Maxwell-Boltzmann
A box of N hard-sphere particles, all with the same initial speed. They collide elastically — energy preserved, momentum redistributed. The speed distribution starts as a delta function. Within seconds, collisions thermalize the gas to the Maxwell-Boltzmann distribution: f(v) ∝ v² exp(-mv²/(2k_BT)). Maxwell 1860, derived from symmetry alone. Boltzmann's H-theorem 1872 explains why this happens irreversibly.
Speed distribution (red = MB theoretical)
how a delta function becomes a smooth distribution irreversibly
The simulation starts with every particle at the same speed (a delta function in speed). Collisions are elastic — they preserve total energy. Yet the distribution rapidly becomes Maxwell-Boltzmann:
f(v) = 4π n (m / (2π k_B T))^(3/2) v² exp(-m v² / (2 k_B T))
Why this happens: collisions randomize relative velocities. After many collisions, the distribution becomes the unique one that maximizes entropy subject to fixed total energy. This is the equilibrium distribution of an ideal gas in 3D (in 2D, the formula is slightly different — see below).
Boltzmann's H-theorem (1872): define H(t) = ∫ f(v, t) log f(v, t) dv. Boltzmann showed that under his collision equation, dH/dt ≤ 0 always — H decreases monotonically toward equilibrium. This was the first derivation of the second law of thermodynamics from microscopic dynamics, and one of the most controversial results in 19th-century physics. (Loschmidt's paradox: how does irreversible H decrease emerge from time-reversible Newtonian collisions? Answer: H decreases for "typical" initial conditions; very rare reversed configurations are statistically negligible.)
Why this matters:
• Equilibrium: the equilibrium of any classical interacting system with conserved energy is Maxwell-Boltzmann. Earth's atmosphere, your blood, the interstellar gas, a cup of coffee — all distribute speeds this way.
• Temperature defined: T is defined by the variance of MB: ⟨v²⟩ = 3k_BT/m. The "average kinetic energy = (3/2)k_BT per particle" follows directly.
• Diffusion / Brownian motion: small particles in a fluid get kicked by MB-distributed molecules. The mean-square displacement grows linearly with time — diffusion.
• Chemical reactions: only the high-velocity tail of MB has enough energy to react. Arrhenius rate k ∝ exp(-E_a / (k_BT)).
• Speed of sound: equals roughly the most-probable molecular speed (≈ 480 m/s in air at 20°C).