Kinetic theory & equipartition

Pressure from collisions, Maxwell–Boltzmann, equipartition, mean free path, transport.

A gas is a swarm of molecules in ceaseless motion, and its macroscopic state variables are statistical summaries of that motion. Kinetic theory is the bridge: pressure from the rate of momentum delivery, temperature from the mean molecular kinetic energy, the heat capacity from a count of degrees of freedom, the speed distribution from statistical equilibrium, and the transport of momentum, heat, and matter from the distance a molecule travels between collisions.

• 2.1 The molecular picture and pressure — pressure as molecular momentum flux, $p = \tfrac13 n m\langle v^2\rangle$, and the identification of $\langle\varepsilon\rangle = \tfrac32 k_B T$.
• 2.2 Temperature and equipartition — the equipartition theorem ($\tfrac12 k_B T$ per quadratic degree of freedom), heat capacity, and the ratio of specific heats $\gamma$.
• 2.3 The Maxwell–Boltzmann distribution — the equilibrium speed distribution and its most-probable, mean, and rms speeds.
• 2.4 The Boltzmann factor and thermal activation — $e^{-E/k_B T}$, two-level populations, the barometric formula, and the Arrhenius law.
• 2.5 Collisions and the mean free path — the collision cross-section, $\ell = 1/(\sqrt2\, n\pi d^2)$, and the Knudsen number.
• 2.6 Transport from kinetic theory — viscosity, conduction, and diffusion from the mean free path; the pressure-independence of viscosity; Brownian motion.