“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

What gets measured, what gets computed in statistical mechanics

Statistical Mechanics

Statistical mechanics sits between thermodynamics on the macroscopic side and microscopic dynamics on the other. Most of its "experiments" are condensed-matter or soft-matter measurements that probe the thermal averages of microscopic quantities — heat capacities, susceptibilities, structure factors, critical exponents. Most of its "calculations" produce those same thermal averages, either by analytical approximation (mean field, RG, series expansions, conformal field theory) or by direct sampling (Monte Carlo, molecular dynamics, transfer matrix). This page catalogs both columns and the bridge between them.

What an experiment measures

Four broad classes of observable that test statistical-mechanical theory:

1. Equation of state and thermodynamic response

Specific heat $C (T) = d Q / d T$ from calorimetry — directly probes the temperature dependence of the energy distribution. Magnetic susceptibility $χ (T)$ from SQUID magnetometry — probes the spin response. Compressibility $κ$ and other generalized susceptibilities. Equation of state in $(P, V, T)$ from gas-cell experiments; in $(M, H, T)$ from magnetometry. The shape of these response functions near phase transitions encodes the universality class.

2. Phase transitions and critical phenomena

Critical temperatures $T_{c}$ from peaks in susceptibilities or specific heats. Critical exponents $α, β, γ, δ, ν, η$ extracted from power-law fits to $C \sim ∣ T - T_{c} ∣^{- α}$ , $M \sim ∣ T_{c} - T ∣^{β}$ , $χ \sim ∣ T - T_{c} ∣^{- γ}$ , etc. Order parameters measured by their natural probe: spontaneous magnetization (magnetometry), staggered magnetization (neutron diffraction), nematic order (birefringence), superfluid fraction (torsional oscillator). Correlation length $ξ (T)$ diverging at $T_{c}$ from peak width in $S (q)$ .

3. Correlations and structure

Pair correlation function $g (r)$ from X-ray and neutron diffraction on liquids and amorphous solids. Static structure factor $S (q) = 1 + ρ \int [g (r) - 1] e^{- i q \cdot r} d r$ — the Fourier dual; this is what scattering experiments directly report. Dynamic structure factor $S (q, ω)$ from inelastic scattering; related to the density-density response function through the fluctuation-dissipation theorem.

4. Dynamics and transport

Relaxation times $τ (T)$ from dielectric, mechanical, or magnetic spectroscopy. Near glass transitions, $τ (T)$ follows VFT or Arrhenius forms that probe energy landscapes. Transport coefficients — viscosity $η$ , thermal conductivity $κ_{T}$ , ionic conductivity $σ$ , diffusion constant $D$ — all computable from time-correlation functions via Green-Kubo relations. Aging and non-equilibrium response in glasses and gels probe deviation from equilibrium fluctuation-dissipation.

What a calculation produces

The fundamental object is the partition function $Z = \sum_{states} e^{- β E_{state}}$ . Everything thermodynamic is a derivative of $F = - k_{B} T ln Z$ ; everything fluctuation-like is a moment of the underlying probability distribution. The computational menu:

Free energy and equation of state

From the partition function: $F (T, V, h) = - k_{B} T ln Z$ . Derivatives give the equation of state, pressure $P = - \partial F / \partial V$ , magnetization $M = - \partial F / \partial h$ , specific heat $C = - T \partial^{2} F / \partial T^{2}$ . Computed analytically (mean field, series expansion) or numerically (Wang-Landau gives $ρ (E)$ directly; thermodynamic integration recovers $F$ from sampled energy distributions).

Order parameters and phase diagrams

Ensemble averages of the order parameter (magnetization, staggered magnetization, density difference, …) as functions of temperature and external fields. Phase boundaries from where the order parameter or its susceptibility becomes singular. The most famous example: the Ising model's second-order transition at $k_{B} T_{c} \approx 2.27 J$ in 2D.

Correlation functions

Spin-spin $⟨ s_{i} s_{j} ⟩$ , density-density $⟨ ρ (0) ρ (r)⟩$ , and higher multi-point correlations. Near criticality these decay as $\sim e^{- r / ξ} / r^{d - 2 + η}$ with the correlation length $ξ$ diverging as $ξ \sim ∣ t ∣^{- ν}$ .

Critical exponents and universality

Renormalization group predicts the critical exponents $α, β, γ, δ, ν, η$ from the universality class — dimension and symmetry of the order parameter. Each universality class has fixed exponents; in 2D the Ising universality class has $β = 1/8, γ = 7/4, ν = 1, η = 1/4$ exactly, from conformal field theory.

Transport coefficients via Green-Kubo

Linear-response transport coefficients are integrals of equilibrium time-correlation functions:

η = \frac{1}{V k _{B} T} \int_{0}^{\infty} ⟨ σ_{x y} (0) σ_{x y} (t)⟩ d t, D = \frac{1}{3} \int_{0}^{\infty} ⟨ v (0) \cdot v (t)⟩ d t .

Computed from molecular-dynamics trajectories or from the appropriate microscopic response function.

Density of states

$ρ (E) = \sum_{states} δ (E - E_{state})$ — the energy histogram of microstates. Wang-Landau sampling targets $ρ (E)$ directly, from which all thermodynamic averages at any temperature follow by reweighting.

