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"source": [
"# Thermal Speed"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib inline\n",
"\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"\n",
"from astropy import units as u\n",
"\n",
"from plasmapy.formulary import (\n",
" Maxwellian_speed_1D,\n",
" Maxwellian_speed_2D,\n",
" Maxwellian_speed_3D,\n",
")\n",
"from plasmapy.formulary.parameters import thermal_speed"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"[thermal_speed]: ../../api/plasmapy.formulary.parameters.thermal_speed.rst#plasmapy.formulary.parameters.thermal_speed\n",
"\n",
"The [thermal_speed] function can be used to calculate the thermal velocity for a Maxwellian velocity distribution. There are three common definitions of the thermal velocity, which can be selected using the \"method\" keyword, which are defined for a 3D velocity distribution as\n",
"\n",
"- 'most_probable'
\n",
"$v_{th} = \\sqrt{\\frac{2 k_B T}{m}}$\n",
"\n",
"- 'rms'
\n",
"$v_{th} = \\sqrt{\\frac{3 k_B T}{m}}$\n",
"\n",
"- 'mean_magnitude'
\n",
"$v_{th} = \\sqrt{\\frac{8 k_B T}{m\\pi}}$\n",
"\n",
"The differences between these velocities can be seen by plotitng them on a 3D Maxwellian speed distribution"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"nbsphinx-thumbnail": {
"tooltip": "Thermal Speeds"
}
},
"outputs": [],
"source": [
"T = 1e5 * u.K\n",
"speeds = np.linspace(0, 8e6, num=600) * u.m / u.s\n",
"\n",
"pdf_3D = Maxwellian_speed_3D(speeds, T=T, particle=\"e-\")\n",
"\n",
"fig, ax = plt.subplots(figsize=(4, 3))\n",
"\n",
"v_most_prob = thermal_speed(T=T, particle=\"e-\", method=\"most_probable\", ndim=3)\n",
"v_rms = thermal_speed(T=T, particle=\"e-\", method=\"rms\", ndim=3)\n",
"v_mean_magnitude = thermal_speed(T=T, particle=\"e-\", method=\"mean_magnitude\", ndim=3)\n",
"\n",
"ax.plot(speeds / v_rms, pdf_3D, color=\"black\", label=\"Maxwellian\")\n",
"\n",
"ax.axvline(x=v_most_prob / v_rms, color=\"blue\", label=\"Most Probable\")\n",
"ax.axvline(x=v_rms / v_rms, color=\"green\", label=\"RMS\")\n",
"ax.axvline(x=v_mean_magnitude / v_rms, color=\"red\", label=\"Mean Magnitude\")\n",
"\n",
"ax.set_xlim(-0.1, 3)\n",
"ax.set_ylim(0, None)\n",
"ax.set_title(\"3D\")\n",
"ax.set_xlabel(\"|v|/|v$_{rms}|$\")\n",
"ax.set_ylabel(\"f(|v|)\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Similar speeds are defined for 1D and 2D distributions. The differences between these definitions can be illustrated by plotting them on their respective Maxwellian speed distributions."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"pdf_1D = Maxwellian_speed_1D(speeds, T=T, particle=\"e-\")\n",
"pdf_2D = Maxwellian_speed_2D(speeds, T=T, particle=\"e-\")\n",
"\n",
"dim = [1, 2, 3]\n",
"pdfs = [pdf_1D, pdf_2D, pdf_3D]\n",
"\n",
"plt.tight_layout()\n",
"fig, ax = plt.subplots(ncols=3, figsize=(10, 3))\n",
"\n",
"for n, pdf in enumerate(pdfs):\n",
" ndim = n + 1\n",
" v_most_prob = thermal_speed(T=T, particle=\"e-\", method=\"most_probable\", ndim=ndim)\n",
" v_rms = thermal_speed(T=T, particle=\"e-\", method=\"rms\", ndim=ndim)\n",
" v_mean_magnitude = thermal_speed(\n",
" T=T, particle=\"e-\", method=\"mean_magnitude\", ndim=ndim\n",
" )\n",
"\n",
" ax[n].plot(speeds / v_rms, pdf, color=\"black\", label=\"Maxwellian\")\n",
"\n",
" ax[n].axvline(x=v_most_prob / v_rms, color=\"blue\", label=\"Most Probable\")\n",
" ax[n].axvline(x=v_rms / v_rms, color=\"green\", label=\"RMS\")\n",
" ax[n].axvline(x=v_mean_magnitude / v_rms, color=\"red\", label=\"Mean Magnitude\")\n",
"\n",
" ax[n].set_xlim(-0.1, 3)\n",
" ax[n].set_ylim(0, None)\n",
" ax[n].set_title(\"{:d}D\".format(ndim))\n",
" ax[n].set_xlabel(\"|v|/|v$_{rms}|$\")\n",
" ax[n].set_ylabel(\"f(|v|)\")\n",
"\n",
"\n",
"ax[2].legend(bbox_to_anchor=(1.9, 0.8), loc=\"upper right\")"
]
}
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