The most transparent definition of this standard comes from the Maxwell – 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.

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 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.

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.

If there are g ( E ) dE states with energy E to E + dE, then the Boltzmann distribution predicts a probability distribution for the energy:

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

In the classical limit, i. e. at large values of 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.

This effect can be illustrated with a Boltzmann distribution and energy profile diagram.

Fluorescence is most effective when there is a larger ratio of atoms at lower energy levels in a Boltzmann distribution.

If the group of atoms is in thermal equilibrium, it can be shown from thermodynamics that the ratio of the number of atoms in each state is given by a Boltzmann distribution:

As a result, the Boltzmann factor for states of systems at negative temperature increases rather than decreases with increasing state energy.

Much of the mathematics behind information theory with events of different probabilities was developed for the field of thermodynamics by Ludwig Boltzmann and J. Willard Gibbs.

where n < sub > i </ sub > stands for the number of particles occupying level i or the number of feasible microstates corresponding to macrostate i ; U < sub > i </ sub > stands for the energy of i ; T stands for temperature ; and k < sub > B </ sub > is the Boltzmann constant.

Yet, one must consider Boltzmann to be the " father " of statistical thermodynamics with his 1875 derivation of the relationship between entropy S and multiplicity Ω, the number of microscopic arrangements ( microstates ) producing the same macroscopic state ( macrostate ) for a particular system.

His professors in Vienna were von Escherich for mathematical analysis, Gegenbauer and Mertens for arithmetic and algebra, Weiss for astronomy, Stefan's student Boltzmann for physics.

The value of the Stefan – Boltzmann constant is derivable as well as experimentally determinable ; see Stefan – Boltzmann law for details.

The algorithm was named after Nicholas Metropolis, who was an author along with Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller of the 1953 paper Equation of State Calculations by Fast Computing Machines which first proposed the algorithm for the specific case of the Boltzmann distribution ; and W. Keith Hastings, who extended it to the more general case in 1970.

* 1872 – Ludwig Boltzmann states the Boltzmann equation for the temporal development of distribution functions in phase space, and publishes his H-theorem

* 1978 – Peter Goldreich and Scott Tremaine present a Boltzmann equation model of planetary-ring dynamics for indestructible spherical ring particles that do not self-gravitate and find a stability requirement relation between ring optical depth and particle normal restitution coefficient

Using the relaxation time approximation for the Boltzmann equation and assuming steady-state conditions, the phonon thermal conductivity λ < sub > L </ sub > can be determined.

Using the Boltzmann expression for the mean electron velocity given above with and setting the ion current to zero, the electron saturation current density would be

The behavior of a quantum Boltzmann gas is the same as that of a classical ideal gas except for the specification of these constants.

where is the Boltzmann constant, T is temperature ( assumed to be a well-defined quantity ), is the degeneracy ( meaning, the number of levels having energy ; 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.

The plasma pressure can be calculated by the state equation of a perfect gas, where is the particle number density, the Boltzmann constant and the plasma temperature.

Boltzmann showed that this definition of entropy was equivalent to the thermodynamic entropy to within a constant number which has since been known as Boltzmann's constant.

The history of quantum chemistry essentially began with the 1838 discovery of cathode rays by Michael Faraday, the 1859 statement of the black body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be discrete, and the 1900 quantum hypothesis by Max Planck that any energy radiating atomic system can theoretically be divided into a number of discrete energy elements ε such that each of these energy elements is proportional to the frequency ν with which they each individually radiate energy and a numerical value called Planck ’ s Constant.

where is the electron plasma frequency, is the number density of electrons and ions, is the classical radius of electron, is its mass, is the Boltzmann constant, and is the speed of light.

The results of the quantum Boltzmann gas are used in a number of cases including the Sackur-Tetrode equation for the entropy of an ideal gas and the Saha ionization equation for a weakly ionized plasma.

The expected number of particles with energy for Maxwell – Boltzmann statistics is where:

with ε < sub > i </ sub > > μ and where n < sub > i </ sub > is the number of particles in state i, g < sub > i </ sub > is the degeneracy of state i, ε < sub > i </ sub > is the energy of the ith state, μ is the chemical potential, k is the Boltzmann constant, and T is absolute temperature.

Following the same procedure used in deriving the Maxwell – Boltzmann statistics, we wish to find the set of for which W is maximised, subject to the constraint that there be a fixed total number of particles, and a fixed total energy.

where S is entropy, k < sub > B </ sub > is the Boltzmann constant, and is the number of microstates consistent with the macroscopic configuration.

Since at temperature T the molecules have energies given by a Boltzmann distribution, one can expect the number of collisions with energy greater than E < sub > a </ sub > to be proportional to.

The basic distribution function uses the Boltzmann constant and temperature with the number density to modify the normal distribution:

From 1872 to 1875, Ludwig Boltzmann reinforced the statistical explanation of this paradox in the form of Boltzmann's entropy formula stating that as the number of possible microstates a system might be in increases, the entropy of the system increases and it becomes less likely that the system will return to an earlier state.

It led Boltzmann to his statistical concept of entropy as a logarithmic tally of the number of microscopic states corresponding to a given thermodynamic state.

It is also possible to write down relativistic Boltzmann equations for systems in which a number of particle species can collide and produce different species.

The Boltzmann distribution describes a system that can exchange energy with a heat bath ( or alternatively with a large number of similar systems ) so that its temperature remains constant.

However, due to a number of issues discussed below, Boltzmann machines with unconstrained connectivity have not proven useful for practical problems in machine learning or inference.

In a plasma, the Boltzmann relation describes the number density of an isothermal charged particle fluid when the thermal and the electrostatic forces acting on the fluid have reached equilibrium.

where n < sub > e </ sub > is the electron number density, T < sub > e </ sub > the temperature of the plasma, and k < sub > B </ sub > is Boltzmann constant.