Appendix B: Key Statistical Mechanics Formulas
B.1 Ensembles
| Ensemble | Fixed variables | Partition function | Free energy |
|---|---|---|---|
| Microcanonical (NVE) | \(N, V, E\) | \(\Omega(E)\) | \(S = k_B \ln \Omega\) |
| Canonical (NVT) | \(N, V, T\) | \(Z = \sum_i e^{-\beta E_i}\) | \(F = -k_BT \ln Z\) |
| Grand canonical (\(\mu\)VT) | \(\mu, V, T\) | \(\mathcal{Z} = \sum_{N,i} e^{-\beta(E_i - \mu N)}\) | \(\Omega = -k_BT \ln \mathcal{Z}\) |
| Isothermal-isobaric (NPT) | \(N, P, T\) | \(\Delta = \sum_i e^{-\beta(E_i + PV_i)}\) | \(G = -k_BT \ln \Delta\) |
B.2 Thermodynamic Relations from MC Data
Given canonical MC samples \({E_k, M_k}\):
$$\langle E \rangle = \frac{1}{N}\sum_k E_k, \qquad \langle M \rangle = \frac{1}{N}\sum_k M_k$$
$$C_V = \frac{\langle E^2 \rangle - \langle E \rangle^2}{k_B T^2} = \frac{\beta^2}{N_{\rm spins}}\left(\langle H^2 \rangle - \langle H \rangle^2\right)$$
$$\chi = \frac{\langle M^2 \rangle - \langle M \rangle^2}{k_B T} = \frac{\beta}{N_{\rm spins}}\left(\langle (MN)^2 \rangle - \langle MN \rangle^2\right)$$
$$P = \frac{Nk_BT}{V} + \frac{\langle \mathcal{W} \rangle}{3V}, \quad \mathcal{W} = -\sum_{i<j} r_{ij} \frac{\partial \phi}{\partial r_{ij}} \quad (\text{virial})$$
B.3 Ising Model Reference Values
2D Square Lattice (Onsager exact solution, \(h = 0\)):
$$k_BT_c = \frac{2J}{\ln(1+\sqrt{2})} \approx 2.2692,J$$
$$M(T) = \left[1 - \sinh^{-4}(2\beta J)\right]^{1/8} \quad (T < T_c)$$
Critical exponents (2D Ising):
| Exponent | Symbol | Value |
|---|---|---|
| Correlation length | \(\nu\) | 1 |
| Magnetisation | \(\beta\) | 1/8 |
| Susceptibility | \(\gamma\) | 7/4 |
| Specific heat | \(\alpha\) | 0 (log divergence) |
| Anomalous dimension | \(\eta\) | 1/4 |
| Dynamic (Metropolis) | \(z\) | ≈ 2.17 |
| Dynamic (Wolff) | \(z\) | ≈ 0.25 |
3D Ising critical exponents (numerical):
| Exponent | Value |
|---|---|
| \(\nu\) | 0.6301 |
| \(\beta\) | 0.3265 |
| \(\gamma\) | 1.2372 |
| \(\eta\) | 0.0364 |
B.4 Heisenberg and XY Models
3D Classical Heisenberg (\(J > 0\)): $$k_B T_c \approx 1.4432,J \quad (\text{simple cubic, MC})$$
2D XY model — Kosterlitz-Thouless transition: $$k_B T_{\rm KT} = \frac{\pi J}{2} \approx 1.5708,J$$
Universal helicity modulus jump: $$\lim_{T \to T_{\rm KT}^-} \Upsilon(T) = \frac{2}{\pi} k_B T_{\rm KT}$$
Mermin-Wagner theorem: \(T_c = 0\) for \(n \geq 2\) in \(d \leq 2\).
B.5 Lennard-Jones Fluid Reference
Lennard-Jones potential: $$\phi(r) = 4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6\right]$$
Critical point (reduced units \(T^* = k_BT/\varepsilon\), \(\rho^* = \rho\sigma^3\)): $$T_c^* \approx 1.326, \quad \rho_c^* \approx 0.316, \quad P_c^* \approx 0.129$$
Triple point: \(T_{\rm tp}^* \approx 0.694\), \(\rho_{\rm tp,liq}^* \approx 0.845\).
Argon parameters: \(\varepsilon/k_B = 119.8\) K, \(\sigma = 3.405\) Å.
B.6 Arrhenius and Transition State Theory
$$k = \nu_0, e^{-E_a/k_BT} \quad \text{(Arrhenius)}$$
$$k = \frac{k_BT}{h} e^{-\Delta G^\ddagger/k_BT} = \frac{k_BT}{h} e^{\Delta S^\ddagger/k_B} e^{-\Delta H^\ddagger/k_BT} \quad \text{(Eyring TST)}$$
Typical attempt frequency for solids: \(\nu_0 \sim 10^{12}\text{–}10^{13}\) s\(^{-1}\).
B.7 Statistical Error Formulas
| Quantity | Formula |
|---|---|
| Sample mean | \(\bar{A} = N^{-1}\sum_i A_i\) |
| Sample variance | \(s^2 = (N-1)^{-1}\sum_i(A_i - \bar{A})^2\) |
| Standard error (i.i.d.) | \(\text{SE} = s/\sqrt{N}\) |
| Integrated autocorr. time | \(\tau_{\rm int} = \frac{1}{2} + \sum_{\tau=1}^\infty C(\tau)\) |
| Effective sample size | \(N_{\rm eff} = N/(2\tau_{\rm int})\) |
| True standard error | \(\text{SE}{\rm true} = s/\sqrt{N{\rm eff}}\) |
| Jackknife variance | \(\sigma_{\rm JK}^2 = \frac{N-1}{N}\sum_k(\hat{\theta}_k - \bar{\theta})^2\) |
B.8 Free Energy Methods Reference
Free energy perturbation: $$\Delta F_{A\to B} = -k_BT \ln \langle e^{-\beta(H_B - H_A)} \rangle_A$$
Thermodynamic integration: $$\Delta F_{A\to B} = \int_0^1 \left\langle \frac{\partial H(\lambda)}{\partial\lambda} \right\rangle_\lambda d\lambda$$
Wang-Landau: \(Z(T) = \sum_E g(E) e^{-\beta E}\) from flat-histogram sampling of \(g(E)\).
Parallel tempering swap acceptance: $$A = \min!\left(1, e^{(\beta_k - \beta_{k+1})(E_k - E_{k+1})}\right)$$