# Boltzmann distribution

Boltzmann distribution

In physics and mathematics, the Boltzmann distribution is a certain distribution function or probability measure for the distribution of the states of a system. It underpins the concept of the canonical ensemble, providing its underlying distribution. A special case of the Boltzmann distribution, used for describing the velocities of particles of a gas, is the Maxwell-Boltzmann distribution. In more general mathematical settings, the Boltzmann distribution is also known as the Gibbs measure.

The Boltzmann distribution for the fractional number of particles "N""i" / "N" occupying a set of states "i" which each respectively possess energy "Ei":

: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

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

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 or $e^\left\{-E_i/\left(kT\right)\right\}$, 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

:

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 . 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:

:

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 $E/\left(kT\right)$ or at small density of states &mdash; when wave functions of particles practically do not overlap &mdash; both the Bose–Einstein or Fermi–Dirac distribution become the Boltzmann distribution.

Derivation

See Maxwell–Boltzmann statistics.

ee also

* Partition function (mathematics)

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Boltzmann distribution — Boltzmann distribution, an equation that expresses the distribution of energies for a collection of particles at a fixed temperature, based on the Boltzmann constant …   Useful english dictionary

• Boltzmann distribution — Bolcmano skirstinys statusas T sritis fizika atitikmenys: angl. Boltzmann distribution; Boltzmann distribution law vok. Boltzmannsche Verteilung, f; Boltzmannsches Verteilungsgesetz, n; Boltzmann Verteilung, f rus. больцмановское распределение,… …   Fizikos terminų žodynas

• Boltzmann distribution law — Bolcmano skirstinys statusas T sritis fizika atitikmenys: angl. Boltzmann distribution; Boltzmann distribution law vok. Boltzmannsche Verteilung, f; Boltzmannsches Verteilungsgesetz, n; Boltzmann Verteilung, f rus. больцмановское распределение,… …   Fizikos terminų žodynas

• Boltzmann distribution law — Bolcmano dėsnis statusas T sritis Standartizacija ir metrologija apibrėžtis Apibrėžtį žr. priede. priedas( ai) Grafinis formatas atitikmenys: angl. Boltzmann distribution law vok. Boltzmannsches Verteilungsgesetz, n rus. закон Больцмана, m; закон …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

• Boltzmann distribution law — noun (physics) a law expressing the distribution of energy among the molecules of a gas in thermal equilibrium • Syn: ↑Maxwell Boltzmann distribution law • Topics: ↑physics, ↑natural philosophy • Hypernyms: ↑distrib …   Useful english dictionary

• Maxwell–Boltzmann distribution — Maxwell–Boltzmann Probability density function Cumulative distribution function parameters …   Wikipedia

• Maxwell-Boltzmann distribution law — noun (physics) a law expressing the distribution of energy among the molecules of a gas in thermal equilibrium • Syn: ↑Boltzmann distribution law • Topics: ↑physics, ↑natural philosophy • Hypernyms: ↑distribution law …   Useful english dictionary

• Distribution De Boltzmann — En physique, la distribution de Boltzmann prédit la fonction de distribution pour le nombre fractionnaire de particules Ni / N occupant un ensemble d états i qui ont chacun pour énergie Ei : où kB est la constante de Boltzmann, T est la… …   Wikipédia en Français

• Distribution de boltzmann — En physique, la distribution de Boltzmann prédit la fonction de distribution pour le nombre fractionnaire de particules Ni / N occupant un ensemble d états i qui ont chacun pour énergie Ei : où kB est la constante de Boltzmann, T est la… …   Wikipédia en Français

• Maxwell-Boltzmann distribution law — ▪ chemistry       a description of the statistical distribution of the energies of the molecules of a classical gas. This distribution was first set forth by the Scottish physicist James Clerk Maxwell in 1859, on the basis of probabilistic… …   Universalium