# Distribution function

VIDEOS 1 TO 50

GO TO RESULTS [1 .. 50]

### WIKIPEDIA ARTICLE

This article describes the distribution function as used in physics. You may be looking for the related mathematical concepts of cumulative distribution function or probability density function.

In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, ${\displaystyle f(x,y,z,t;v_{x},v_{y},v_{z})}$, which gives the number of particles per unit volume in single-particle phase space. It is the number of particles per unit volume having approximately the velocity ${\displaystyle (v_{x},v_{y},v_{z})}$ near the place ${\displaystyle (x,y,z)}$ and time ${\displaystyle (t)}$. The usual normalization of the distribution function is

${\displaystyle n(x,y,z,t)=\int f\,dv_{x}\,dv_{y}\,dv_{z},}$
${\displaystyle N(t)=\int n\,dx\,dy\,dz,}$

where, 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, ${\displaystyle f(x,y,z;p_{x},p_{y},p_{z})}$ 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 ${\displaystyle (p_{x},p_{y},p_{z})}$.

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 ${\displaystyle k}$ and temperature ${\displaystyle T}$ with the number density to modify the normal distribution:

${\displaystyle f=n\left({\frac {m}{2\pi kT}}\right)^{3/2}\exp \left({-{\frac {m(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}{2kT}}}\right).}$

Related distribution functions may allow bulk fluid flow, in which case the velocity origin is shifted, so that the exponent's numerator is ${\displaystyle m((v_{x}-u_{x})^{2}+(v_{y}-u_{y})^{2}+(v_{z}-u_{z})^{2})}$, where ${\displaystyle (u_{x},u_{y},u_{z})}$ 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.

### Disclaimer

None of the audio/visual content is hosted on this site. All media is embedded from other sites such as GoogleVideo, Wikipedia, YouTube etc. Therefore, this site has no control over the copyright issues of the streaming media.

All issues concerning copyright violations should be aimed at the sites hosting the material. This site does not host any of the streaming media and the owner has not uploaded any of the material to the video hosting servers. Anyone can find the same content on Google Video or YouTube by themselves.

The owner of this site cannot know which documentaries are in public domain, which has been uploaded to e.g. YouTube by the owner and which has been uploaded without permission. The copyright owner must contact the source if he wants his material off the Internet completely.