Bridge between microscopic dynamics and macroscopic thermodynamics: partition function, ensembles, Maxwell-Boltzmann / Fermi-Dirac / Bose-Einstein distributions, equipartition theorem, fluctuation-dissipation, ergodic hypothesis.
statistical-mechanics
Partition function
Z = Σ_i e^(−β E_i) where β = 1/(k_B T). All equilibrium thermodynamic quantities follow by differentiation of ln Z.
Maxwell-Boltzmann distribution
Velocity distribution of classical, distinguishable, non-interacting particles in thermal equilibrium: f(v) ∝ v²·e^(−mv²/(2k_BT)).
Fermi-Dirac distribution
Occupation of single-particle energy states for identical fermions: f(E) = 1/(e^((E−μ)/k_BT) + 1). Enforces Pauli exclusion statistically.
Bose-Einstein distribution
Occupation for identical bosons: f(E) = 1/(e^((E−μ)/k_BT) − 1). Diverges at E = μ → Bose-Einstein condensation.
Equipartition theorem
In classical thermal equilibrium, each quadratic degree of freedom contributes ½ k_B T to the average energy.
Fluctuation-dissipation theorem
Near equilibrium, the linear response of a system to a small perturbation is given by the equilibrium fluctuations of the corresponding…
Ergodic hypothesis
Over long times, a dynamical system's trajectory spends equal time in equal phase-space volumes, so time averages equal ensemble averages.…
Microcanonical ensemble
Ensemble of isolated systems at fixed (E, V, N); equiprobable microstates on the energy shell. S = k_B ln Ω with Ω the number of…
Canonical ensemble
System in contact with a heat bath at fixed (T, V, N). Probability p_i = e^{−βE_i}/Z with Z = Σ e^{−βE_i}, β = 1/k_B T.
Grand canonical ensemble
Open system exchanging energy and particles with a reservoir at fixed (T, V, μ). Grand partition function Ξ = Σ e^{−β(E − μN)}.
Detailed balance
Microscopic reversibility: at equilibrium, probability currents between any pair of states cancel pairwise. P_eq(i) W(i→j) = P_eq(j)…
Langevin equation
Stochastic ODE m ẍ = −γẋ + ξ(t) with white-noise ξ; models Brownian motion in a viscous medium. Equivalent in the overdamped limit to the…
Fokker–Planck equation
Deterministic PDE for the time-evolving probability density of an Itô diffusion: ∂_t p = −∂_x (A p) + ½ ∂_{xx}(B² p).
Onsager reciprocal relations
Near equilibrium, the linear response coefficients L_ij coupling thermodynamic forces X_j to fluxes J_i are symmetric: L_ij = L_ji…
Renormalisation group
Systematic coarse-graining that maps effective couplings at scale ℓ to those at ℓ'. Explains universality of critical exponents;…
Dulong–Petit law
Classical limit for the molar heat capacity of monatomic crystalline solids: C_V = 3R ≈ 24.94 J K⁻¹ mol⁻¹, independent of the element. …
Gibbs ensembles (μC/C/GC)
Microcanonical (E,V,N), canonical (T,V,N), grand canonical (T,V,μ); equivalence in thermodynamic limit for short-range interactions.
Partition function Z and free energy
Z = Σe^(-βE_s); F = -k_BT ln Z generates all thermo via derivatives; path-integral representation in QM.
Fluctuation-dissipation theorem
Response function χ'' related to equilibrium correlation via Kubo: χ''(ω) = (1-e^(-βℏω))/2ℏ ∫e^(iωt)⟨A(t)A(0)⟩dt.
Onsager reciprocal relations
L_ij = L_ji for linear-response coefficients; consequence of microscopic reversibility; 1968 Nobel.
Jarzynski equality
⟨e^(-βW)⟩ = e^(-βΔF) for any driving protocol; recovers ΔF from nonequilibrium work distributions; DNA pulling experiments.
Crooks fluctuation theorem
P_F(+W)/P_R(-W) = e^(β(W-ΔF)); refines Jarzynski; verified in single-molecule pulling.
Mermin–Wagner theorem
No spontaneous continuous-symmetry breaking in d≤2 at T>0 due to long-range fluctuations; softened for Ising (discrete).
Widom scaling & critical exponents
Free energy has homogeneous singular part f(t,h) = |t|^(2-α) f̂(h/|t|^βδ); 6 exponents ↔ 2 independent via scaling relations.
Universality classes
Exponents depend only on d, symmetry of OP, short/long-range interactions; independent of microscopic detail (3D Ising, XY, Heisenberg).
Onsager exact 2D Ising solution
Exact free energy on square lattice; T_c = 2/ln(1+√2); critical exponents α=0 (log), β=1/8, γ=7/4. Transfer-matrix method.
q-state Potts model
Generalization of Ising with q states; 1st-order for q>4 in 2D; Kasteleyn–Fortuin random cluster rep; relates to percolation.
Edwards–Anderson spin glass
Random ±J bonds on lattice; frozen disorder replicated; Parisi's infinite-step RSB solution of SK mean-field (2021 Nobel).
Boltzmann transport equation
∂f/∂t + v·∇f + F·∇_p f = (∂f/∂t)_coll; H-theorem entropy increase; derives hydrodynamics via Chapman-Enskog.
Master equation (Markov processes)
dP_i/dt = Σ_j(W_ji P_j - W_ij P_i); detailed balance equivalent to reversibility; Pauli master equation from quantum.
Mode-coupling theory of glass transition
Ideal MCT predicts dynamical arrest at T_c > T_g via nonlinear coupling of density modes; α-relaxation stretched-exponential.
Large-deviation rate functions
P(A_N ≈ a) ~ e^(-N I(a)) with convex rate function I; Gartner–Ellis theorem; Legendre transform of cumulant-generating function.
Percolation & geometric phase transition
Critical bond/site probability p_c (square lattice bond p_c=1/2); scale-invariant cluster fractal dimension d_f = 91/48 in 2D.
Machine learning in physics
Application of machine-learning methods to physics: ML-based interatomic potentials, neural-network wavefunctions (FermiNet, PauliNet),…
Sinai–Ruelle–Bowen measure (statistical-mechanics)
Statistical-mechanical application of L0 Pesin entropy formula and mixing. For a C^{1+α} dissipative chaotic map f on a compact manifold…
Gallavotti–Cohen fluctuation theorem (statistical-mechanics)
Statistical-mechanical application of L0 Shannon–McMillan–Breiman AEP and mixing dynamics. In a reversible chaotic (Anosov /…