Distribution function (physics): Difference between revisions

Revision as of 10:37, 24 August 2024

Revision as of 23:51, 1 April 2025

Line 3:

{{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]]. 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,</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>.

Line 11 ⟶ 13:

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 [[Maxwell–Boltzmann distribution|basic distribution function]] uses the [[Boltzmann constant]] <math>kk_\text{B}</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^2 + v_yv^2 + v_z^2)}{2kT2 k_\text{B} 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)}{2k_\text{B}T}\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.