Boltzmann distribution, the statistically equilibrium distribution function over the momenta p and the coordinates r of the particles of an ideal gas whose molecules move according to the laws of classical mechanics, in an external potential field:
Here p2/2m — the kinetic energy of a molecule of mass m, U(ν) — its potential energy in the external field, T is the absolute temperature of the gas. The constant A is determined from the condition that the total number of particles in the various possible states is equal to the full number of particles in the system (the normalisation condition).
The Boltzmann distribution is a particular case of the canonical Gibbs distribution for an ideal gas in an external potential field, since in the absence of interaction between the particles the Gibbs distribution breaks up into a product of Boltzmann distributions for the individual particles. At U=0 the Boltzmann distribution gives the Maxwell distribution. The distribution function (1) is sometimes called the Maxwell — Boltzmann distribution, while the Boltzmann distribution is taken to be the distribution function (1) integrated over all the momenta of the particles, which represents the number density of particles at the point ν:
The Boltzmann distribution is a particular case of the canonical Gibbs distribution for an ideal gas in an external potential field, since in the absence of interaction between the particles the Gibbs distribution breaks up into a product of Boltzmann distributions for the individual particles. At U=0 the Boltzmann distribution gives the Maxwell distribution. The distribution function (1) is sometimes called the Maxwell — Boltzmann distribution, while the Boltzmann distribution is taken to be the distribution function (1) integrated over all the momenta of the particles, which represents the number density of particles at the point ν:
where n0 is the number density of the particles of the system in the absence of an external field. The ratio of the number densities of particles at different points depends on the difference between the values of the potential energy at those points
where ΔU= U(ν1)-U(ν2). In particular, the barometric formula follows from (3), which gives the distribution of a gas with height in the gravitational field above the earth's surface. In this case ΔU=mgh, where g is the acceleration due to gravity, m is the mass of a particle, h is the height above the earth's surface. For a mixture of gases with particles of different mass the Boltzmann distribution shows that the distribution of the partial densities of the particles of each component is independent of the other components. For a gas in a rotating vessel U (r) defines the potential of the field of centrifugal forces U (r)=-mω2r2/2, where ω is the angular velocity of rotation. The separation of isotopes and of highly disperse systems by means of an ultracentrifuge is based on this effect.
For quantum ideal gases the state of the individual particles is determined not by momenta and coordinates but by the quantum energy levels Εi of a particle in the field U(r). In this case the mean number of particles in the i-th quantum state, or the mean occupation number, is equal to:
For quantum ideal gases the state of the individual particles is determined not by momenta and coordinates but by the quantum energy levels Εi of a particle in the field U(r). In this case the mean number of particles in the i-th quantum state, or the mean occupation number, is equal to:
where μ is the chemical potential, determined from the condition that the total number of particles on all the quantum levels Εi is equal to the full number of particles N in the system: Σini=N. Formula (4) holds at those temperatures and densities for which the mean distance between the particles is considerably greater than the de Broglie wavelength corresponding to the mean thermal velocity, that is, when not only the force interaction of the particles can be neglected but also their mutual quantum-mechanical influence (there is no quantum degeneracy of the gas. (see Degenerate gas). Thus the Boltzmann distribution is the limiting case both of the Fermi — Dirac distribution and of the Bose — Einstein distribution for gases of low density.