Distribution function (physics): Difference between revisions

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{{Short description|Function of seven variables}}

:''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]].''}}

{{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,t; 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| url-access=subscription }}</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

:''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]].''

<math display="block">\begin{align}

:<math>n(x,y,z\mathbf{r},t) &= \int f(\mathbf{r}, \mathbf{v}, t) \,dv_x \,dv_y \,dv_z,</math> \\

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(t) &= \int n(\mathbf{r}, t) \,dx \,dy \,dz, </math>

\end{align} </math>

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

where, {{math|''N''}} is the total number of particles, and {{math|''n''}} is the [[number density]] of particles – the number of particles per unit volume, or the [[density]] divided by the mass of individual particles.

:<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>.

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The [[Maxwell–Boltzmann distribution|basic distribution function]] uses the [[Boltzmann constant]] <math>k</math> and temperature <math>T</math> with the number density to modify the [[normal distribution]]:

<math display="block">\begin{align}

:<math> f &= n\left(\frac{m}{2 \pi kT}\right)^{3/2} \exp\left({-\frac{m(v_x^2 + v_yv^2}{2 +k v_z^2)}{2kT}T}\right). </math>\\[2pt]

&= 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).

\end{align} </math>

Related distribution functions may allow bulk fluid flow, in which case the velocity origin is shifted, so that the exponent's numerator is <math>m((v_x - u_x)^2 + (v_y - u_y)^2 + (v_z - u_z)^2)</math>, where <math>(u_x, u_y, u_z)</math> 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.

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The mathematical analogue of a distribution is a [[measure (mathematics)|measure]]; the time evolution of a measure on a phase space is the topic of study in [[dynamical systems]].

==References==

{{reflist}}

[[Category:Statistical mechanics]]