Ising Model Solutions.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
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   "outputs": [],
   "source": [
    "import numpy as np  # package for arrays\n",
    "import matplotlib.pyplot as plt  # package for plotting\n",
    "import time  # for timing\n",
    "# display plots inside of notebook\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have an $M$ by $N$ lattice of spins. Let $\\sigma_i \\in \\{-1, 1\\}$ be the spin of lattice site $i$. The energy of the system is:\n",
    "\n",
    "$\\sum_{(i,j)} ' -J \\sigma_i \\sigma_j - \\sum_j h \\sigma _j$,\n",
    "    \n",
    "where the prime denotes a sum over nearest neighbors on the $M$ by $N$ lattice.\n",
    "\n",
    "We implement periodic boundary conditions to mimic a system of infinite extent.\n",
    "\n",
    "The following class, `IsingModel`, is a data structure to store attributes of an Ising Model and provide abstractions to manipulate the lattice for simulations.\n",
    "\n",
    "For example, to construct an instance of an `IsingModel` class as a 20 by 10 lattice with $J=1.0$ and $h=0$:\n",
    "\n",
    "    model = IsingModel(20, 10, 1.0, 0.0)\n",
    "    model.print_attributes()\n",
    "\n",
    "The lattice state is stored in an $M$ by $N$ numpy array, `lattice_state`; each entry is either -1 and 1. Each site of the lattice is initialized as -1 or 1 with equal probability in the initial construction of an instance. Retreive the lattice state by:\n",
    "\n",
    "    model.lattice_state\n",
    "    \n",
    "Plot the lattice state by:\n",
    "\n",
    "    model.plot_lattice()\n",
    "    \n",
    "We can conveniently manipulate the lattice for simulations. For example, to flip the spin in row 2 and column 3 and calculate the energy of this spin:\n",
    "\n",
    "    model.flip_spin(2, 3)\n",
    "    model.calculate_energy_of_spin(2, 3)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class IsingModel:\n",
    "    \"\"\"\n",
    "    Store attributes of an Ising lattice model\n",
    "    Provide abstractions to conveniently manipulate lattice for simulations\n",
    "    \"\"\"\n",
    "    def __init__(self, M, N, J, h):\n",
    "        \"\"\"\n",
    "        Initialization.\n",
    "        \n",
    "        parameters:\n",
    "            lattice is M by N sites\n",
    "            J: interaction between neighbors (units: kT)\n",
    "            h: background field (units: kT)\n",
    "        \"\"\"\n",
    "        # store parameters for convenience:\n",
    "        #   energetic parameters\n",
    "        self.J = J\n",
    "        self.h = h\n",
    "        \n",
    "        #   size of lattice\n",
    "        self.M = M\n",
    "        self.N = N\n",
    "        \n",
    "        # store lattice state with M by N array of -1 or 1\n",
    "        # initialize each site as -1 or 1 with equal probability\n",
    "        lattice_state = np.random.randint(-1, high=1, size=(M, N))\n",
    "        lattice_state[lattice_state == 0] = 1\n",
    "        self.lattice_state = lattice_state\n",
    "    \n",
    "    def print_params(self):\n",
    "        \"\"\"\n",
    "        Print lattice attributes\n",
    "        \"\"\"\n",
    "        print(\"\\t%d by %d lattice\" % (self.M, self.N))\n",
    "        print(\"\\tJ = %f   (+ve means preferable )\" % self.J)\n",
    "        print(\"\\th = %f\" % self.h)\n",
    "    \n",
    "    def critical_kT(self):\n",
    "        \"\"\"\n",
    "        Calculate critical temperature of this lattice model. Depends on J.\n",
    "        \"\"\"\n",
    "        return 2.0 * self.J / (np.log(1.0 + np.sqrt(2.0)))\n",
    "    \n",
    "    def flip_spin(self, i, j):\n",
    "        \"\"\"\n",
    "        Flip spin (i, j)\n",
    "        i.e. -1 ---> 1\n",
    "              1 ---> -1\n",
    "        \"\"\"\n",
