{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "72c8af0a-c3db-4571-85a5-44cd19ab9208",
   "metadata": {
    "tags": []
   },
   "source": [
    "# Introduction to Thermodynamics and Statistical Physics"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d4d593e4-c975-4577-8743-869bfa076ff4",
   "metadata": {},
   "source": [
    "In this lecture, we are going to discuss:\n",
    "\n",
    "* Applications of the energy equipartition theorem.\n",
    "* The partition function."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "861e3316-370e-43cd-ad7e-24025c014e1e",
   "metadata": {
    "tags": []
   },
   "source": [
    "## Applications of Equipartition of Energy\n",
    "\n",
    "**Vibrational energy in a diatomic gas**: Now imagine that same gas, but where the bond is no longer rigid, but instead acts like a spring with spring constant k. This adds gives another 2 terms:\n",
    "$$\n",
    "    \\frac{1}{2}\\mu ({\\bf \\dot{r_1}-\\dot{r_2}})^2 + \\frac{1}{2} k (|{\\bf \\dot{r_1}-\\dot{r_2}}|-l_0)^2\n",
    "$$\n",
    "where k is an imagined spring constant, $l_0$ is the equilibrium molecular bond length, and $\\mu$ is the reduced mass of the system. This means the total energy is \n",
    "$$\n",
    "    E = \\frac{1}{2} mv_x^2+\\frac{1}{2} mv_y^2+\\frac{1}{2} mv_z^2+\\frac{L_1^2}{2I_1} + \\frac{L_2^2}{2I_2}+\\frac{1}{2}\\mu ({\\bf \\dot{r_1}-\\dot{r_2}})^2 + \\frac{1}{2} k (|{\\bf \\dot{r_1}-\\dot{r_2}}|-l_0)^2\n",
    "$$\n",
    "and thus\n",
    "$$\n",
    "    <E> = \\frac{7}{2} k_{\\rm B} T\n",
    "$$\n",
    "![Entropy_of_mixing](Figures/Diatomic_Vibration.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "03cfd75f-874b-4f85-8d86-e6725f23f71f",
   "metadata": {},
   "source": [
    "## Dulong-Petite rule\n",
    "\n",
    "A relatively good model for a solid is that we have a rigid lattice of atoms, with each atom attached to it's nearest neighbour by a chemical bond that allows vibrations like. If there are $N$ atoms in the solid, then we have approimately 3$N$ atoms. Each spring has two quadratic modes of energy (1 kinetic and 1 potential), which means each spring contributes $k_{\\rm B} T$ to the mean energy. As such, the total mean energy of the solid should be $3N k_{\\rm B} T$, and has a heat capacity of $3 N k_{\\rm B}$. This is known as the Dulong-Petit rule, and is something we'll come back to in future lectures."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb76e7cc-1c2b-4ce4-a211-2649c2d2d590",
   "metadata": {
    "tags": []
   },
   "source": [
    "# Revisiting the partition function"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e06b162d-7082-4399-8cf4-0ec327cdb55b",
   "metadata": {},
   "source": [
    "In this lecture, we are going to revisit the partition function which we discussed during the introductory part of the course, and figure out how it relates to the thermodynamic quantities which we introduced in classical thermal physics.\n",
    "\n",
    "The partition function is defined as \n",
    "$$\n",
    "    Z = \\sum_i {\\rm e}^{-\\beta E_i}\n",
    "$$\n",
    "It looks pretty boring, but is a very powerful tool, as we'll shortly see. Typically, when solving a statistical mechanics issues, there are two steps.\n",
    "\n",
    "1. Write down the partition function.\n",
    "2. Follow the standard procedures needed to get the relevant quantity out of the partition function.\n",
    "\n",
    "For this lecture, we are going to be focusing on the single particle partition function - that is, we will work out what $Z$ is for a single particle. We'll generalise to many particles later."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "777b0e13-a47f-4fa3-ac98-9035dd249655",
