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Boltzmann Distribution

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In physics, the '''Boltzmann distribution''' predicts the distribution function for the fractional number of particles ''N_i / N'' occupying a set of states ''i'' which each have energy ''E_i'': :

{{N_i}\over{N}} = {{g_i e^{-E_i/k_BT}}\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(-βE_i)'' or ''exp(-E_i/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} \frac{1}{2} \end{matrix}} mv^{2}

, then the distribution correctly gives the Maxwell-Boltzmann distribution of gas molecule speeds, previously predicted by James Clerk Maxwell 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} \frac{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 identical particles 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 sttistics.

Derivation
See Derivation of the partition function - first presented by Boltzmann in 1877. Category:Particle statistics Category:Statistical mechanics {{physics-stub}} pl:Rozkład Boltzmanna see Maxwell-Boltzmann distribution

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