Boltzmann Statistics.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Boltzmann Statistics\n",
    "A review of the Boltzmann distribution, exploration of the $kT$ factor, and observations of it's behavior in various systems.\n",
    "\n",
    "As usual, run the cell below to import necessary packages. Note that we import $k$ from the `scipy.constants` package. There are many other constants available too."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "from scipy.constants import k"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def b(E,T):\n",
    "    return np.exp(-E/(k*T))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1) What range of energies corresponds to kT at human-scale temperatures?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2) Plot the Boltzmann factor at room temperature and over an energy range that shows the full behavior of the function (hint: guess the shape of the function first, and work to find that in your plot)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3) Plot the Boltzmann factor at one Energy and over a temperature range that shows the full behavior of the function (hint: it's not just an exponential)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4) Use this template to write a function that returns the partition function for a system with a list of allowed energies:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def Z(Elist,T):\n",
    "    \"\"\"Note the Elist must be a list for this to work. For example, use:\n",
    "    Z([1e-21,2e-21],300)\n",
    "    to calculate Z at 300K for a system with states at 1e-21 J and 2e-21 J of energy.\"\"\"\n",
    "    Z_value = 0\n",
    "    for E in Elist:\n",
    "        Z_value += ...\n",
    "        \n",
    "    return Z_value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.4025150702306313"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z([1e-21,2e-21],300)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot Z vs E value and compare to Eq. 6.20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x118dd7320>]"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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YSbbjqAY0WvwiEiMiK0UkW0Q2icg99YwREXlKRHJEJENERtRaVi0iaa4vPXPI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\nGAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x118dd7518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot Z vs allowed E\n",
    "Elist = np.linspace(-2e-21,2e-21)\n",
    "T = 300\n",
    "\n",
    "plt.plot(Elist,Z([...,...],T))\n",
    "plt.plot(..., 'o')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5) For a two-state system (i.e. the paramagnet) graph the average energy vs. temperature."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6) Problem 6.27"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}