{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "fff45747-34fc-4752-a4c0-e891bc0c5148",
   "metadata": {
    "tags": []
   },
   "source": [
    "# Introduction to Thermodynamics and Statistical Physics"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43a36220-4f03-47c9-9eee-b1679859cac4",
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   "source": [
    "In this lecture, we are going to discuss entropy. In particular, we'll cover:\n",
    "\n",
    "* Entropy changes within closed systems.\n",
    "* Entropy an the 1st law.\n",
    "* Entropy from a statistical point of view.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "41a8267b-43e7-4514-ae34-3800b76ee286",
   "metadata": {},
   "source": [
    "## Example"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "26143156-fe7b-41b3-b94a-e0e7e41c880f",
   "metadata": {},
   "source": [
    "Imagine we have a large reservoir at temperature $T_R$ which is placed into thermal contact with a small system at temperature $T_S$. They both end up at the temperature of the reservoir, $T_R$. The heat transferred from the reservoir to the system is given by\n",
    "$$\n",
    "    \\Delta Q = C(T_S-T_R)\n",
    "$$\n",
    "where C is the heat capacity of the system. There are two different outcomes here:\n",
    "\n",
    "1. $T_S$>$T_R$. This means that heat would be transferred from the system to the reservoir. As such, the entropy of the system would decrease as it gets colder and the entropy of the reservoir would increase as it warms up.\n",
    "2. $T_S$<$T_R$. This means that heat would be transferred from the reservoir to the system. As such, the entropy of the reservoir would decrease as it gets colder and the entropy of the system would increase as it warms up.\n",
    "\n",
    "Let's explicitly calculate the entropy change of both the system and reservoir:\n",
    "\\begin{align}\n",
    "    \\Delta S_{\\rm reservoir} &= \\int \\frac{{\\rm d} Q}{T}\\\\\n",
    "                             &= \\frac{1}{T_R} \\int {\\rm d} Q\\\\\n",
    "                             &= \\frac{\\Delta Q}{T_R}\\\\\n",
    "                             &= \\frac{C(T_S-T_R)}{T_R}\\\\\n",
    "\\end{align}\n",
    "The entropy of the system is given by\n",
    "\\begin{align}\n",
    "    \\Delta S_{\\rm system} &= \\int \\frac{{\\rm d} Q}{T}\\\\\n",
    "                             &= \\int ^{T_R} _{T_S} \\frac{C {\\rm d} T}{T}\\\\\n",
    "                             &= C \\ln \\frac{T_R}{T_S}\n",
    "\\end{align}\n",
    "So the entropy of the system is allowed to increase or decrease, as is that of the reservoir. But the entropy of the Universe, which is given by\n",
    "\\begin{align}\n",
    "    \\Delta S_{\\rm Universe} &= \\Delta S_{\\rm reservoir}+\\Delta S_{\\rm system}\\\\\n",
    "                            &=\\frac{C(T_S-T_R)}{T_R}+C \\ln \\frac{T_R}{T_S}\\\\\n",
    "                            &=C\\left[\\frac{T_S}{T_R} +\\ln \\frac{T_R}{T_S} - 1 \\right]\n",
    "\\end{align}\n",
    "The below figure plots this expression and shows that even though the entropy of the system and reservoir may increase or decrease individually, their combined change is always positive!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "c44a272b-c402-4592-9832-9920304c8fee",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 600x450 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "C = 1 #Just setting the heat capacity to a positive number.\n",
    "T_S_over_T_R = np.arange(0.01,4.05,0.01)\n",
    "S_R = C*(T_S_over_T_R-1)\n",
