# Boltzmann factor

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

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

## See also

- Boltzmann relation (plasma physics)

## References

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