Boltzmann distribution

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In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution^[1]) is a probability distribution, probability measure, or frequency distribution over various possible states of a system, with the form

$F({{\rm {state}}})\propto e^{{-{\frac {E}{kT}}}}$

where $E$ is state energy (which varies from state to state), and $kT$ (a constant of the distribution) is the product of Boltzmann's constant and thermodynamic temperature.

The ratio of a Boltzmann distribution computed for two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference.

${\frac {F({{\rm {state1}}})}{F({{\rm {state2}}})}}=e^{{{\frac {E_{2}-E_{1}}{kT}}}}.$

The Boltzmann distribution is named after Ludwig Boltzmann who first formulated it in 1868 during his studies of the statistical mechanics of gases in thermal equilibrium.^{[citation needed]} The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902.^[2]^:Ch.IV

In statistical mechanics

Main articles: Canonical ensemble and Maxwell–Boltzmann statistics

The Boltzmann distribution appears in statistical mechanics when considering isolated (or nearly-isolated) systems of fixed composition that are in thermal equilibrium (equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble, but also some special cases (derivable from the canonical ensemble) also show the Boltzmann distribution in different aspects:

Canonical ensemble (general case): The canonical ensemble gives the probabilities of the various possible states of an isolated system of fixed composition, in thermal equilibrium with a heat bath. The canonical ensemble is a probability distribution with the Boltzmann form.
Statistical frequencies of subsystems' states (in a non-interacting collection): When the system of interest is a collection of many non-interacting copies of a smaller subsystem, it is sometimes useful to find the statistical frequency of a given subsystem state, among the collection. The canonical ensemble has the property of separability when applied to such a collection: as long as the non-interacting subsystems have fixed composition, then each subsystem's state is independent of the others and is also characterized by a canonical ensemble. As a result, the expected statistical frequency distribution of subsystem states has the Boltzmann form.
Maxwell–Boltzmann statistics of classical gases (systems of non-interacting particles): In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space. Maxwell–Boltzmann statistics give the expected number of particles found in a given single-particle state, in a classical gas of non-interacting particles at equilibrium. This expected number distribution has the Boltzmann form.

Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed:

When a system is in thermodynamic equilibrium with respect to both energy exchange and particle exchange, the requirement of fixed composition is relaxed and a grand canonical ensemble is obtained rather than canonical ensemble. On the other hand if both composition and energy are fixed, then a microcanonical ensemble applies instead.
If the subsystems within a collection do interact with each other, then the expected frequencies of subsystem states no longer follow a Boltzmann distribution, and even may not have an analytical solution.^[3] The canonical ensemble can however still be applied to the collective states of the entire system considered as a whole, provided the entire system is isolated and in thermal equilibrium.
With quantum gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow Maxwell–Boltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the grand canonical ensemble the state-filling statistics of quantum gases are described by Fermi–Dirac statistics or Bose–Einstein statistics, depending on whether the particles are fermions or bosons respectively.

In mathematics

Main articles: Gibbs measure and Log-linear model

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.

References

↑ Landau, Lev Davidovich; and Lifshitz, Evgeny Mikhailovich (1980) [1976]. Statistical Physics. Course of Theoretical Physics 5 (3 ed.). Oxford: Pergamon Press. ISBN 0-7506-3372-7. Translated by J.B. Sykes and M.J. Kearsley. See section 28
↑ Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons.
↑ A classic example of this is magnetic ordering. Systems of non-interacting spins show paramagnetic behaviour that can be understood with a single-particle canonical ensemble (resulting in the Brillouin function). Systems of interacting spins can show much more complex behaviour such as ferromagnetism or antiferromagnetism.

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