    "        self.lattice_state[i, j] = - self.lattice_state[i, j]\n",
    "    \n",
    "    def calculate_energy_of_spin(self, i, j):\n",
    "        \"\"\"\n",
    "        Calculate energy of spin (i, j)\n",
    "        \n",
    "        Periodic boundary conditions implemented\n",
    "        \"\"\"\n",
    "        spin_here = self.lattice_state[i, j]  # value of spin here\n",
    "        \n",
    "        # value of spin above, below, left, and right of spin (i, j)\n",
    "        # for each, if on boundary, we wrap around to the other side\n",
    "        # of the lattice for periodic boundary conditions\n",
    "        if j == 0:\n",
    "            spin_above = self.lattice_state[i, self.N - 1]\n",
    "        else:\n",
    "            spin_above = self.lattice_state[i, j - 1]\n",
    "        \n",
    "        if j == self.N - 1:\n",
    "            spin_below = self.lattice_state[i, 0]\n",
    "        else:\n",
    "            spin_below = self.lattice_state[i, j + 1]\n",
    "            \n",
    "        if i == self.M - 1:\n",
    "            spin_right = self.lattice_state[0, j]\n",
    "        else:\n",
    "            spin_right = self.lattice_state[i + 1, j]\n",
    "        \n",
    "        if i == 0:\n",
    "            spin_left = self.lattice_state[self.M - 1, j]\n",
    "        else:\n",
    "            spin_left = self.lattice_state[i - 1, j]\n",
    "        \n",
    "        return - self.h * spin_here - self.J * spin_here *\\\n",
    "            (spin_above + spin_below + spin_left + spin_right)\n",
    "        \n",
    "    def plot_lattice(self):\n",
    "        \"\"\"\n",
    "        Plot lattice configuration\n",
    "        \"\"\"\n",
    "        plt.figure()\n",
    "        \n",
    "        imgplot = plt.imshow(self.lattice_state)\n",
    "        imgplot.set_interpolation('none')\n",
    "        \n",
    "        plt.xticks(range(self.N))\n",
    "        plt.yticks(range(self.M))\n",
    "        \n",
    "        plt.show()\n",
    "    \n",
    "    def calculate_lattice_energy_per_spin(self):\n",
    "        \"\"\"\n",
    "        Calculate energy of lattice normalized by the number of spins\n",
    "        \"\"\"\n",
    "        E = 0.0\n",
    "        for i in range(self.M):\n",
    "            for j in range(self.N):\n",
    "                E += self.calculate_energy_of_spin(i, j)\n",
    "        # factor of two for overcounting neighboring interactions.\n",
    "        # but then need to add back -1/2 h \\sum s_i \n",
    "        return E / 2.0 / (self.M * self.N) -\\\n",
    "            self.h * np.sum(self.lattice_state) / 2.0 / (self.M * self.N)\n",
    "       \n",
    "    def calculate_net_magnetization_per_spin(self):\n",
    "        \"\"\"\n",
    "        Calculate net magentization normalized by the number of spins\n",
    "        \"\"\"\n",
    "        return 1.0 * np.sum(self.lattice_state) / (self.M * self.N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example use"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "<matplotlib.figure.Figure at 0x7f8ee4b9aed0>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "model = IsingModel(3, 5, 1.0, 0.0)\n",
    "model.plot_lattice()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model.calculate_energy_of_spin(2, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8ee4b7aa90>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "model.flip_spin(0, 0)\n",
    "model.plot_lattice()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Autocorrelation functions to determine how many sweeps needed before samples become statistically independent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def autocorrelation_k_steps(data, k):\n",
    "    \"\"\"\n",
    "    Calculate autocorrelation of data after k steps\n",
    "    https://en.wikipedia.org/wiki/Autocorrelation#Estimation\n",
    "    \"\"\"\n",
    "    mu = np.mean(data)  # mean\n",
    "    n = np.size(data)  # number of data points\n",
    "    variance = np.var(data)  # variance\n",
    "    \n",
    "    corr = 0.0  # autocorrelation\n",
    "    for t in range(n - k):\n",
    "        corr += (data[t] - mu) * (data[t + k] - mu)\n",
    "    return corr / (variance * (n - k))\n",
    "\n",
    "def plot_autocorrelation(data, kmax):\n",
    "    \"\"\"\n",
    "    Plot autocorrelation function of data up to kmax steps\n",
    "    \"\"\"\n",
    "    corr = np.zeros((kmax, ))\n",
    "    for k in range(kmax):\n",
    "        corr[k] = autocorrelation_k_steps(data, k)\n",
    "        \n",
    "    fig = plt.figure()\n",
    "    \n",
    "    plt.xlabel(\"Monte Carlo cycle\")\n",
    "    plt.ylabel(\"Autocorrelation\")\n",
    "    \n",
    "    plt.ylim([0, 1])\n",
    "    \n",
    "    plt.plot(np.arange(kmax), corr, marker='s')\n",
    "    \n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Function to sweep the lattice in a Monte Carlo simulation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our function `sweep_lattice` will perform $NM$ Monte Carlo spin flip trials. We call it `sweep_lattice` because, on average, each spin will be selected for a Monte Carlo trial once during a 'sweep'."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def sweep_lattice(isingmodel, kT):\n",
    "    \"\"\"\n",
    "    Perform N*M Monte Carlo spin flip attempts\n",
    "    \n",
    "    parameters:\n",
    "        isingModel: Ising lattice model object instance\n",
    "        kT: Boltzmann constant times temperature, $kT$\n",
    "    returns:\n",
    "        fraction of Monte Carlo proposals accepted\n",
    "    \"\"\"\n",
    "    num_flips = isingmodel.M * isingmodel.N\n",
    "    # keep track of number of Monte Carlo propsoals that are accepted\n",
    "    n_accepted = 0\n",
    "    \n",
    "    for flip in range(num_flips):\n",
    "        # choose a spin to flip at random\n",
    "        i = np.random.randint(0, high=isingmodel.M)\n",
    "        j = np.random.randint(0, high=isingmodel.N)\n",
    "        \n",
    "        # calculate old energy\n",
    "        E_old = isingmodel.calculate_energy_of_spin(i, j)\n",
    "        \n",
    "        # flip the spin\n",
    "        isingmodel.flip_spin(i, j)\n",
    "        \n",
    "        # calculate new energy\n",
    "        E_new = isingmodel.calculate_energy_of_spin(i, j)\n",
    "        \n",
    "        # Monte Carlo: accept this proposal with probability e^{-\\beta \\Delta E}\n",
    "        if np.random.uniform(0, 1) > np.exp(-(E_new - E_old) / kT):\n",
    "            # reject move, flip spin back to original state\n",
    "            isingmodel.flip_spin(i, j)\n",
    "        else:\n",
    "            # else: accept, do nothing\n",
    "            n_accepted += 1\n",
    "    return 1.0 * n_accepted / (model.M * model.N)  # return fraction of MC moves accepted"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Function for Monte Carlo simulations of Ising Model in Canonical Ensemble"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def simulate(isingmodel, kT, num_sweeps, num_burn_sweeps, sample_frequency):\n",
    "    \"\"\"\n",
    "    Monte Carlo simulations of Ising model in Canonical ensemble\n",
    "    \n",
    "    parameters:\n",
    "        isingModel: Ising lattice model object instance\n",
    "        kT: Boltzmann constant times temperature, $kT$\n",
    "        num_sweeps: total number of Monte Carlo sweeps to perform in the simulation\n",
    "        num_burn_sweeps: number of sweeps for equilibration (no samples taken during this period)\n",
    "        sample_frequency: frequency at which to take samples\n",
    "    \"\"\"  \n",
    "    t0 = time.time()  # keep track of timing\n",
    "    \n",
    "    # Print info\n",
    "    print(\"Simulating Ising model:\")\n",