    "S_S = C*np.log(1/T_S_over_T_R)\n",
    "\n",
    "plt.figure(figsize=[4,3],dpi=150)\n",
    "plt.plot(T_S_over_T_R,S_R,'--',color='grey',label='Entropy of Reservoir')\n",
    "plt.plot(T_S_over_T_R,S_S,'-.',color='grey',label='Entropy of System')\n",
    "plt.plot(T_S_over_T_R,S_S+S_R,'-',color='k',label='Combined Entropy')\n",
    "plt.ylabel(\"S\")\n",
    "plt.xlabel(r\"$\\frac{T_S}{T_R}$\")\n",
    "plt.legend()\n",
    "plt.xlim(0,4)\n",
    "plt.savefig(\"Figures/Entropy_Universe_plot.png\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6b170dff-8dd7-451d-a89b-7b44cab3044f",
   "metadata": {},
   "source": [
    "## Revisiting the first law"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c9cc411d-7e97-4287-886b-201060437c3a",
   "metadata": {},
   "source": [
    "Ok, armed with this definition of entropy, let's consider again the first law of thermodynamics, which states that\n",
    "$$\n",
    "    {\\rm d} U = {\\rm d} Q + {\\rm d} W\n",
    "$$\n",
    "For a reversible change, we now have that\n",
    "$$\n",
    "    {\\rm d} Q = T {\\rm d} S\n",
    "$$\n",
    "by rearranging Clausius' law. We also know that \n",
    "$$\n",
    "    {\\rm d} W = -P {\\rm d} V\n",
    "$$\n",
    "This means that \n",
    "$$\n",
    "    {\\rm d} U = T{\\rm d} S - P {\\rm d} V\n",
    "$$\n",
    "We have derived this assuming everything is reversible. However, all of the quantities in the previous equation are functions of state - that is, they are path independant. This means that this equation also holds for irreversible changes. This might not seem immediately obvious. As a quick check, remember that for irreversible changes we have that\n",
    "$$\n",
    "    {\\rm d} Q \\leq T {\\rm d} S\n",
    "$$\n",
    "while\n",
    "$$\n",
    "    {\\rm d} W \\geq - P {\\rm d} V.\n",
    "$$\n",
    "So for irreversible changes, ${\\rm d} Q $ decreases while ${\\rm d} W $ increases, meaning ${\\rm d} U$ is balanced. As such \n",
    "$$\n",
    "    {\\rm d} U = T{\\rm d} S - P {\\rm d} V\n",
    "$$\n",
    "is true **for any process**!\n",
    "Now, using one of the tricks we used earlier, given that U is now only a function of S and V, we can write U as\n",
    "$$\n",
    "    {\\rm d} U = \\left( \\frac{\\partial U}{\\partial S}\\right)_V {\\rm d} S + \\left( \\frac{\\partial U}{\\partial V}\\right)_S {\\rm d} V\n",
    "$$\n",
    "let's us identify pressure and temperature as\n",
    "\\begin{align}\n",
    "    T &= \\left( \\frac{\\partial U}{\\partial S}\\right)_V\\\\\n",
    "    P &= -\\left( \\frac{\\partial U}{\\partial V}\\right)_S\\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "82b344c3-393c-4ba2-8bf6-dc9f97db4b6f",
   "metadata": {},
   "source": [
    "## Summary"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a7b9f1b5-9e68-4535-af9a-e9da08efc389",
   "metadata": {},
   "source": [
    "Thus, the important equations for us are\n",
    "\n",
    "1. ${\\rm d} U = {\\rm d} Q + {\\rm d} W$, which is always true.\n",
    "2. ${\\rm d} Q = T{\\rm d} S$, which is true for reversible changes.\n",
    "3. ${\\rm d} W = -P{\\rm d} V$, which is true for reversible changes.\n",
    "4. ${\\rm d} U = T{\\rm d} S-P{\\rm d} V$, which is always true.\n",
    "\n",
    "For irreversible changes,  ${\\rm d} Q \\leq T{\\rm d} S$ and ${\\rm d} W \\geq -P{\\rm d} V$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7a2fe90-568e-4cdd-a9ea-b151ec684a04",
   "metadata": {},
   "source": [
