# 1D Maxwellian distribution function¶

We import the usual modules, and the hero of this notebook, the Maxwellian 1D distribution:

import numpy as np
from astropy import units as u
import matplotlib.pyplot as plt
from astropy.constants import (m_e, k_B)

from plasmapy.formulary import Maxwellian_1D


Given we’ll be plotting, import astropy’s quantity support:

from astropy.visualization import quantity_support
quantity_support()


Out:

<astropy.visualization.units.quantity_support.<locals>.MplQuantityConverter object at 0x7f0aef3d4c88>


As a first example, let’s get the probability density of finding an electron with a speed of 1 m/s if we have a plasma at a temperature of 30 000 K:

p_dens = Maxwellian_1D(v=1 * u.m / u.s,
T=30000 * u.K,
particle='e',
v_drift=0 * u.m / u.s)
print(p_dens)


Out:

5.916328704912824e-07 s / m


Note the units! Integrated over speed, this will give us a probability. Let’s test that for a bunch of particles:

T = 3e4 * u.K
dv = 10 * u.m / u.s
v = np.arange(-5e6, 5e6, 10) * u.m / u.s


Check that the integral over all speeds is 1 (the particle has to be somewhere):

for particle in ['p', 'e']:
pdf = Maxwellian_1D(v, T=T, particle=particle)
integral = (pdf).sum() * dv
print(f"Integral value for {particle}: {integral}")
plt.plot(v, pdf, label=particle)
plt.legend() Out:

Integral value for p: 1.0000000000000002
Integral value for e: 0.9999999999998787

<matplotlib.legend.Legend object at 0x7f0aeedc7be0>


The standard deviation of this distribution should give us back the temperature:

std = np.sqrt((Maxwellian_1D(v, T=T, particle='e') * v ** 2 * dv).sum())
T_theo = (std ** 2 / k_B * m_e).to(u.K)

print('T from standard deviation:', T_theo)
print('Initial T:', T)


Out:

T from standard deviation: 29999.999999792235 K
Initial T: 30000.0 K


Total running time of the script: ( 0 minutes 2.639 seconds)

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