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 plasmapy
import matplotlib.pyplot as plt
from plasmapy.constants import (m_p, m_e, c, mu0, k_B, e, eps0, pi, e)

from plasmapy.physics.distribution import Maxwellian_1D

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

from astropy.visualization import quantity_support
quantity_support()

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:

5.916329687405703e-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()
../_images/sphx_glr_plot_distribution_001.png

Out:

Integral value for p: 0.9999999999999999
Integral value for e: 0.9999999999998787

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: 29999.999999792246 K
Initial T: 30000.0 K

Total running time of the script: ( 0 minutes 2.304 seconds)

Gallery generated by Sphinx-Gallery