In molecular kinetic theory in physics, a particle's **distribution function** is a function of seven variables, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time . The usual normalization of the distribution function is

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, and multiply by the total space volume, to give the momentum distribution i.e. the number of particles in the momentum phase space having approximately the momentum .

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 and temperature with the number density to modify the normal distribution:

Related distribution functions may allow bulk fluid flow, in which case the velocity origin is shifted, so that the exponent's numerator is ; is the bulk velocity of the fluid. Distribution functions may also feature non-isotropic temperatures, in which each term in the exponent is divided by a different temperature.

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.

### Famous quotes containing the words distribution and/or function:

“Classical and romantic: private language of a family quarrel, a dead dispute over the *distribution* of emphasis between man and nature.”

—Cyril Connolly (1903–1974)

“The information links are like nerves that pervade and help to animate the human organism. The sensors and monitors are analogous to the human senses that put us in touch with the world. Data bases correspond to memory; the information processors perform the *function* of human reasoning and comprehension. Once the postmodern infrastructure is reasonably integrated, it will greatly exceed human intelligence in reach, acuity, capacity, and precision.”

—Albert Borgman, U.S. educator, author. Crossing the Postmodern Divide, ch. 4, University of Chicago Press (1992)