tomkreker/cs166-13.1-pcw.ipynb

## cs166-13.1-pcw.ipynb
{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import scipy\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "class IsingSimulation:\n",
    "\n",
    "    J = 6.34369e-21  # Interaction constant for iron [Joule]\n",
    "    kB = 1.38065e-23  # Boltzmann constant [Joule / Kelvin]\n",
    "\n",
    "    def __init__(self, size, temperature=300):\n",
    "        '''\n",
    "        Create a new Ising model simulation. The state is initialized\n",
    "        randomly.\n",
    "\n",
    "        Arguments:\n",
    "\n",
    "            size (int) The number of cells along each dimension of the\n",
    "              square grid.\n",
    "\n",
    "            temperature (float) The temperature in Kelvin at which to\n",
    "              run the simulation. The Curie temperature for iron is\n",
    "              1043 K. Room temperature is about 300 K. Default: 300.\n",
    "        '''\n",
    "        self.size = size\n",
    "        self.temperature = temperature\n",
    "        self.state = scipy.random.choice(\n",
    "            [-1, +1], size=(self.size, self.size))\n",
    "        self.step = 0\n",
    "\n",
    "    def set_temperature(self, temp):\n",
    "        '''\n",
    "        Set the temperature of the model in Kelvin.\n",
    "        '''\n",
    "        self.temperature = float(temp)\n",
    "\n",
    "    def energy(self):\n",
    "        '''\n",
    "        Compute the energy of the current state of the model.\n",
    "        '''\n",
    "        return -self.J * (self.state * (\n",
    "            scipy.roll(self.state, 1, axis=0) +\n",
    "            scipy.roll(self.state, 1, axis=1))).sum()\n",
    "\n",
    "    def average_magnetism(self):\n",
    "        '''\n",
    "        Compute the average per-cell magnetism of the current state\n",
    "        of the model.\n",
    "        '''\n",
    "        return self.state.mean()\n",
    "\n",
    "    def update(self):\n",
    "        '''\n",
    "        Do a single cell update of the model state.\n",
    "        '''\n",
    "        # Pick a random cell to flip\n",
    "        i, j = scipy.random.randint(self.size, size=2)\n",
    "\n",
    "        # Change in energy from current state to next state\n",
    "        delta_E = 2 * self.J * self.state[i,j] * (\n",
    "            self.state[(i + 1) % self.size, j] +\n",
    "            self.state[(i - 1) % self.size, j] +\n",
    "            self.state[i, (j + 1) % self.size] +\n",
    "            self.state[i, (j - 1) % self.size])\n",
    "\n",
    "        # Log probability of changing state\n",
    "        log_p = -delta_E / (self.temperature * self.kB)\n",
    "        if scipy.log(scipy.random.uniform(0, 1)) < log_p:\n",
    "            self.state[i,j] = -self.state[i,j]\n",
    "\n",
    "        self.step += 1\n",
    "\n",
    "    def display(self):\n",
    "        '''\n",
    "        Plot the state of the model.\n",
    "        '''\n",
    "        plt.figure()\n",
    "        plt.imshow(self.state, vmin=-1, vmax=+1)\n",
    "        plt.title('Step: ' + str(self.step) + ', M = ' + str(self.average_magnetism()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\tomkr\\AppData\\Roaming\\Python\\Python36\\site-packages\\ipykernel_launcher.py:66: DeprecationWarning: scipy.log is deprecated and will be removed in SciPy 2.0.0, use numpy.lib.scimath.log instead\n"
     ]
    },
    {
     "data": {
      "image/png": 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EDknHAesl3Zf2SA6RwmMFwJxZhzccv9PMxlbLVzEknQcsBt4dEQ3/oCNiR3reBawG5ra6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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2020d395cc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "size = 20\n",
    "sim = IsingSimulation(size, temperature=1043)  # Ising model at the Curie temperature\n",
    "for _ in range(1000):\n",
    "    sim.update()\n",
    "sim.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\tomkr\\AppData\\Roaming\\Python\\Python36\\site-packages\\ipykernel_launcher.py:66: DeprecationWarning: scipy.log is deprecated and will be removed in SciPy 2.0.0, use numpy.lib.scimath.log instead\n"
