Boltzmann factor

In physics, the Boltzmann factor is a weighting factor that determines the relative probability of a state ${\displaystyle i}$ in a multi-state system in thermodynamic equilibrium at temperature ${\displaystyle T}$.

${\displaystyle e^{-{\frac {E_{i}}{k_{B}\,T}}}}$

Where ${\displaystyle k_{B}}$ is Boltzmann's constant, and ${\displaystyle E_{i}}$ is the energy of state ${\displaystyle i}$. The ratio of the probabilities of two states is given by the ratio of their Boltzmann factors.

The Boltzmann factor is not a probability by itself, because it is not normalized. To normalize the Boltzmann factor into a probability, one divides it by the sum Z of the Boltzmann factors of all possible states of a system, which is called the partition function. This gives the Boltzmann distribution.

From the Boltzmann factor it is possible to derive the Maxwell-Boltzmann statistics, Bose-Einstein statistics, and Fermi-Dirac statistics that govern classical particles as well as quantum mechanical bosons, and fermions, respectively.