    "## A statistical basis for entropy - The Joule Expansion"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f96c9ee-5b5a-4b69-82f9-beac5a23129e",
   "metadata": {},
   "source": [
    "Consider the following isolated system. Imagine we have a 2 containers, each of volume $V_0$, and which are initially isolated from each other by a valve. In the left container, we have an ideal gas with initial pressure $P_i$, and in the right, we have a vacuum. The ideal gas is described by the equation of state $P_i V_0 = N k_{\\rm B} T_i$.\n",
    "\n",
    "![Joule_Expansion](Figures/Joule_Expansion.png)\n",
    "\n",
    "We then open the valve, and the gas rapidly expands to fill both containers. At this stage, the gas will fill a volume of $2V_0$ and have a temperature and pressure of $T_f$ and $P_f$ respectively. It's equation of state is thus $P_f 2V_0 = N k_{\\rm B} T_f$.\n",
    "\n",
    "Since the entire system is thermally isolated, we have that $\\Delta U = 0$. Recalling from earlier that the internal energy, U, of an ideal gas only depends on temperature, this then means that $\\Delta T = 0$, and so $T_i=T_f$. As such, we get\n",
    "\\begin{align}\n",
    "    P_i V_0 &= N k_{\\rm B} T_i =  P_f 2 V_0\\\\\n",
    "    P_f &= \\frac{P_i}{2}\n",
    "\\end{align}\n",
    "\n",
    "However, calculating the entropy associated with this change is difficult. Immediately after opening the valve, the gas will expand rapidly, meaning it is not in equilibrium and will not be undergoing a reversible process. However, entropy is a function of state. This means that the difference in entropy between the gas prior to the valve opening and after opening the valve and letting the gas reach equilibrium after expanding irreversibly **is the same** as if we had reverisble expanded the gas, as the change in entropy is path independant.\n",
    "\n",
    "As such, we can pretend that the gas were expanded, say, isothermally, calculate the change in the entropy for this expansion, and it will be the same as for the irreversible process. this gives\n",
    "\\begin{align}\n",
    "    \\Delta S &= \\int ^{f}_{i} {\\rm d} S \\\\\n",
    "             &= \\int ^{V_f}_{V_i} \\frac{P}{T}{\\rm d} V \\\\\n",
    "             &= \\frac{1}{T}\\int ^{2V_0}_{V_0} P{\\rm d} V \\\\\n",
    "             &= \\frac{1}{T}\\int ^{2V_0}_{V_0} \\frac{Nk_{\\rm B}T}{V}{\\rm d} V \\\\\n",
    "             &= Nk_{\\rm B}\\int ^{2V_0}_{V_0} \\frac{1}{V}{\\rm d} V \\\\\n",
    "             &= Nk_{\\rm B} \\ln 2\\\\\n",
    "\\end{align}"
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    "The first law of thermodynamics can be written as\n",
    "$$\n",
    "    {\\rm d} U = T {\\rm d}S - P{\\rm d} V\n",
    "$$\n",
    "and is always true. From this, in the last lecture we arrived at the following definition for temperature\n",
    "$$\n",
    "    T = \\left(\\frac{\\partial S}{\\partial U}\\right)_V\n",
    "$$\n",
    "Several lectures ago, we also definied temperature in terms of the statistical weight of a system in a given macrostate to be\n",
    "$$\n",
    "    \\frac{1}{k_{\\rm B} T} = \\frac{{\\rm d} \\ln \\Omega}{{\\rm d} E}.\n",
    "$$\n",
    "Comparing these, we can then identify another definition of $S$ as\n",
    "$$\n",
    "    S = k_{\\rm B} \\ln \\Omega.\n",
    "$$\n",
    "At this point, it's important to note that this definition assumes that the system is in a macrostate with a fixed energy (this comes from the section entitled \"Temperature\" back in Lecture 2). We'll generalise the result later."
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