{
 "cells": [
  {
   "cell_type": "markdown",
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   "source": [
    "# Thermal Speed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "from astropy import units as u\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from plasmapy.formulary import  Maxwellian_speed_1D, Maxwellian_speed_2D, Maxwellian_speed_3D\n",
    "from plasmapy.formulary.parameters import thermal_speed"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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",
    "\n",
    "- 'most_probable' <br>\n",
    "$v_{th} = \\sqrt{\\frac{2 k_B T}{m}}$\n",
    "\n",
    "- 'rms' <br>\n",
    "$v_{th} = \\sqrt{\\frac{3 k_B T}{m}}$\n",
    "\n",
    "- 'mean_magnitude' <br>\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(-.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": [
    "\n",
    "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(T=T, particle='e-', method='mean_magnitude', ndim=ndim)\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(-.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, .8), loc='upper right')\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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