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…