the Boltzmann factor
/ BOLTS-mahn /
This is perhaps the single most useful exponential in physics. How much less likely is a state that costs an energy E? Less likely by the factor e^(-E / k_B T). Every additional chunk of energy k_B T in cost cuts the probability by a factor of e. It is why high-energy states are rare when it is cold and become common when it is hot.
In the canonical ensemble the relative probability of a microstate of energy E is proportional to e^(-E / k_B T) = e^(-beta E), with beta = 1/(k_B T). The ratio of probabilities of two microstates is therefore p_1 / p_2 = e^(-(E_1 - E_2)/k_B T). Summing the Boltzmann factor over all microstates yields the partition function Z that normalizes it. In the grand canonical ensemble it generalizes to the Gibbs factor e^(-(E - mu N)/k_B T).
It governs an astonishing range of phenomena: reaction rates through the Arrhenius factor e^(-E_a / k_B T), atmospheric density through the barometric law e^(-mgh / k_B T), semiconductor carrier concentrations, and spin populations. An honest subtlety: the Boltzmann factor weights a single microstate. To get the probability of an entire energy level of degeneracy g(E) you must multiply by that count, g(E) e^(-E / k_B T); degeneracy and Boltzmann weight together shape observed distributions such as the Maxwell-Boltzmann speeds.
At room temperature k_B T is about 1/40 eV, roughly 0.025 eV; a state 0.5 eV higher in energy is suppressed by e^(-0.5/0.025) = e^-20, about 2 x 10^-9.
A gap of twenty times k_B T makes a state a billion times rarer.
The bare Boltzmann factor is a weight for one microstate, not a probability; probabilities require dividing by Z, and a level's population also carries its degeneracy factor g(E).