# Boltzmann factor

In physics, the Boltzmann factor is a weighting factor determining the relative probability of a system in thermodynamic equilibrium at a temperature T being in a state with energy E:

[itex]e^{-\frac{E}{k_B T}}[itex]

(kB is Boltzmann's constant.) This is not a probability function by itself, because it is not normalized. However, the ratio of the probabilities of two states is given by the ratio of their Boltzmann factors. To normalize the Boltzmann factor into a probability, one simply divides it by the sum Z of the Boltzmann factors of all possible states of a system, which sum is called the partition function.

From the Boltzmann factor, one can then 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.

## References

• Charles Kittel and Herbert Kroemer, Thermal Physics, 2nd ed. (Freeman & Co.: New York, 1980).

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