It was first defined and used for describing particle speeds in idealized gases, where the particles move freely inside a stationary container without interacting with one another, except for very brief collisions in which they exchange energy and momentum with each other or with their thermal environment.

Mathematically, the Maxwell–Boltzmann distribution is the chi distribution with three degrees of freedom (the components of the velocity vector in Euclidean space), with a scale parameter measuring speeds in units proportional to the square root of

[2] The Maxwell–Boltzmann distribution is a result of the kinetic theory of gases, which provides a simplified explanation of many fundamental gaseous properties, including pressure and diffusion.

In real gases, there are various effects (e.g., van der Waals interactions, vortical flow, relativistic speed limits, and quantum exchange interactions) that can make their speed distribution different from the Maxwell–Boltzmann form.

This is also true for ideal plasmas, which are ionized gases of sufficiently low density.

[5][6] Boltzmann later, in the 1870s, carried out significant investigations into the physical origins of this distribution.

The distribution can be derived on the ground that it maximizes the entropy of the system.

A list of derivations are: For a system containing a large number of identical non-interacting, non-relativistic classical particles in thermodynamic equilibrium, the fraction of the particles within an infinitesimal element of the three-dimensional velocity space d 3v, centered on a velocity vector

, one can integrate over solid angle and write a probability distribution of speeds as the function[7]

This probability density function gives the probability, per unit speed, of finding the particle with a speed near v. This equation is simply the Maxwell–Boltzmann distribution (given in the infobox) with distribution parameter

With the Darwin–Fowler method of mean values, the Maxwell–Boltzmann distribution is obtained as an exact result.

The evolution of a system towards its equilibrium state is governed by the Boltzmann equation.

To the right is a molecular dynamics (MD) simulation in which 900 hard sphere particles are constrained to move in a rectangle.

[8] For diatomic nitrogen (N2, the primary component of air)[note 1] at room temperature (300 K), this gives In summary, the typical speeds are related as follows:

is the adiabatic index, f is the number of degrees of freedom of the individual gas molecule.

The Maxwell–Boltzmann distribution assumes that the velocities of individual particles are much less than the speed of light, i.e. that

The original derivation in 1860 by James Clerk Maxwell was an argument based on molecular collisions of the Kinetic theory of gases as well as certain symmetries in the speed distribution function; Maxwell also gave an early argument that these molecular collisions entail a tendency towards equilibrium.

[5][6][9] After Maxwell, Ludwig Boltzmann in 1872[10] also derived the distribution on mechanical grounds and argued that gases should over time tend toward this distribution, due to collisions (see H-theorem).

Maxwell–Boltzmann statistics gives the average number of particles found in a given single-particle microstate.

Under certain assumptions, the logarithm of the fraction of particles in a given microstate is linear in the ratio of the energy of that state to the temperature of the system: there are constants

All that is needed is to discover the density of microstates in energy, which is determined by dividing up momentum space into equal sized regions.

We may therefore rewrite Equation (1) as: where: This distribution of Ni : N is proportional to the probability density function fp for finding a molecule with these values of momentum components, so: The normalizing constant can be determined by recognizing that the probability of a molecule having some momentum must be 1.

Integrating the exponential in equation 4 over all px, py, and pz yields a factor of

The Maxwell–Boltzmann distribution for the momentum (or equally for the velocities) can be obtained more fundamentally using the H-theorem at equilibrium within the Kinetic theory of gases framework.

is the infinitesimal phase-space volume of momenta corresponding to the energy interval dE.

Making use of the spherical symmetry of the energy-momentum dispersion relation

Using the equipartition theorem, given that the energy is evenly distributed among all three degrees of freedom in equilibrium, we can also split

At equilibrium, this distribution will hold true for any number of degrees of freedom.

For example, if the particles are rigid mass dipoles of fixed dipole moment, they will have three translational degrees of freedom and two additional rotational degrees of freedom.

Integration of the probability density function of the velocity over the solid angles