Boltzmann distribution

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 has energy E_i:

{{N_{i}} \over {N}}={{g_{i}e^{-E_{i}/k_{B}T}} \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_{B}T}.

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 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.

Then g(E) is called the density of states if the energy spectrum is continuous.

Classical particles with this energy distribution are said to obey Maxwell–Boltzmann statistics.

In the classical limit, i.e. at large values of E/kT or at small density of states—when wave functions of particles practically do not overlap, both the Bose–Einstein or Fermi–Dirac distribution become the Boltzmann distribution.

Derivation

See Maxwell–Boltzmann statistics.