Distribution function (physics): Difference between revisions

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{{Short description|Function of seven variables}}

{{about|the distribution function as used in physics|the related mathematical concepts|cumulative distribution function|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(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| 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

{{Unreferenced|date=December 2009}}

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

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]]. 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">n(\mathbf{r},t) &= \int f(\mathbf{r}, \mathbf{v}, t) \,dv_x \,dv_y \,dv_z,</math> \\

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

\end{align} </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.

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}
f &= n\left(\frac{m}{2 \pi kT}\right)^{3/2} \exp\left({-\frac{m(v_x v^2 + v_y^}{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.