Maxwell–Boltzmann distribution
The probability distribution of molecular speeds in a gas at thermal equilibrium — the first statistical law in the history of physics.
Definition
The Maxwell–Boltzmann distribution gives the probability density of molecular speeds in a classical gas at temperature T: f(v) = 4π (m/2πk_BT)^(3/2) v² exp(−mv²/2k_BT). It is the product of two competing factors — a v² term from the growing volume of velocity space available at higher speed, and a Gaussian exp(−mv²/2k_BT) that suppresses very fast molecules. The result is a skewed bell shape that rises from zero, peaks, and trails off in a long high-speed tail.
The distribution is fixed by three characteristic speeds, always in the order v_mp < ⟨v⟩ < v_rms: the most-probable speed √(2k_BT/m) at the peak, the mean speed √(8k_BT/πm), and the root-mean-square speed √(3k_BT/m). Raising the temperature or lowering the molecular mass shifts the whole curve to higher speeds and flattens it.
Maxwell derived it in 1859 from a symmetry argument — that the three velocity components are independent and identically distributed — making it the first probability distribution introduced into physics. Boltzmann generalised it in 1871 to the distribution of energies, exp(−E/k_BT), the foundation of statistical mechanics.
History
Derived by James Clerk Maxwell in 1859 and generalised by Ludwig Boltzmann in 1871; verified directly by molecular-beam experiments, notably Otto Stern's rotating-disc method in the 1920s and Zartman and Ko in 1930–31.