Maxwellian_velocity_3D

plasmapy.physics.distribution.Maxwellian_velocity_3D(vx, vy, vz, T, particle='e', vx_drift=0, vy_drift=0, vz_drift=0, vTh=nan, units='units')

Probability distribution function of velocity for a Maxwellian distribution in 3D.

Return the probability density function for finding a particle with velocity components vx, vy, and vz in m/s in an equilibrium plasma of temperature T which follows the 3D Maxwellian distribution function. This function assumes Cartesian coordinates.

Parameters:
  • vx (Quantity) – The velocity in x-direction units convertible to m/s.
  • vy (Quantity) – The velocity in y-direction units convertible to m/s.
  • vz (Quantity) – The velocity in z-direction units convertible to m/s.
  • T (Quantity) – The temperature, preferably in Kelvin.
  • particle (str, optional) – Representation of the particle species (e.g., 'p' for protons, 'D+' for deuterium, or 'He-4 +1' for \(He_4^{+1}\) (singly ionized helium-4)), which defaults to electrons.
  • vx_drift (Quantity, optional) – The drift velocity in x-direction units convertible to m/s.
  • vy_drift (Quantity, optional) – The drift velocity in y-direction units convertible to m/s.
  • vz_drift (Quantity, optional) – The drift velocity in z-direction units convertible to m/s.
  • vTh (Quantity, optional) – Thermal velocity (most probable) in m/s. This is used for optimization purposes to avoid re-calculating vTh, for example when integrating over velocity-space.
  • units (str, optional) – Selects whether to run function with units and unit checks (when equal to “units”) or to run as unitless (when equal to “unitless”). The unitless version is substantially faster for intensive computations.
Returns:

f – Probability density in Velocity^-1, normalized so that \(\iiint_{0}^{\infty} f(\vec{v}) d\vec{v} = 1\).

Return type:

Quantity

Raises:

Notes

In 3D, the Maxwellian speed distribution function describing the distribution of particles with speed \(v\) in a plasma with temperature \(T\) is given by:

\[f = (\pi v_{Th}^2)^{-3/2} \exp \left [-(\vec{v} - \vec{V}_{drift})^2 / v_{Th}^2 \right ]\]

where \(v_{Th} = \sqrt{2 k_B T / m}\) is the thermal speed.

See also

Maxwellian_1D()

Example

>>> from plasmapy.physics import Maxwellian_velocity_3D
>>> from astropy import units as u
>>> v=1 * u.m / u.s
>>> Maxwellian_velocity_3D(vx=v,
... vy=v,
... vz=v,
... T=30000 * u.K,
... particle='e',
... vx_drift=0 * u.m / u.s,
... vy_drift=0 * u.m / u.s,
... vz_drift=0 * u.m / u.s)
<Quantity 2.07089033e-19 s3 / m3>