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%matplotlib inline

1D Maxwellian distribution function

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

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import matplotlib.pyplot as plt
import numpy as np

from astropy import units as u
from astropy.constants import k_B, m_e

from plasmapy.formulary import Maxwellian_1D

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

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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:

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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)

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

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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):

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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()

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

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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)