Distribution function (physics)

Maxwellian distribution

The basic distribution function uses the Boltzmann constant ${\displaystyle k}$ and temperature ${\displaystyle T}$ with the number density to modify the normal distribution: $${\displaystyle \begin{array}{cc}f& =n{\left(\frac{m}{2\pi kT}\right)}^{3/2}\exp \left(-\frac{m{v}^2}{2kT}\right)\\ & =n{\left(\frac{m}{2\pi kT}\right)}^{3/2}\exp \left(-\frac{m(v_x^2+v_y^2+v_z^2)}{2kT}\right).\end{array}}$$

Related distribution functions may allow bulk fluid flow, in which case the velocity origin is shifted, so that the exponent's numerator is ${\displaystyle m((v_x-u_x{)}^2+(v_y-u_y{)}^2+(v_z-u_z{)}^2)}$, where ${\displaystyle (u_x,u_y,u_z)}$ 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. 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 plasmas are rarely in thermal equilibrium.

The mathematical analogue of a distribution is a measure; the time evolution of a measure on a phase space is the topic of study in dynamical systems.