In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables,
Here, N is the total number of particles and n is the number density of particles - the number of particles per unit volume, or the density divided by the mass of individual particles.
A distribution function may be specialised with respect to a particular set of dimensions. E.g. take the quantum mechanical six-dimensional phase space,
Particle distribution functions are often used in plasma physics to describe wave–particle interactions and velocity-space instabilities. Distribution functions are also used in fluid mechanics, statistical mechanics and nuclear physics.
The basic distribution function uses the Boltzmann constant
Related distribution functions may allow bulk fluid flow, in which case the velocity origin is shifted, so that the exponent's numerator is
Plasma theories such as magnetohydrodynamics may assume the particles to be in thermodynamic equilibrium. In this case, the distribution function is Maxwellian. This distribution function allows fluid flow and different temperatures in the directions parallel to, and perpendicular to, the local magnetic field. More complex distribution functions may also be used since plasmas are rarely in thermal equilibrium.
The mathematical analog of a distribution is a measure; the time evolution of a measure on a phase space is the topic of study in dynamical systems.