Distribution function (physics): Difference between revisions

Revision as of 19:15, 13 July 2018

Revision as of 14:17, 13 December 2018

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:''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]].''

In molecular [[kinetic theory of gases|kinetic theory]] in [[physics]], a particle'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 placeposition <math>\mathbf{r}=(x,y,z)</math> and time <math>(t)</math>. The usual normalization of the distribution function is

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

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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>.

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>k</math> and temperature <math>T</math> with the number density to modify the [[normal distribution]]:

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:<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>, 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.

[[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]]''. 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 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]].

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[[Category:Statistical mechanics]]