Boltzmann factor
exp(−E/k_BT) — the exponential weight that sets how likely a state of energy E is at temperature T, and the engine of statistical mechanics.
Definition
The Boltzmann factor exp(−E/k_BT) gives the relative probability that a system in thermal contact with a reservoir at temperature T occupies a microstate of energy E. States of higher energy are exponentially less likely; the thermal energy scale k_BT sets how steeply the probability falls. Ratios of Boltzmann factors give the relative populations of two energy levels without needing the overall normalisation.
It is the single most reused expression in statistical physics. Summing Boltzmann factors over all states defines the partition function, from which every thermodynamic quantity can be derived. The factor governs the population of excited states, the rate of thermally activated processes, the barometric fall-off of atmospheric density with height, and the saturation of magnetic moments.
The Boltzmann factor is the energy-space generalisation of the Maxwell–Boltzmann speed distribution: where Maxwell weighted molecular speeds, Boltzmann weighted arbitrary microstate energies. It underlies the Arrhenius law of reaction rates and the exponential tail that makes chemistry, evaporation and escape from a gravitational well possible at all.
History
Introduced by Ludwig Boltzmann in the 1870s as the generalisation of Maxwell's speed distribution; central to the statistical mechanics developed by Boltzmann and later by Gibbs.