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(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

<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>, \\

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

\end{align} </math>

Here,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.

InA moleculardistribution [[kineticfunction theory]]may inbe [[physics]],specialised with respect to a particle'sparticular '''distributionset function'''of isdimensions. aE.g. functiontake ofthe sevenquantum mechanical six-dimensional phase variablesspace, <math>f(x,y,z,t;v_xp_x,v_yp_y,v_zp_z)</math>, whichand multiply givesby the numbertotal ofspace particlesvolume, perto unitgive volumethe inmomentum [[phasedistribution, space]]i.e. It is the number of particles perin unitthe volumemomentum phase space having approximately the [[velocitymomentum]] <math>(v_xp_x,v_yp_y,v_z)</math> near the place <math>(x,y,z)</math> and time <math>(tp_z)</math>. The usual normalization of the distribution function is

Particle distribution functions are often used in [[plasma physics]] to describe wave-particlewave–particle interactions and velocity-space instabilities. Distribution functions are also used in [[fluid mechanics]], [[statistical mechanics]] and [[nuclear physics]].

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

The [[Maxwell-BoltzmannMaxwell–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>N(t) = \int n \,dx \,dy \,dz. </math>

<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.

Here, 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.

[[Plasma (physics)|Plasma]] theories such as [[magnetohydrodynamics]] may assume the particles to be in [[thermodynamic equilibrium]]. In this case, the distribution function is ''[[Maxwell–Boltzmann distribution|Maxwellian]]{{dn|date=July 2012}}''. This distribution function allows fluid flow and different temperatures in the directions parallel to, and perpendicular to, the local magnetic field. More complex distribution functions may also be used, since [[Plasma (physics)|plasmas]] are rarely in thermal equilibrium.

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>.

The mathematical analoganalogue 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]].

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]].

==References==

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]]:

{{reflist}}

:<math> f = 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). </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>; <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.

Plasma theories such as [[magnetohydrodynamics]] may assume the particles to be in [[thermodynamic equilibrium]]. In this case, the distribution function is ''[[Maxwellian]]{{dn|date=July 2012}}''. This distribution function allows fluid flow and different temperatures in the directions parallel to, and perpendicular to, the local magnetic field. More complex distribution functions may also be used since [[Plasma (physics)|plasmas]] are rarely in thermal equilibrium.

The mathematical analog 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]].

{{DEFAULTSORT:Distribution Function}}

[[Category:Statistical mechanics]]

[[Category:Dynamical systems]]

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[[nl:Verdelingsfunctie]]

[[pl:Funkcja rozkładu]]

[[pt:Função distribuição]]