Maxwellian_1D

plasmapy.formulary.distribution.Maxwellian_1D(v, T, particle: str | Integral | Particle | CustomParticle | Quantity = 'e', v_drift=0, vTh=nan, units='units', *, mass_numb=None, Z=None)[source]

Probability distribution function of velocity for a Maxwellian distribution in 1D.

Returns the probability density function at the velocity v in m/s to find a particle particle in a plasma of temperature T following the Maxwellian distribution function.

Parameters:

  • v (Quantity) – The velocity in units convertible to m/s.

  • T (Quantity) – The temperature in kelvin.

  • particle (str, optional) – Representation of the particle species(e.g., 'p' for protons, 'D+' for deuterium, or 'He-4 +1' for singly ionized helium-4), which defaults to electrons.

  • v_drift (Quantity, optional) – The drift velocity in units convertible to m/s.

  • vTh (Quantity, optional) – Thermal velocity (most probable velocity) 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.

  • mass_numb (integer, keyword-only, optional) – The mass number associated with particle.

  • Z (real number, keyword-only, optional) – The charge number associated with particle.

Returns:

f – Probability density in units of velocity-1, normalized so that \(\int_{-∞}^{+∞} f(v) dv = 1\).

Return type:

Quantity

Raises:

  • TypeError – The parameter arguments are not Quantities and cannot be converted into Quantities.

  • UnitConversionError – If the parameters are not in appropriate units.

  • ValueError – If the temperature is negative, or the particle mass or charge state cannot be found.

Notes

In one dimension, the Maxwellian distribution function for a particle of mass m, velocity v, a drift velocity V and with temperature T is:

\[f = \sqrt{\frac{m}{2 \pi k_B T}} e^{-\frac{m}{2 k_B T} (v-V)^2} \equiv \frac{1}{\sqrt{\pi v_{Th}^2}} e^{-(v - v_{drift})^2 / v_{Th}^2}\]

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

Examples

>>> import astropy.units as u
>>> v=1*u.m/u.s
>>> Maxwellian_1D(v=v, T=30000 * u.K, particle='e', v_drift=0 * u.m / u.s)
<Quantity 5.9163...e-07 s / m>