Boltzmann distribution

In chemistry, physics, and mathematics, the Boltzmann distribution (also called the Gibbs Distribution) is a certain distribution function or probability measure for the distribution of the states of a system. The distribution was discovered in the context of classical statistical mechanics by J.W. Gibbs in 1901. It underpins the concept of the canonical ensemble, providing the 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. In statistics and machine learning it is called a log-linear model.

Definition

The Boltzmann distribution for the fractional number of particles N_i / N occupying a set of states i possessing energy E_i is:

${N_i \over N} = {g_i e^{-E_i/(k_BT)} \over Z(T)}$

where $k_B$ is the Boltzmann constant, T is temperature (assumed to be a well-defined quantity), $g_i$ is the degeneracy (meaning, the number of levels having energy $E_i$; sometimes, the more general 'states' are used instead of levels, to avoid using degeneracy in the equation), N is the total number of particles and Z(T) is the partition function.

$N=\sum_i N_i,$

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

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 $e^{-\beta E_i}$ or $e^{-E_i/(kT)}$, 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:

$p(E)\,dE = {g(E) e^{-\beta E} \over \int g(E') e^{-\beta E'}\,dE'}\, dE.$

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.

Inverted Boltzmann distribution

In January 2013, German scientists reported having achieved an "inverted Boltzmann distribution" with the ultracooling of atomic gas, creating negative absolute temperature. The experiments may shed light on the nature of dark energy, and indicate that a 100 percent energy-efficient internal combustion engine, which had previously been considered impossible, might in fact be achievable.

See also

• Boltzmann factor
• Gibbs measure
• Partition function (mathematics)
• Log-linear model

External links

• Derivation of the distribution for microstates of a system