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# Boltzmann factor

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

$e^{-\frac{E_i}{k_B\,T}}$

Where kB is Boltzmann's constant, and Ei is the energy of state 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.