Boltzmann distribution

The Boltzmann distribution gives the probability that a system in thermal equilibrium at temperature T occupies a particular microstate of energy E_i: p_i = exp(−E_i/k_BT)/Z. Equivalently, the ratio of probabilities of two states depends only on their energy difference, p_i/p_j = exp(−(E_i − E_j)/k_BT). A state costing one k_BT more in energy is a factor e ≈ 2.7 less likely; one ten k_BT higher is rarer by some twenty-two thousand.

The distribution follows from a single assumption — that a large isolated system explores all its microstates with equal probability — applied to a small subsystem in contact with a much larger reservoir. Expanding the reservoir's entropy to first order in the subsystem's energy produces the exponential weight, with the inverse temperature β = 1/k_BT appearing as the expansion coefficient.

It is the foundation on which the partition function, the Maxwell–Boltzmann speed distribution, the Arrhenius law of reaction rates, and the barometric formula all rest. Wherever a system can be modelled as a set of energy levels in contact with a heat bath, their populations follow the Boltzmann distribution.

Developed by Ludwig Boltzmann in the 1860s–1870s and given its general statistical-mechanical setting by Gibbs in 1902. It generalises Maxwell's 1860 velocity distribution from kinetic energies to arbitrary microstate energies.