Boltzmann distribution

In classical statistical physics, the Boltzmann distribution expresses the relative probability that a subsystem of a physical system has a certain energy. The subsystem must part of a physical system that is in thermal equilibrium, that is, the system must have a well-defined (absolute) temperature. For instance, a single molecule can be a subsystem of, say, one mole of ideal gas and the Boltzmann distribution applies to the energies of the individual gas molecules, provided the total amount of the ideal gas is in thermal equilibrium.

The Boltzmann distribution (also known as the Maxwell-Boltzmann distribution) was proposed in 1859 by the Scotsman James Clerk Maxwell for the statistical distribution of the kinetic energies of ideal gas molecules. The Maxwell-Boltzmann law can be formulated as the ratio of two numbers. Consider an ideal gas of temperature T. Let n₁ be the number of molecules with energy E₁ and n₂ be the number with energy E₂, then according to the Maxwell-Boltzmann distribution law,

${\frac {n_{1}}{n_{2}}}={\frac {e^{-E_{1}/(kT)}}{e^{-E_{2}/(kT)}}},\qquad \qquad \qquad \qquad (1)$

where k is the Boltzmann constant. Most noticeable in this expression are (i) the energy in an exponential, (ii) the inverse temperature in the exponent, and (iii) the appearance of the natural constant k. Note that an argument of an exponential must be dimensionless and that accordingly kT has the dimension energy.

The molecular energies may, in addition to kinetic energies, contain interactions with an external field. For instance, if a system, as for instance a column of air, is in the gravitational field of the Earth, each molecular energy may contain the additional term mgh, where m is the molecular mass, g the gravitational acceleration and h the height of the molecule above the surface of the Earth.

Generalization

Maxwell's finding was generalized in 1871 by the Austrian Ludwig Boltzmann, who gave the energy distribution of gas molecules having both kinetic and potential energies.

A few years later (ca. 1877) the American Josiah Willard Gibbs gave a further formalization and generalization. Gibbs introduced an ensemble consisting of a statistically large number of identical subsystems. For instance, the subsystems may be identical vessels containing the same number N of the same real gas molecules at the same temperature and pressure. Further Gibbs assumed that the ensemble, just like a system of gas molecules, is in thermal equilibrium, i.e., the subsystems are in thermal contact so that they can exchange heat. For example, the subsystems may be gas-filled vessels with heat-conducting walls.

Gibbs assumed that the Maxwell-Boltzmann law, equation (1), holds for the energies of the subsystems in the ensemble. In this generalization—and the example of an ensemble consisting of gas-filled vessels—the energy of a single gas molecule is replaced by the energy of the N molecules in a single vessel; a one-molecule energy is promoted to a one-vessel energy. Because of molecular interactions (that are absent in an ideal gas), a one-vessel energy cannot be written as a sum of one-molecule energies. This is the main reason for Gibbs' generalization of a single vessel of gas to an ensemble of vessels, or, in more general terms, the generalization from an ideal gas to an ensemble of (rather arbitrary) subsystems. Like the molecules in an ideal gas, the subsystems do not interact other than by exchanging heat.

The absolute probability for a subsystem to have total energy E_k can be obtained from the relative probability [equation (1)] by normalizing the probability distribution. Let us assume, for convenience sake, that the one-subsystem energies are discrete (as they often are in quantum mechanics), then

${\mathcal {P}}(E_{k})={\frac {e^{-E_{k}/(kT)}}{Q}}\quad {\hbox{with}}\quad Q\equiv \sum _{i=0}^{\infty }e^{-E_{i}/(kT)}.$

This probability can be written as

${\mathcal {P}}(E_{k})={\frac {M_{k}}{M}}$

where M_k is the number of subsystems that have energy E_k and M is the total number of subsystems in the ensemble (M must be large, say on the order of Avogadro's number for statistics to apply).

The quantity Q is known as the partition function of the subsystem. It is a sum over all energies that the subsystem can have. In the older literature Q is called Zustandssumme, which is German for "sum over states", and is often denoted by Z. When we consider again as an example a vessel with N interacting molecules at certain pressure and temperature, the sum is over the possible total energies of all N molecules in the vessel.

In classical statistical physics, where energies are not discrete, the partition function of a system of N molecules is an integral over the 6^N-dimensional phase space (space of momenta and positions). In the framework of the "old quantum theory" it was was around 1913 discovered that the classical partition function of N molecules must multiplied by a quantum factor. The classical partition function is,

$Q_{\mathrm {class} }={\frac {1}{N!h^{3N}}}\int E^{-E(p,q)/(kT)}\mathrm {d} \mathbf {q} _{1}\cdots \mathrm {d} \mathbf {q} _{N}\,\mathrm {d} \mathbf {p} _{1}\cdots \mathrm {d} \mathbf {p} _{N},$

where h is Planck's constant and N! = 1 × 2× ... × N (the factorial of N).