The bridge

Row by row:

Experimental observable	Model-side counterpart	Notes / accuracy
Specific heat $C (T)$	$C = - T \partial^{2} F / \partial T^{2}$ , or $⟨ δ E^{2} ⟩ / k_{B} T^{2}$ via fluctuations	Lambda-shaped peak at $T_{c}$ ; exponent $α$ from the diverging tail
Magnetic susceptibility $χ (T)$	$χ = β ⟨ δ M^{2} ⟩$ from sampled magnetization fluctuations	Diverges at $T_{c}$ with exponent $γ$
Spontaneous magnetization $M (T)$	$⟨ ∣ M ∣ ⟩$ from MC sampling below $T_{c}$	Vanishes as $∣ T_{c} - T ∣^{β}$ at second-order transitions
Critical exponents ( $α, β, γ, δ, ν, η$ )	RG prediction; high-precision MC + finite-size scaling; CFT (exact in 2D)	Universality class predicted from dimension + symmetry
Pair correlation $g (r)$ , structure factor $S (q)$ (XRD, neutron)	Average over equilibrium configurations; sampled from MC or MD	For liquids, $g (r)$ directly tests integral-equation theories (Percus-Yevick, HNC)
Dynamic structure factor $S (q, ω)$ (INS, IXS)	FT of density-density time correlation; from MD or response theory	FDT links to imaginary part of response function
Correlation length $ξ (T)$	Inverse exponential decay rate of $⟨ s_{0} s_{r} ⟩$	Diverges as $∣ t ∣^{- ν}$ at $T_{c}$
Relaxation time $τ (T)$	Decay timescale of equilibrium autocorrelation $⟨ A (0) A (t)⟩$	Diverges at glass transition; Arrhenius/VFT/MCT forms
Viscosity $η$ , diffusion $D$ , conductivity $σ$	Green-Kubo integral of stress / velocity / current autocorrelations	Computed from MD trajectories
Latent heat / first-order transition signatures	Discontinuity in $\partial F / \partial T$ ; bimodal energy histogram in MC	Wang-Landau or umbrella sampling reads off the free-energy barrier

Which method for which quantity

Statistical mechanics has one of the most varied method landscapes in physics — the same Ising model can be solved by a dozen different techniques, each illuminating a different aspect. Pick by the regime of the problem (near criticality? deep in the ordered phase? high-dimensional? disordered?) and the quantity you need:

Method	Best for	Doesn't handle	Cost / scope
Mean field / Bragg-Williams / Curie-Weiss	Qualitative phase diagrams; exact in $d \to \infty$ ; intuition	Critical exponents (classical values only); 2D and below	Analytical; trivial
Series expansions (high-T, low-T)	Critical-point location; coefficients enable resummation	Generic finite-temperature observables far from expansion point	Analytical, then numerical resummation (Padé, Borel)
Renormalization group (Wilson, Kadanoff)	Critical exponents; universality classes; structural understanding	Non-universal quantities; far-from-criticality details	Analytical / $ϵ$ -expansion / numerical RG
Transfer matrix	1D + quasi-1D exact partition functions; finite-strip 2D for critical exponents	True 2D bulk; 3D	Exact in 1D; matrix size grows exponentially with strip width
Monte Carlo (Metropolis, Glauber, heat bath)	Generic equilibrium sampling at any temperature	Critical slowing down near $T_{c}$ ; broken-ergodicity systems	Scales as system size × Markov chain length
Cluster algorithms (Wolff, Swendsen-Wang)	Critical-region sampling; near continuous transitions	First-order transitions (need different cluster moves)	Defeats critical slowing down — $z \approx 0$
Wang-Landau sampling	Direct computation of $ρ (E)$ ; full free energy at all temperatures	Very large systems (logarithmic energy range)	Iterative refinement; one calculation, all temperatures
Parallel tempering / replica exchange	Disordered systems, glasses, first-order transitions	Overhead of running many replicas in parallel	Multiplied cost but defeats trapping
Conformal field theory + bootstrap	High-precision critical exponents in 2D (CFT exact) and 3D (bootstrap)	Non-critical thermodynamic quantities	Analytical (2D); numerical convex optimization (bootstrap)
DMRG / tensor networks	1D quantum lattice models; 2D quantum spin systems (some progress)	Fully 3D + 1D quantum problems with high entanglement	Exponential in bond dimension
Replica method / cavity method / belief propagation	Disordered systems, spin glasses, optimization problems	Replica symmetry breaking is subtle; sparse-system requirement	Analytical (replica); message-passing iteration (cavity)
Molecular dynamics	Liquids, soft matter, transport coefficients via Green-Kubo	Long-time correlation; quantum effects	$O (N lo g N)$ per step with mesh Ewald or fast multipole

One-sentence summary

A statistical-mechanics calculation almost always answers "what thermal average or fluctuation does an experiment measure?" — and the menu of observables collapses to derivatives of the free energy and moments of the underlying probability distribution. Pick the experimental quantity, find its row in the bridge table, choose a method appropriate to the regime (near critical? in the ordered phase? disordered? quantum?), and run.

Related on this site

The Ising model — the prototype of statistical mechanics, with multiple methods worked out concretely.
Hubbard dimer FCI — the two-site quantum stat-mech toy with exact answers.
Solid state quantities of interest — heavy overlap on transport, structure, magnetic susceptibility, and phase transitions.
Parallel pages: comp neuroscience, quantum chemistry, nuclear physics.