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: = \over{Z(T)}}

where k_B is the Boltzmann Constant , ''T'' is temperature (assumed to be a sharply well-defined quantity), g_i is the degeneracy, or number of states having energy E_i, ''N'' is the total number of particles:

:N=\sum_i N_i\,

and ''Z(T)'' is called the Partition Function , which can be seen to be equal to

:Z(T)=\sum_i g_i e^{-E_i/k_BT}

Alternatively, for a single system at a well-defined temperature, it gives the probability that the system is in the specified state. The Boltzmann distribution applies only to particles at a high enough temperature and low enough density that quantum effects can be ignored, and the particles are obeying Maxwell-Boltzmann Statistics . (See that article for a derivation of the Boltzmann distribution.)

The Boltzmann distribution is often expressed in terms of β=''1/kT'' where β is referred to as Thermodynamic Beta . The term ''exp(-βEi)'' or ''exp(-Ei/kT)'', which gives the (unnormalised) relative probability of a state, is called the Boltzmann Factor and appears often in the study of physics and chemistry.

When the energy is simply the kinetic energy of the particle

:E_i = {\begin{matrix} rac{1}{2} \end{matrix}} mv^{2},

then the distribution correctly gives the Maxwell-Boltzmann Distribution of gas molecule speeds, previously predicted by Maxwell in 1859 . The Boltzmann distribution is, however, much more general. For example, it also predicts the variation of the particle density in a gravitational field with height, if E_i = {\begin{matrix} rac{1}{2} \end{matrix}} mv^{2} + mgh. In fact the distribution applies whenever quantum considerations can be ignored.

In some cases, a continuum approximation can be used. If there are ''g(E)dE'' states with energy ''E'' to ''E+dE'', then the Boltzmann distribution predicts a probability distribution for the energy:

:p(E)dE = {g(E) \exp({-\beta E})\over {\int g(E') \exp {(-\beta E')}}dE'} dE

''g(E)'' is then called the Density Of States if the energy spectrum is continuous.

Classical particles with this energy distribution are said to obey Maxwell-Boltzmann Statistics .

For quantum particles, quantum Indistinguishability must be taken into account, giving corresponding Bose-Einstein Statistics for Bosons , and Fermi-Dirac Statistics for Fermions .

Boltzmann distribution then follows from either Bose-Einstein or Fermi-Dirac distribution at large values of E/kT (or at small Density Of States - when wave functions of particles practically do not overlap). So Boltzmann distribution can be considered as a classical limit of Quantum Statistics .


DERIVATION


See Derivation Of The Partition Function - first presented by Boltzmann in 1877 .