Statistical mechanics at the chemistry grain — partition functions, ensembles, and Monte Carlo methods applied to chemical systems. Distinct from L1:statistical-mechanics at the physics grain: here the emphasis is on chemical-specific…
statistical-mechanics-chemistry
Boltzmann distribution: N_i/N = g_i exp(-E_i/kT) / Z (equilibrium populations)
Foundational framework of chemical statistical mechanics: the Boltzmann distribution (Boltzmann 1872) gives the equilibrium population of…
Equipartition: ⟨(1/2) k T⟩ per classical quadratic DoF; γ = 1 + 2/f
Framework for classical-limit statistical mechanics: the equipartition theorem (Maxwell 1860, Boltzmann 1871) assigns (1/2) kT of energy…
QHO partition function: Z = 1/(2 sinh(β ℏω/2)) per normal mode
Framework for vibrational thermodynamics: the quantum harmonic oscillator (QHO) partition function Z_{QHO} = 1 / (2 sinh(β ℏω / 2)) per…
ΔE=kT log(2) ⇒ N₂/N₁ = 1/2, log ratio = -log(2) (one-bit population split)
Sympy-exact symbolic witness of the one-bit (factor-of-2) population split between two quantum levels at the log-unit Boltzmann-factor…
Equipartition: monatomic ⟨U⟩/NkT = 3/2, C_V/Nk = 3/2, γ = 5/3; diatomic γ = 7/5
Sympy-exact symbolic witness of the canonical adiabatic-index values for monatomic and classical-diatomic ideal gases. Setup: apply…
QHO Z = 1/(2 sinh(βℏω/2)) ≡ exp/geom form; βℏω=log 4 ⇒ Z = 2/3
Sympy-exact symbolic witness of the QHO partition-function identity and a canonical intermediate-temperature evaluation. Setup: Z_sinh = 1…
Canonical ensemble (NVT)
Gibbs 1902: closed system at fixed T has Boltzmann-distributed states P_i = exp(-E_i/k_BT)/Z where Z = Σ exp(-E_i/k_BT). Helmholtz free…
Grand canonical ensemble (μVT)
Variable particle-number ensemble: P(N, E_i) ∝ exp[-β(E_i - μN)]/Ξ. Grand-potential Ω = -k_BT ln Ξ; mean particle-number ⟨N⟩ = ∂ln…
Virial equation of state
Onnes 1901: pV/(nRT) = 1 + B(T)/V + C(T)/V² + ... expansion in 1/V. Coefficients B, C from cluster integrals over molecular pair, triplet…
Mayer cluster expansion
Mayer 1937: partition function Z = (V/N!)·∫d³r₁...d³r_N exp(-βU) expanded via Mayer-function f_ij = exp(-βu_ij) - 1. Diagrammatic…
Fluctuation-dissipation theorem
Callen-Welton 1951: linear response χ''(ω) of system to weak perturbation = imaginary part of equilibrium correlation-spectrum at…
Monte Carlo (Metropolis algorithm)
Metropolis-Rosenbluth-Teller 1953: importance-sampling MCMC simulation of canonical distribution. Detailed-balance acceptance criterion…
Partition function (Boltzmann)
Boltzmann 1877 + Gibbs 1902 Q = sum_i exp(-E_i/kT); molecular: Q = q_trans q_rot q_vib q_elec; thermodynamic functions from Q via -kT ln Q.
FEP (Zwanzig 1954)
R Zwanzig 1954: Delta-A = -kT ln <exp(-Delta-U/kT)>_0; basis of MD-FEP for binding-affinity + relative free-energies; modern alchemical…
Metropolis Monte Carlo (1953)
Metropolis-Rosenbluth 1953 acceptance min(1, exp(-Delta-U/kT)); detailed balance + ergodicity guarantee Boltzmann sampling.
OZ + closures (Hansen-McDonald)
Ornstein-Zernike 1914: h(r) = c(r) + rho integral c(|r-r'|) h(r') dr'; closures (HNC/PY/MSA) yield g(r); benchmark of liquid-state theory.
Kirkwood-Buff (1951)
Kirkwood-Buff 1951 fluctuation theory: G_ij = integral [g_ij(r) - 1] 4 pi r^2 dr; thermodynamic derivatives from MD-derived KB integrals;…
Jarzynski equality (1997)
C Jarzynski 1997 <exp(-W/kT)> = exp(-Delta-A/kT); free-energy from non-equilibrium pulling; tested in DNA single-molecule pulling Liphardt…
Boltzmann H-theorem (1872)
L Boltzmann 1872 H-theorem; modern modern foundational text + ergodic-hypothesis + Boltzmann-distribution + canonical ensemble.
Gibbs ensemble (1902)
J W Gibbs 1902 'Elementary Principles' canonical / grand-canonical / microcanonical; modern modern foundational text statistical mechanics.
Kramers (1940)
H A Kramers 1940 reaction-rate-theory friction-corrections; modern modern Pollak-Talkner 2005 + nonequilibrium SDE chemical-kinetics.
Metropolis MC (1953)
N Metropolis 1953 + Hastings 1970 Monte-Carlo; modern modern foundational text + replica-exchange + parallel-tempering chemistry.
MD (Alder-Wainwright 1957)
B Alder-T Wainwright 1957 first molecular-dynamics hard-sphere; modern modern AMBER + GROMACS + ML-force-fields ~10^9 atoms 2024.
Car-Parrinello (1985)
R Car-M Parrinello 1985 ab-initio MD; modern modern CP-MD + BO-MD + meta-dynamics-Laio-Parrinello 2002 free-energy sampling.