Statistical Mechanics Partition Function Calculator

What Is the Partition Function? The canonical partition function Z is the central object of statistical mechanics: Z = Σ exp(−Eᵢ / kT) = Σ exp(−βEᵢ) where the sum runs over all quantum states i, β = 1/(kT), k = Boltzmann constant, T = temperature. Everything in equilibrium thermodynamics follows from Z: energy, entropy, heat capacity, free energy.

Why Z Matters Helmholtz free energy: F = −kT ln Z Average energy: ⟨E⟩ = −∂(ln Z)/∂β = kT² ∂(ln Z)/∂T Heat capacity: Cv = ∂⟨E⟩/∂T Entropy: S = (⟨E⟩ − F)/T = k[ln Z + β⟨E⟩] All thermodynamic properties follow from a single function Z.

Quantum Harmonic Oscillator Energy levels: Eₙ = ℏω(n + 1/2), n = 0, 1, 2, … Z = Σ exp(−β ℏω(n+1/2)) = exp(−βℏω/2) / (1 − exp(−βℏω)) Average energy: ⟨E⟩ = ℏω[1/2 + 1/(exp(βℏω) − 1)] At high T (classical limit): ⟨E⟩ → kT (equipartition theorem). At low T: ⟨E⟩ → ℏω/2 (zero-point energy).

Einstein Model of Solids A solid with N atoms has 3N independent harmonic oscillators. Einstein heat capacity: Cv = 3Nk(βℏω)² exp(βℏω) / (exp(βℏω)−1)² At high T: Cv → 3Nk (Dulong-Petit law, classical result). At low T: Cv → 0 exponentially (quantum suppression). Observed by Einstein (Switzerland, 1907) — explained why Cv drops at low temperature.

Two-Level System Energy levels: 0 and ε (e.g., spin-1/2 in magnetic field, or two-state molecule). Z₂ = 1 + exp(−βε) ⟨E⟩ = ε / (exp(βε) + 1) — Fermi-Dirac-like distribution Heat capacity shows a Schottky anomaly — a peak at kT ≈ 0.42ε. This peak in Cv is diagnostic of a two-level system.

Boltzmann Factor and Occupation Probability Probability of state with energy E: P(E) = exp(−βE) / Z Ratio of populations: N₁/N₀ = exp(−Δε/kT) At T → 0: only the ground state is populated. At T → ∞: all states equally populated.

Physical Constants Used k (Boltzmann): 1.380649 × 10⁻²³ J/K = 8.617333 × 10⁻⁵ eV/K. ℏ (reduced Planck): 1.054572 × 10⁻³⁴ J·s. A frequency of 1 THz corresponds to ℏω = 0.00414 eV (quantum effects matter below ~50 K).