Distribution function (physics): Difference between revisions

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{{Short description|Function of seven variables}}

{{Unreferenced|date=December 2009}}

:''This article describes {{about|the ''distribution function'' as used in physics. You may be looking for |the related mathematical concepts of [[|cumulative distribution function]] or [[|and|probability density function]].''}}

In molecular [[kinetic theory of gases|kinetic theory]] in [[physics]], a system's '''distribution function''' is a function of seven variables, <math>f(x,y,z,t; v_x,v_y,v_z)</math>, 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]] <math>\mathbf{v}=(v_x,v_y,v_z)</math> near the position <math>\mathbf{r}=(x,y,z)</math> and time <math>t</math>. The usual normalization of the distribution function is

:<math>n(x,y,z,t) = \int f \,dv_x \,dv_y \,dv_z,</math>

:<math>N(t) = \int n \,dx \,dy \,dz, </math>

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, <math>f(x,y,z;p_x,p_y,p_z)</math> 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]] <math>(p_x,p_y,p_z)</math>.