    "    isingmodel.print_params()\n",
    "    print(\"\\tTemperature (kT) = %f\" % kT)\n",
    "    print(\"\\t%d total sweeps, %d of them burn sweeps\" % (\n",
    "            num_sweeps, num_burn_sweeps))\n",
    "    print(\"\\t\\tSampling every %d sweeps\" % sample_frequency)\n",
    "    \n",
    "    # store energy and net magnetization samples here\n",
    "    energy_samples = np.zeros(((num_sweeps - num_burn_sweeps) / sample_frequency, ))\n",
    "    magnetization_samples = np.zeros(((num_sweeps - num_burn_sweeps) / sample_frequency, ))\n",
    "    \n",
    "    # keep track of number of samples taken\n",
    "    n_samples = 0\n",
    "    # keep track of fraction of proposals accepted\n",
    "    fraction_accepted_during_burn_sweeps = 0\n",
    "    fraction_accepted_during_sampling_sweeps = 0\n",
    "\n",
    "    for sweep in range(num_sweeps):\n",
    "        # sweep lattice\n",
    "        fraction_accepted = sweep_lattice(model, kT)\n",
    "\n",
    "        # if burn sweeps are done, take samples\n",
    "        if sweep >= num_burn_sweeps:\n",
    "            fraction_accepted_during_sampling_sweeps += fraction_accepted\n",
    "            \n",
    "            # take a sample every sample_frequency sweeps\n",
    "            if ((sweep - num_burn_sweeps) % sample_frequency) == 0:\n",
    "                energy_samples[n_samples] = model.calculate_lattice_energy_per_spin()\n",
    "                \n",
    "                magnetization_samples[n_samples] = model.calculate_net_magnetization_per_spin()\n",
    "                \n",
    "                n_samples += 1\n",
    "        else:\n",
    "            # if still in burn regime, keep track of fraction accepted\n",
    "            fraction_accepted_during_burn_sweeps += fraction_accepted\n",
    "        \n",
    "    print(\"\\t\\tFraction proposals accepted during burn sweeps = %f\" % (\n",
    "        1.0 * fraction_accepted_during_burn_sweeps / num_burn_sweeps))\n",
    "    print(\"\\t\\tFraction proposals accepted during sampling regime = %f\" % (\n",
    "        1.0 * fraction_accepted_during_sampling_sweeps / (num_sweeps - num_burn_sweeps)))\n",
    "    \n",
    "    print(\"\\t<E> = %f +/- %f\" % (np.mean(energy_samples),\n",
    "                                 1.96 * np.std(energy_samples) / np.sqrt(n_samples)))\n",
    "    print(\"\\t<m> = %f +/- %f\" % (np.mean(magnetization_samples), \n",
    "                                 1.96 * np.std(magnetization_samples) / np.sqrt(n_samples)))\n",
    "    \n",
    "    print(\"\\tSimulation finished. Took %s sec.\" % (time.time() - t0))\n",
    "    assert((num_sweeps - num_burn_sweeps) / sample_frequency)\n",
    "    \n",
    "    return energy_samples, magnetization_samples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# An example Monte Carlo simulation of the Ising Model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "model = IsingModel(20, 20, 1.0, 0.10)\n",
    "critical_kT = model.critical_kT()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Simulating Ising model:\n",
      "\t20 by 20 lattice\n",
      "\tJ = 1.000000   (+ve means preferable )\n",
      "\th = 0.100000\n",
      "\tTemperature (kT) = 2.382645\n",
      "\t2000 total sweeps, 1000 of them burn sweeps\n",
      "\t\tSampling every 1 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.166808\n",
      "\t\tFraction proposals accepted during sampling regime = 0.172552\n",
      "\t<E> = -1.561524 +/- 0.007723\n",
      "\t<m> = 0.785645 +/- 0.004830\n",
      "\tSimulation finished. Took 4.49448895454 sec.\n"
     ]
    },
    {
     "data": {
      "image/png": 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0jdx7j6QlNeVrZi0pVd4bEcdMT0v6HPC7+lMzsybl6fC7WtLcjJB3AK+vLyUza0OlDj9J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\nZmPKK7/ZmPLKbzam/h9bJ1AUcdghXwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8ee2a5a390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "energy_samples, magnetization_samples = simulate(model, model.critical_kT() * 1.05, 2000, 1000, 1)\n",
    "model.plot_lattice()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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/nHBXKn4HZlZlZi8Aa4A6d19ESu5/ntihwHufhsRQCT7l7h8HPgtcFjZ3pF15\nt0Fm+w1wkLsPJ/gfJg3NGb2BPwFXhX95t7zfZX3/c8Sfmt+Bu29392MIamqfMbMaUnL/W8R+opmd\nRAfufRoSwypg/4ztfcN9qeHub4T/rgf+SutpQdJgbfMcVmE78rqE44nM3df7js60m4ARScbTHjPr\nTvCleru73x/uTs39zxV/2n4HAO6+EXgYOI4U3X/4KPaHgOM6cu/TkBg+GiRnZj0IBslNTzimyMxs\nl/CvJ8xsV+B04OVko4rEyG6XnA58OXz/JeD+lh8oI1mxh/8jNzuX8r//twKL3P0XGfvSdP9bxZ+W\n34GZ9W9uajGznYHTgBdIwf3PE/v8jtz7sn8qCT5a0+EX7BgkNyXhkCIzswMJaglOMKDwznKP38zu\nAmqAPQkmM5wM/A24D9gPWA78p7u/m1SM+eSJ/WSCtu7twGvApblG15cDM/sUMBN4ieC/GQeuAeYC\n0yj/+58v/vGk4HdgZkcRdC43PzByu7v/xMz2oMzvfxux30aB9z4ViUFEREonDU1JIiJSQkoMIiKS\nRYlBRESyKDGIiEgWJQYREcmixCAiIlmUGKQsmdn28Pnr5u1uZrbezDo0uNHMdjOzr3Xgc7ua2Y1m\ntjSc6+pJMyto1K6ZzTCzjxd67c4ws1fDZ+9FCqbEIOVqMzDUzHqG26eRPZliofoBX+/A524mmG75\nEHcfQTD9df+oHw6njU+CBihJhykxSDl7GDgzfD8OuLv5QLhwyl8tWADpX2Y2NNw/2YLFemaEf+Vf\nHn7kWuCgcKGS68Ky3zKzueGsk5NbXtzMDiKY1+r/Nu8Lp39/JDz+17AW8ZKZfTXjc5vM7CfhLJef\nbHHOcWa2IHxdm+uHNrMRZjYrjGu2mfU2s6fMbFhGmX+a2VFhjebW8Hzzzeyc5iIZZb9gwQIuz5vZ\nb80sFdNfS3KUGKRcOcHaG+PCWsMwdkzfDPA94Hl3Pxr4LnB7xrHDCGoYnwBqw0VLJgGvuPvH3f1q\nMzsNONTdRwLHAMflmPX2SGC+558eYEJYixgBXGVm/cL9uwLPuPsx7j6rubCZfQyYQjBlx3BgpJl9\nPvOEZrZT+HNfEc6GOQp4n6DmMiEsUw30dPeXgP8B3nX3YWH5J1ucbwjBIjknhDP8bge+kOfnEQGU\nGKSMufvLwAEEtYWHyJ7U79OEycDdZwB7NE9WCDzk7lvd/S2C+ZIG5Dj96cBpZvY88DxBMjm0wBC/\nYWbzCdY7Di9JAAACBklEQVRL2Dfj81uBv+QoPwKY4e5vu/t24E7gxBZlDgNWu/vz4c/2nrtvI5it\n9MwwyU0Afh+WHwX8uvnD7r6h+W3476nAx4F5YQ3mFOCgAn9O6WK6Jx2ASDumAz8m+Cs7atv+Bxnv\nt5P7v3MDrnX3m9o4z0LgaDOzlrWGcJ77U4BPuPsHZjYD6BUebmqjlhGlGadVGXd/38weJ1g57Hzg\n2OZDEc71R3f/boTrigCqMUj5av5yvBX4nrsvbHH8n8BFAOFCKm+GC8Lkswnok7H9d+Ar4VTomNk+\nZrZX5gfcfRnwLEGzFWG5wWb2WWA34J0wKQwBjs8Re0tzCRZP2SP8y38c8FSLMkuAgWZ2bHi93hkd\n2LcAvwTmZtQMHgcuy4hv9xYxPAGc1/yzhX0zmeubiLSixCDlygHcfZW735DjeC1wrJm9CPwI+GI7\n53kbmBV20l7n7o8TdGY/Y2YLCKYU753j818l+KJeGpb7PUHz1KPATma2MLz+My2vmSOGNQR9HXUE\nc/zPc/cHsgoG65pfANwQNlM9RlgTCZuXNrKjGQnghwTNaC+FTUU1La5ZT9B5/lh4rx4DMufnF2lF\n026LpISZ7QM86e5Dko5FKptqDCIpYGYXE9RKrkk6Fql8qjGIiEgW1RhERCSLEoOIiGRRYhARkSxK\nDCIikkWJQUREsigxiIhIlv8PSxavqdMilroAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8ee4b9ae90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_autocorrelation(energy_samples, 35)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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txJ73vS/5FkM4Sa6OjElywLmZk+RKnZm9Axzp7h/FHUsuzOxY4FPgAXc/LCy7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\nICIiDfx/DEaGNIr/HsgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8ee283f0d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_autocorrelation(magnetization_samples, 35)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Complete elliptic integrals of the first and second kind"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the Scipy website, note that these differ from the definition in Koonin and Meredith.\n",
    "\n",
    "http://docs.scipy.org/doc/scipy/reference/generated/scipy.special.ellipk.html#scipy.special.ellipk\n",
    "http://docs.scipy.org/doc/scipy/reference/generated/scipy.special.ellipe.html#scipy.special.ellipe\n",
    "\n",
    "`ellipk(kappa**2)`\n",
    "returns $K_1(\\kappa)$\n",
    "\n",
    "`ellipe(kappa**2)`\n",
    "returns $E_1(\\kappa)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.special import ellipk, ellipe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.6596235986105281"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ellipk(0.2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.4890350580958529"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ellipe(0.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 8.6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$J=0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If $J=0$, $E(s)=-h\\sum s_i$.\n",
