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{{about|the distribution function as used in physics|the related mathematical concepts|cumulative distribution function|and|probability density function}}
{{Unreferenced|date=December 2009}}
In molecular [[kinetic theory of gases|kinetic theory]] in [[physics]], a system's '''distribution function''' is a function of seven variables, <math>f(t, x,y,z, v_x,v_y,v_z)</math>, which gives the number of particles per unit volume in single-particle [[phase space]].<ref name="m713">{{cite journal | last=Hillery | first=M. | last2=O'Connell | first2=R.F. | last3=Scully | first3=M.O. | last4=Wigner | first4=E.P. | title=Distribution functions in physics: Fundamentals | journal=Physics Reports | volume=106 | issue=3 | date=1984 | doi=10.1016/0370-1573(84)90160-1 | pages=121–167 | url=https://linkinghub.elsevier.com/retrieve/pii/0370157384901601 | access-date=2025-07-25}}</ref> 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 display="block">\begin{align}
n(\mathbf{r},t) &= \int f(\mathbf{r}, \mathbf{v}, t) \,dv_x \,dv_y \,dv_z, \\
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