Distribution function (physics): Difference between revisions

Revision as of 20:28, 6 March 2014

Revision as of 20:26, 9 March 2014

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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-particlewave–particle interactions and velocity-space instabilities. Distribution functions are also used in [[fluid mechanics]], [[statistical mechanics]] and [[nuclear physics]].

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> 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>

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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]]{{dndisambiguation needed|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]].