   "metadata": {},
   "source": [
    "## Writing down the partition function"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69773ba1-12be-4a56-8d94-d55d8e7f033e",
   "metadata": {},
   "source": [
    "This is not too difficult a step, and is something you have encountered in your problem sets previously. Let's take a look at some explicit examples.\n",
    "\n",
    "### The two level system\n",
    "Let the energy of a system be $\\pm\\epsilon/2$. The partition function for such a system is thus\n",
    "$$\n",
    "    Z = {\\rm e}^{\\beta \\epsilon /2}+{\\rm e}^{-\\beta \\epsilon /2} = 2 \\cosh \\left(\\frac{\\beta \\epsilon}{2}\\right)\n",
    "$$\n",
    "\n",
    "### The simple harmonic oscillator\n",
    "Let the energy of a system be $(n+1/2)\\hbar \\omega$, where $\\hbar=h/(2\\pi)$ and n can go up to infinity. The partition function for such a system is thus\n",
    "$$\n",
    "    Z = \\sum_{n=0}^{\\infty} {\\rm e}^{\\beta (n+1/2)\\hbar \\omega} = {\\rm e}^{-\\beta \\hbar \\omega/2} \\sum_{n=0}^{\\infty} {\\rm e}^{-\\beta n \\hbar \\omega}=\\frac{{\\rm e}^{-\\beta \\hbar \\omega/2}}{1-{\\rm e}^{-\\beta \\hbar \\omega}}\n",
    "$$\n",
    "where we've used the result for an infinite geometric progression of\n",
    "$$\n",
    "    a\\sum_{n=0}^{\\infty} r^n = \\frac{a}{1-r}\n",
    "$$\n",
    "where $a$ is a constant to simplify the expression."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "df3b972d-e734-45aa-aa29-5c9ae197f282",
   "metadata": {},
   "source": [
    "## Deriving useful quantities from the partition function"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "63e88661-09f1-45cf-8577-1249de450af9",
   "metadata": {},
   "source": [
    "Ok, now that we can write down what the partition function, let's see what we can derive from it.\n",
    "\n",
    "### The internal energy (U)\n",
    "As discussed in previous lectures, the internal energy of a system is given by\n",
    "$$\n",
    "    U = \\sum_i P_i E_i\n",
    "$$\n",
    "$$\n",
    "    U = \\frac{\\sum_i E_i {\\rm e}^{-\\beta E_i}}{\\sum_i {\\rm e}^{-\\beta E_i}}\n",
    "$$\n",
    "Given that $Z = \\sum_i {\\rm e}^{-\\beta E_i}$, then $\\frac{{\\rm d} Z}{{\\rm d} \\beta} = - \\sum_i E_i {\\rm e}^{-\\beta E_i}$. This leaves us with\n",
    "$$\n",
    "    U = -\\frac{1}{Z}\\frac{{\\rm d} Z}{{\\rm d} \\beta} = -\\frac{{\\rm d} \\ln Z}{{\\rm d} \\beta}\n",
    "$$\n",
    "as we've seen previously. It can also sometimes be useful to rewrite this in terms of temperature. Using\n",
    "$$\n",
    "    \\beta = \\frac{1}{k_{\\rm B}T} \\to \\frac{{\\rm d} \\beta}{{\\rm d} T} = -\\frac{1}{k_{\\rm B}T^2}\n",
    "$$\n",
    "gives\n",
    "$$\n",
    "    U = k_{\\rm B}T^2 \\frac{{\\rm d} \\ln Z}{{\\rm d} T}\n",
    "$$\n",
    "\n",
    "### Heat Capacities\n",
    "Recalling that\n",
    "$$\n",
    "    C_{\\rm V} = \\left(\\frac{\\partial U}{\\partial T}\\right)_V\n",
    "$$\n",
    "we get\n",
    "$$\n",
    "    C_{\\rm V} = 2 k_{\\rm B}T \\frac{{\\rm d} \\ln Z}{{\\rm d} T} + k_{\\rm B}T^2 \\frac{{\\rm d^2} \\ln Z}{{\\rm d} T^2}\n",
    "$$\n",
    "\n",
    "### Entropy (S)\n",
    "The probability of a state being in some energy j is given by\n",
    "$$\n",
    "    P(E_j) = \\frac{{\\rm e}^{-\\beta E_j}}{Z}\n",
    "$$\n",
    "Taking the log gives\n",
    "$$\n",
    "    \\ln P(E_j) = -\\beta E_j - \\ln (Z)\n",
    "$$\n",
    "If we now recall our statistical definition of entropy, which was\n",
    "$$\n",
    "    S = -k_{\\rm B} \\sum_i P_i \\ln P_i\n",
    "$$\n",
    "(see Lecture 7) then we get\n",
    "$$\n",
    "    S = k_{\\rm B} \\sum_i P_i [\\beta E_i + \\ln (Z)]= k_{\\rm B} \\beta \\sum_i P_i E_i + \\ln (Z)\\sum_i P_i]\n",
    "$$\n",
    "We can now substitute $U = \\sum_i P_i E_i$ and $\\sum_i P_i E_i = 1.0$ to get\n",
    "$$\n",
    "    S = k_{\\rm B} [\\beta U + \\ln(Z)]\n",
    "$$\n",
    "which in terms of temperature works out as\n",
    "$$\n",
    "    S =  \\frac{U}{T} + k_{\\rm B} \\ln(Z)\n",
    "$$\n",
    "\n",
    "### The Helmholtz Free Energy (F)\n",
    "The Helmholtz Free Energy is given by\n",
    "$$\n",
    "    F = U-TS\n",
    "$$\n",
    "Using the above substituion for $S$ then leads to\n",
    "$$\n",
    "    F = -k_{\\rm B} T \\ln(Z)\n",
    "$$\n",
    "or, in terms of Z is\n",
    "$$\n",
    "    Z = {\\rm e}^{-\\beta F}\n",
    "$$\n",
    "The Helmholtz Free Energy turns out to be very useful in deriving pretty much everything else we've defined over the last 4 weeks. For example, we can get the entropy by recalling that\n",
    "$$\n",
    "    S = -\\left(\\frac{\\partial F}{\\partial T}\\right)_V\n",
    "$$\n",
    "(again, Lecture 7) which gives\n",
    "$$\n",
    "    S = k_{\\rm B} \\ln(Z) + k_{\\rm B} T \\left(\\frac{\\partial \\ln(Z)}{\\partial T}\\right)_V\n",
    "$$\n",
    "which is the same as the above expression if we use the relation between $U$ and $\\frac{{\\rm d} \\ln Z}{{\\rm d} T}$ derived earlier.\n",
    "\n",
    "### Pressure\n",
    "From Lecture 7 we have that\n",
    "$$\n",
    "    P = -\\left(\\frac{\\partial F}{\\partial V}\\right)_T = k_{\\rm B} T \\left( \\frac{\\partial \\ln(Z)} {\\partial V} \\right)_T\n",
    "$$\n",
    "\n",
    "### Enthalpy\n",
    "This lets us write down the enthalpy as\n",
    "$$\n",
    "    H = U+PV =  k_{\\rm B}T^2 \\frac{{\\rm d} \\ln Z}{{\\rm d} T}+k_{\\rm B} T V \\left( \\frac{\\partial \\ln(Z)} {\\partial V} \\right)_T\n",
    "$$\n",
    "\n",
    "### Gibbs Free Energy\n",
    "$$\n",
    "    G = F+PV =  -k_{\\rm B} T \\ln(Z) +k_{\\rm B} T V \\left( \\frac{\\partial \\ln(Z)} {\\partial V} \\right)_T\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72566a8d-2f15-4a38-9d69-5fe7f018ab7b",
   "metadata": {},
   "source": [
    "# Working through an example"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f9691e7-9af0-4b6a-9a63-90485b50f1b7",
   "metadata": {},
   "source": [
    "Let's take the 2 level system which was described earlier. This system has a partition function of\n",
    "$$\n",
    "    Z = 2 \\cosh \\left(\\frac{\\beta \\epsilon}{2}\\right)\n",
    "$$\n",
    "The internal energy of the system is\n",
    "$$\n",
    "    U = -\\frac{{\\rm d} \\ln Z}{{\\rm d} \\beta}= -\\frac{\\epsilon}{2} \\tanh\\left(\\frac{\\beta \\epsilon}{2}\\right)\n",
    "$$\n",
    "The heat capacity of the system is given by\n",
    "$$\n",
    "    C_{\\rm V} = \\left(\\frac{\\partial U}{\\partial T}\\right)_V = k_{\\rm B} \\left(\\frac{\\beta \\epsilon}{2}\\right)^2 {\\rm sech}^2\\left(\\frac{\\beta \\epsilon}{2}\\right)\n",
    "$$\n",
    "The Helholtz Free energy is\n",
    "$$\n",
    "    F = -k_{\\rm B}T \\ln Z = -k_{\\rm B}T \\ln\\left[2 \\cosh \\left(\\frac{\\beta \\epsilon}{2}\\right)\\right]\n",
    "$$\n",
    "and so the entropy is\n",
    "$$\n",
    "    S = \\frac{U-F}{T} = -\\frac{\\epsilon}{2T} \\tanh\\left(\\frac{\\beta \\epsilon}{2}\\right) + k_{\\rm B} \\ln\\left[2 \\cosh \\left(\\frac{\\beta \\epsilon}{2}\\right)\\right]\n",
    "$$\n",
    "Let's now look at what each of these functions are doing."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "c4e3bf0f-09e4-4011-8baa-e0d91bc460c4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1800x450 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "kT = np.arange(0.001,10,0.05) #in units of eps\n",
    "\n",
    "U = -0.5*np.tanh(1/(kT*2))\n",
    "Cv = (1/(2*kT))**2 * (1/np.cosh(1/(kT*2)))**2\n",
    "S = U/kT + np.log(2*np.cosh(1/(kT*2)))\n",
    "\n",
    "fig, ax = plt.subplots(ncols=3,figsize=[12,3],dpi=150)\n",
    "ax[0].plot(kT,U)\n",
    "ax[0].set_xlabel(r\"$k_{\\rm B}T/\\epsilon$\")\n",
    "ax[0].set_ylabel(r\"$U/\\epsilon$\")\n",
    "ax[1].plot(kT,Cv)\n",
    "ax[1].set_xlabel(r\"$k_{\\rm B}T/\\epsilon$\")\n",
    "ax[1].set_ylabel(r\"$C_V/k_{\\rm B}$\")\n",
    "ax[2].plot(kT,S)\n",
    "ax[2].set_xlabel(r\"$k_{\\rm B}T/\\epsilon$\")\n",
    "ax[2].set_ylabel(r\"$S/k_{\\rm B}$\")\n",
    "plt.tight_layout()\n",
    "plt.savefig(\"Figures/Thermal_properties_2_level.png\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f40b87c1-41ff-4bc7-8d2d-f3c57d586dbc",
   "metadata": {},
   "source": [
    "These plots are very instructive. Let's consider 2 distinct temperature regimes and see what the plots tell us:\n",
    "### $T \\to 0$\n",
    "As the temperature drops to 0, the internal energy converges to $-\\epsilon/2$. That is, the system settles into the ground state. The heat capacity and the entropy converge to 0, as they should in accordance with the third law.\n",
    "\n",
    "### $T \\to \\infty$\n",
    "As the temperature goes to $\\infty$, the internal energy converges to $0$. This is because at these high temperatures, the probabilities that either state is populated are both 0.5, and the internal energy is $U=\\sum_i P_i E_i=-0.5 \\frac{\\epsilon}{2}+0.5\\frac{\\epsilon}{2}=0$. The heat capacity goes to 0. This can be understood by considering the fact that adding any energy to the system as this stage does not affect which state the system is likely to be in.\n",
    "\n",
    "The heat capacity does have a maximum - that is, there is a temperature at which adding more energy substantially changes the proability of the internal energy changing."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9e3661ec-99d0-4829-ad1f-0d7679e82651",
   "metadata": {},
   "source": [
    "This example demonstrates the big picture behind statistical mechanics. You write down $Z$, derive the relevant quantities, and then examine how those quantities behave for various temperatures. However, writing $Z$ down isn't always easy - there are only a handful of systems that we are explicitly able to solve for the discrete energy levels."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7233ab18-29de-4255-8d6c-3fc945c18cdd",
   "metadata": {},
   "source": [
    "## Combining Partition Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c42d9dc7-807c-46e4-8900-9aa41dfaddb8",
   "metadata": {},
   "source": [
    "Let's now consider a system where the energy has various **independent** contributions, where the contributions are distinguishable. For example, imagine the energy is given by\n",
    "$$\n",
    "    E_{i,j} = E_i^{(a)}+E_j^{(b)}\n",
    "$$\n",
    "The partition function for such a combination would be\n",
    "$$\n",
    "    Z = \\sum_i\\sum_j {\\rm e}^{-\\beta (E_i^{(a)}+E_j^{(b)})} = \\sum_i{\\rm e}^{-\\beta E_i^{(a)}}\\sum_j {\\rm e}^{-\\beta E_j^{(b)}}=Z_a Z_b\n",
    "$$\n",
    "so in this case, the partition functions of the independent contributions multiply. It is simple to generalise this to N **independent** contributions, in which case we get\n",
    "$$\n",
    "    Z = \\prod_{i=1} ^N Z_i\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e6ab079d-d4f2-4f5b-8a0e-edeaeb7c0a8a",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.8.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}