Maxwell–Boltzmann distribution

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. The term "particle" in this context refers to gaseous particles only (atoms or molecules), and the system of particles is assumed to have reached thermodynamic equilibrium.^[1] The energies of such particles follow what is known as Maxwell–Boltzmann statistics, and the statistical distribution of speeds is derived by equating particle energies with kinetic energy.


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 ${\displaystyle T/m}$ (the ratio of temperature and particle mass).^[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.^[3] The Maxwell–Boltzmann distribution applies fundamentally to particle velocities in three dimensions, but turns out to depend only on the speed (the magnitude of the velocity) of the particles. A particle speed probability distribution indicates which speeds are more likely: a randomly chosen particle will have a speed selected randomly from the distribution, and is more likely to be within one range of speeds than another. The kinetic theory of gases applies to the classical ideal gas, which is an idealization of real gases. 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. However, rarefied gases at ordinary temperatures behave very nearly like an ideal gas and the Maxwell speed distribution is an excellent approximation for such gases. This is also true for ideal plasmas, which are ionized gases of sufficiently low density.^[4]


The distribution was first derived by Maxwell in 1860 on heuristic grounds.^[5] 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 d³v, centered on a velocity vector ${\displaystyle {\vec {v}}}$ of magnitude ${\displaystyle v}$, is given by


One can write the element of velocity space as ${\displaystyle d^{3}v=dv_{x}\,dv_{y}\,dv_{z}}$, for velocities in a standard Cartesian coordinate system, or as ${\displaystyle d^{3}v=v^{2}\,dv\,d\Omega }$ in a standard spherical coordinate system, where ${\displaystyle d\Omega =\sin {\theta }d\phi d\theta }$ is an element of solid angle and ${\displaystyle v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}$


The Maxwellian distribution function for particles moving in only one direction, if this direction is x, is


Recognizing the symmetry of ${\displaystyle f(v)}$, one can integrate over solid angle and write a probability distribution of speeds as the function^[6]


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 ${\textstyle a={\sqrt {kT/m}}\,.}$ The Maxwell–Boltzmann distribution is equivalent to the chi distribution with three degrees of freedom and scale parameter ${\textstyle a={\sqrt {kT/m}}\,.}$


The simplest ordinary differential equation satisfied by the distribution is:


or in unitless presentation: