{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 02.1\n",
    "- Compute the following 1D integral via Monte Carlo \n",
    "$$I = \\int_0^1 \\frac{\\pi}{2}\\cos(\\pi x/2) dx = 1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. sampling a uniform distribution in $[0,1]$\n",
    "2. using importance sampling (i.e. sampling a non-uniform probability in $[0,1]$)\n",
    "\n",
    "<span style=\"color:red\">Show a picture of both your estimations of $I$</span> and their uncertainties with a large number of *throws* $M$ (e.g. $M\\ge 10^4$) as a function of the number of blocks, $N$</font>."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The taylor Expansion of the cosine is:\n",
    "\n",
    "$$\\cos(x) = 1 - \\frac{x^2}{2} + O(x^3).$$\n",
    "\n",
    "So I choose to sample the integral with distribution:\n",
    "$$d(x) = 1 - x^2.$$\n",
    "This must be normalized so that $\\int_0^1 d(x) \\,dx = 1$:\n",
    "$$\\int_0^1 \\left(1 - x^2\\right) \\,dx = \\left(x - \\frac{x^3}{3}\\right)\\bigg|_0^1 = \\frac{2}{3} \\quad \\implies \\quad d(x) = \\frac{3}{2}\\left(1 - x^2\\right).$$\n",
    "Inverting the cumulative distribution requires solving a cubic equation, so I resort to accept-reject to sample $d(x)$.\n",
    "$$y = F(x) = \\int_0^x dx' d(x') \\rightarrow x = F^{-1}(y)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f16e1d37f40>]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0, 1,100)\n",
    "y = 1 - x**2\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('$d(x)$')\n",
    "plt.plot(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Monte Carlo integration (uniform sampling)')"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "uniform = np.loadtxt('data/exercise-02.1-uniform.dat')\n",
    "plt.errorbar(uniform[:,0], uniform[:,1], yerr=uniform[:,2], fmt='bo')\n",
    "plt.axhline(y = 1, color = 'red', linestyle = '-')\n",
    "plt.text(60, 1+0.005, 'Exact value', color='red')\n",
    "plt.xlabel('N')\n",
    "plt.ylabel('I')\n",
    "plt.title('Monte Carlo integration (uniform sampling)')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Monte Carlo integration (importance sampling)')"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "importance = np.loadtxt('data/exercise-02.1-importance.dat')\n",
    "plt.errorbar(importance[:5,0], importance[:5,1], yerr=importance[:5,2], fmt='bo')\n",
    "plt.axhline(y = 1, color = 'red', linestyle = '-')\n",
    "plt.text(60, 1+0.0005, 'Exact value', color='red')\n",
    "plt.xlabel('N')\n",
    "plt.ylabel('I')\n",
    "plt.title('Monte Carlo integration (importance sampling)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 02.2\n",
    "- <span style=\"color:red\">3D Random Walks (RW) on a cubic lattice and in the continuum:</span> Repeat many times (e.g. say $10^4$) the simulation of a random walk in 3D always starting at the origin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. on a cubic lattice with lattice constant $a=1$; at each discrete time the walker makes a forward or backward step of length equal to $a$ in one of the 3 principal directions of the lattice: $x$, $y$ or $z$\n",
    "2. in the continuum; at each discrete time the walker makes a step of length equal to $a(=1)$ along a **random direction** obtained by sampling **uniformly** the solid angle: $\\theta \\in [0,\\pi]$ and $\\phi \\in [0,2\\pi]$\n",
    "\n",
    "<span style=\"color:red\">Show a picture of $\\sqrt{\\langle |\\vec{r}_N|^2 \\rangle_{RW}}$ for both RWs</span>, with their statistical uncertainties, as function of the step $i\\in [0,10^2]$.\n",
    "\n",
    "Suggestion: divide your $M$ throws into a reasonable number of blocks, compute $\\sqrt{\\langle |\\vec{r}_N|^2 \\rangle_{RW}}$ as function of the step $i\\in [0,10^2]$ in each block and use these values (for fixed $i$) to obtain the average value and its statistical uncertainty.\n",
    "\n",
    "Note that you could try to fit both results with a function like $f(N)=k\\sqrt{N}$. Do your results indicate a diffusive behavior?"
   ]
  }
 ],
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