Distribution function (physics): Difference between revisions

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In many physical situations, a system of particles may show bulk fluid flow. In this case, the particles are moving together with an average velocity. This average velocity is known as the bulk velocity. The bulk velocity is represented as <math> u = (u_x, u_y, u_z) </math>. When such a bulk flow exists, the distribution function must be adjusted to account for this overall motion of the system.

'''FormulaModified Distribution Function''': <math> f(v) = n \left(\frac{m}{2\pi kT}\right)^{3/2} \exp\left(-\frac{m}{2kT}[(v_x-u_x)^2 + (v_y-u_y)^2 + (v_z-u_z)^2]\right) </math>

Where <math>u_x,u_y,u_z</math>​ are the components of the bulk velocity.

=== Non-Isotropic Temperatures ===

Some systems, like plasmas or anisotropic materials, may not have the same temperature in all directions. These systems have non-isotropic temperatures. The distribution function must take into account the different temperatures along the different spatial axes.

'''Modified Distribution Function''': For a system with non-isotropic temperatures <math> Tx,Ty,Tz </math> along the <math> x, y, </math> and <math> z </math> axes, the distribution function takes the form: <math> f(v) = n \left(\frac{m}{2\pi kT_x}\right)^{1/2} \left(\frac{m}{2\pi kT_y}\right)^{1/2} \left(\frac{m}{2\pi kT_z}\right)^{1/2} \exp\left(-\frac{mv_x^2}{2kT_x} - \frac{mv_y^2}{2kT_y} - \frac{mv_z^2}{2kT_z}\right)

</math>

Where <math>T_x, T_y,T_z</math> are the temperatures corresponding to the <math>x, y,</math> and <math>z</math> directions, respectively.

==References==