     ]
    }
   ],
   "source": [
    "# Task 1 - run with temp 1 multiple time\n",
    "size = 20\n",
    "steps = 100000\n",
    "results = []\n",
    "runs = 100\n",
    "for i in range(runs):\n",
    "    sim = IsingSimulation(size, temperature=1)  # Ising model at the Curie temperature\n",
    "    for _ in range(steps):\n",
    "        sim.update()\n",
    "    #sim.display()\n",
    "    results.append(sim.average_magnetism())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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uchHwA2BJRHy70j61MtnJwAPdL8/MzFrVylUuhwCfAO6XtDC3fQk4VdI0IIDlwKd7UqGZ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\nkzwAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2020d522a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(results)\n",
    "plt.title('Average magnetism with T=1, {} runs'.format(runs))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As expected, mostly 1/-1 with some results in between from suboptimal configurations. The temperature cannot be zero because then the denominator of the 'log_p' expression will be zero and division by zero is a crime (and will crash our algorithm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\tomkr\\AppData\\Roaming\\Python\\Python36\\site-packages\\ipykernel_launcher.py:66: DeprecationWarning: scipy.log is deprecated and will be removed in SciPy 2.0.0, use numpy.lib.scimath.log instead\n"
     ]
    }
   ],
   "source": [
    "# task 2 - annealing\n",
    "size = 20\n",
    "steps = 100000\n",
    "results2 = []\n",
    "runs = 100\n",
    "start_t = 2000\n",
    "temp_step = 400\n",
    "\n",
    "for i in range(runs):\n",
    "    sim = IsingSimulation(size, temperature=start_t)  # Ising model at the Curie temperature\n",
    "    for j in range(steps):\n",
    "        if j % temp_step == 0 and sim.temperature > 1:\n",
    "            sim.temperature -= 1\n",
    "        sim.update()\n",
    "    results2.append(sim.average_magnetism())\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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zRHFbSPos0AnsNaA1ajEHjBoRsW+jNEmLJG0ZEQtzQFjcIN97gRuBb0bEgwNU\n1QHVinZYg/Xlszjz1/DP4pS0xbtFUVtI2pd04bVXRPxukOrWEu6S6pvqp04mAj+rzZA/kzINuDQi\nrh3Eug2mXtthDefP4nQraYt3i17bQtJfABcCh0TE6nehFRH+Ff5IfdB3As/mf0fm6Z3Aj/LwZ4G3\ngRmV37h2132w2yGP3wssAd4kXX19ot11b2EbTACeIT2fOiVPO4N0IgBYD7gWmA08DHyg3XVuY1t8\nKG//3wKvALPaXec2tsUdwKLKueGGdte5Lz9/GsTMzIq4S8rMzIo4YJiZWREHDDMzK+KAYWZmRRww\nzMysiAOGmZkVccAwM7Mi/wXOIRp4rZ1XEAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2020d739518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(results2)\n",
    "plt.title('Average magnetism annealing, reduce temp every 200 steps')\n",
    "plt.show()\n",
    "# this makes sense since the temperature could only drop as low as 1750 which is still very high"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\tomkr\\AppData\\Roaming\\Python\\Python36\\site-packages\\ipykernel_launcher.py:66: DeprecationWarning: scipy.log is deprecated and will be removed in SciPy 2.0.0, use numpy.lib.scimath.log instead\n"
     ]
    }
   ],
   "source": [
    "# task 2  -  faster annealing\n",
    "results3 = []\n",
    "temp_step = 50\n",
    "\n",
    "for i in range(runs):\n",
    "    sim = IsingSimulation(size, temperature=start_t)  # Ising model at the Curie temperature\n",
    "    for j in range(steps):\n",
    "        if j % temp_step == 0 and sim.temperature > 1:\n",
    "            sim.temperature -= 1\n",
    "        sim.update()\n",
    "    results3.append(sim.average_magnetism())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x2020d7f49e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(results3)\n",
    "plt.title('Average magnetism annealing, reduce temp every 50 steps')\n",
    "plt.show()\n",
    "# now it had time to reach low temperature and magnetism happens! "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "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": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}
No results found