	±μB	2·cosh(βμB)	Schottky anomaly at kT ≈ μB
Harmonic oscillator (quantum)	(n + ½)ħω, n = 0,1,…	1/(2·sinh(βħω/2))	Einstein C: k at high T; ∝ T at low T
Particle in 3D box	continuous	V/λ³	(3/2)k per particle (equipartition)
Rotational diatomic (rigid rotor)	ℓ(ℓ+1)·B_rot, deg 2ℓ+1	~ kT/B_rot at high T	+k per molecule above the rotational temperature
Bose gas (mode k)	n·εₖ, n = 0,1,…	1/(1 − exp(−βεₖ))	~T³ phonons at low T
Fermi gas (mode k)	0 or εₖ	1 + exp(−βεₖ)	Sommerfeld: linear in T

Concrete numerical anchor: a spin with μB = 1 meV in a field has its Schottky peak at T = μB/k ≈ 11.6 K. At room temperature (kT ≈ 25.7 meV) the same spin is in the high-T regime — strongly paramagnetic, weakly Curie.

Worked example — Curie susceptibility

For a paramagnet of N independent spin-½'s, magnetization is:

M = N·μ·tanh(βμB)
At kT ≫ μB:  M ≈ N·μ²·B/(kT)
Susceptibility χ = M/(B/μ₀) → χ = μ₀·N·μ²/(kT) = C/T
   (Curie's law; C = μ₀·N·μ²/k)

For 1 mole of Bohr-magneton spins (μ = 9.27 × 10⁻²⁴ J/T), at 300 K:

χ ≈ (4π × 10⁻⁷) · (6.022 × 10²³) · (9.27 × 10⁻²⁴)² / (1.381 × 10⁻²³ · 300)
χ ≈ 1.6 × 10⁻⁵  (SI dimensionless)

This is the order of magnitude of typical paramagnetic salts — directly derived from one cosh.

JavaScript — partition function arithmetic

const k_B = 1.380649e-23;  // J/K

// Two-level (spin-½) partition function
function Z_twoLevel(eps, T) {
  const beta = 1 / (k_B * T);
  return 2 * Math.cosh(beta * eps);
}
function E_twoLevel(eps, T) {
  const beta = 1 / (k_B * T);
  return -eps * Math.tanh(beta * eps);
}

// Energy in J ↔ μ·B in J  (μ_B ≈ 9.274e-24 J/T)
const muB = 9.274e-24;
const B = 1;  // 1 tesla
console.log(`Z(spin, 1 T, 300 K) = ${Z_twoLevel(muB * B, 300).toFixed(8)}`);
console.log(`⟨E⟩ = ${E_twoLevel(muB * B, 300).toExponential(2)} J`);

// Quantum harmonic oscillator
function Z_qho(hw, T) {
  const beta = 1 / (k_B * T);
  return 1 / (2 * Math.sinh(beta * hw / 2));
}
function E_qho(hw, T) {
  const beta = 1 / (k_B * T);
  return (hw / 2) / Math.tanh(beta * hw / 2);
}
// vibration of N–N: ħω ≈ 2.36 × 10⁻²⁰ J → T_vib ≈ 1710 K
const hw_N2 = 2.36e-20;
console.log(`Cv(N₂ vib, 300 K) factor: ${(hw_N2 / (k_B * 300))**2 / Math.sinh(hw_N2 / (k_B * 300))**2 * Math.exp(hw_N2 / (k_B * 300))}`);
// Effectively frozen out at 300 K — Cv contribution ≈ 0

// Free energy
function F_from_Z(Z, T) { return -k_B * T * Math.log(Z); }

// Ideal gas partition function (per particle) Z₁ = V / λ³
function lambda_thermal(m, T) {
  const h = 6.626e-34;
  return h / Math.sqrt(2 * Math.PI * m * k_B * T);
}
function Z1_idealGas(V, m, T) {
  const lam = lambda_thermal(m, T);
  return V / (lam * lam * lam);
}
// N₂ molecule at STP: m ≈ 4.65e-26 kg, V = 22.4 L per mol of single particle would be huge
console.log(`λ(N₂, 300 K) = ${lambda_thermal(4.65e-26, 300).toExponential(2)} m`); // ~1.9e-11

// Curie susceptibility from spin partition function
function curieSusceptibility(N, mu, T) {
  const mu0 = 4 * Math.PI * 1e-7;
  return mu0 * N * mu * mu / (k_B * T);
}
console.log(`χ(1 mol Bohr magnetons, 300 K) ≈ ${curieSusceptibility(6.022e23, muB, 300).toExponential(2)}`);

Where the partition function shows up

• Equilibrium thermodynamics. Every textbook calculation of heat capacity, free energy, equation of state, magnetization, polarization — all reducible to Z and its logarithm.
• Phase transitions. Singularities in F (and hence ln Z) signal phase transitions. The Ising model's partition function on a 2D square lattice was solved exactly by Onsager — a tour de force whose critical exponents agree with experiment.
• Polymer physics. Partition function for random walks gives the radius of gyration, persistence length, elasticity (entropic spring constant).
• Chemistry — reaction rates. Transition state theory writes k_rxn = (kT/h)·(Z‡/Z_reactant)·exp(−ΔE/kT). The partition function of the transition state controls the prefactor.
• Astrophysics. Stellar atmospheres use partition functions of atomic species to compute level populations via the Saha equation.
• Field theory. The path integral generalizes Z to functional sums; quantum field theory and statistical mechanics in d+1 dimensions are Wick-rotated cousins of each other.
• Machine learning. Energy-based models (Boltzmann machines, RBMs, contrastive divergence) train on log-partition gradients. The Hopfield network's stability is partition-function thermodynamics.

Common mistakes

• Forgetting the N! for indistinguishable particles. For identical particles you divide by N! to avoid Gibbs's paradox. Skipping it gives the wrong entropy (extensive contradictions) for ideal gases.
• Using classical Z when quantization matters. Below the Einstein/Debye temperature, vibrational modes freeze out and classical equipartition overestimates Cv by factors of 2-3. Always check β·ħω vs 1.
• Mixing up canonical and microcanonical. Canonical: fixed T, energy fluctuates. Microcanonical: fixed E, no heat bath. They agree at large N but you must use Z (canonical) for T-derivatives and the multiplicity Ω(E) (microcanonical) for E-derivatives.
• Truncating the sum too aggressively. States above β·E ≫ 10 contribute negligibly, but if you stop at β·E ≈ 1 you can miss important contributions — particularly for systems with dense high-E spectra like rotors.
• Confusing partition function with probability. Z is the NORMALIZATION; the probability is exp(−βE)/Z. Don't write P(E) = exp(−βE) without dividing by Z (a common shortcut that breaks at low T).
• Differentiating Z when you should differentiate ln Z. Thermodynamic relations use ∂ln Z/∂β, ∂ln Z/∂V, etc., not ∂Z directly. The log is what makes it extensive.