The Boltzmann distribution is a distribution function or probability measure for the distribution of the states of a system. It underpins the concept of the canonical ensemble, providing the underlying distribution. There is a special case of the Boltzman distribution used for describing the velocities of particles of a gas. It is called the Maxwell-Boltzmann distribution but is commonly referred to as the Gibbs measure.
The Boltzmann distribution for the fractional number is
Ni/N = (gie-Ei/(kBT))/E(T)
Where kb is the Boltzmann constant, T is temperature, gi is the degeneracy, N is the total number of particles and Z(T) is the partition function.
The Boltzmann distribution was discovered in the context of classical statistical mechanics by J.W. Gibbs in 1901. It underpins the concept of the canonical ensemble.
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