Boltzmann factor

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

<math>e^{-\frac{E_i}{k_B\,T}}</math>

Where <math>k_B</math> is Boltzmann's constant, and <math>E_i</math> is the energy of state <math>i</math>. 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).