    "\n",
    "Now, each spin is independent of the configuration of the others. So we can write the single-particle partition function $q$ as:\n",
    "\n",
    "$q=e^{-\\beta h} + e^{\\beta h}=2 \\cosh(\\beta h)$\n",
    "\n",
    "The partition function for the entire lattice, $Q$, is the product of these single-particle partition functions:\n",
    "\n",
    "$Q=q^N,$ where $N$ is the total number of lattice sites.\n",
    "\n",
    "i.e. $Q=(2\\cosh(\\beta h))^N$\n",
    "\n",
    "From this partition function $Q$, we can calculate average thermodynamic quantities:\n",
    "\n",
    "$\\langle E \\rangle =- \\frac{\\partial \\log Q}{\\partial \\beta}$\n",
    "\n",
    "$\\langle m \\rangle = \\frac{\\partial \\log Q}{\\partial \\beta h}$\n",
    "\n",
    "and then the magenetic susceptibility $\\Xi$ and heat capacity $c_v$, defined as:\n",
    "\n",
    "$ \\chi = \\frac{\\partial \\langle m \\rangle}{\\partial h}$\n",
    "\n",
    "$ c_v  = \\frac{\\partial \\langle E \\rangle}{\\partial T}$\n",
    "\n",
    "With some calculus:\n",
    "\n",
    "$\\langle E \\rangle = -N h \\tanh(\\beta h)$\n",
    "\n",
    "$\\langle m \\rangle = h \\tanh(\\beta h)$\n",
    "\n",
    "$\\chi = \\beta (\\langle m^2 \\rangle - \\langle m \\rangle^2)= \\beta (1-\\tanh^2(\\beta h)) / N_{sites}$\n",
    "\n",
    "$c_v/k = \\beta^2 (\\langle e^2 \\rangle - \\langle e \\rangle^2)= (\\beta h) ^2 (1-\\tanh^2(\\beta h)) / N_{sites}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try different $h$'s and simulate the Ising model to compare to this analytical formula. The `linspace` function in numpy will generate a span of $h$'s."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Simulating Ising model:\n",
      "\t20 by 20 lattice\n",
      "\tJ = 0.000000   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t3000 total sweeps, 2000 of them burn sweeps\n",
      "\t\tSampling every 4 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 1.000000\n",
      "\t\tFraction proposals accepted during sampling regime = 1.000000\n",
      "\t<E> = 0.000000 +/- 0.000000\n",
      "\t<m> = -0.004880 +/- 0.006155\n",
      "\tSimulation finished. Took 9.93126893044 sec.\n",
      "Simulating Ising model:\n",
      "\t20 by 20 lattice\n",
      "\tJ = 0.000000   (+ve means preferable )\n",
      "\th = 0.750000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t3000 total sweeps, 2000 of them burn sweeps\n",
      "\t\tSampling every 4 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.364840\n",
      "\t\tFraction proposals accepted during sampling regime = 0.365087\n",
      "\t<E> = -0.477810 +/- 0.003528\n",
      "\t<m> = 0.637080 +/- 0.004704\n",
      "\tSimulation finished. Took 10.8095188141 sec.\n",
      "Simulating Ising model:\n",
      "\t20 by 20 lattice\n",
      "\tJ = 0.000000   (+ve means preferable )\n",
      "\th = 1.500000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t3000 total sweeps, 2000 of them burn sweeps\n",
      "\t\tSampling every 4 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.095137\n",
      "\t\tFraction proposals accepted during sampling regime = 0.094340\n",
      "\t<E> = -1.359960 +/- 0.003875\n",
      "\t<m> = 0.906640 +/- 0.002583\n",
      "\tSimulation finished. Took 11.0972828865 sec.\n",
      "Simulating Ising model:\n",
      "\t20 by 20 lattice\n",
      "\tJ = 0.000000   (+ve means preferable )\n",
      "\th = 2.250000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t3000 total sweeps, 2000 of them burn sweeps\n",
      "\t\tSampling every 4 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.022461\n",
      "\t\tFraction proposals accepted during sampling regime = 0.022382\n",
      "\t<E> = -2.201445 +/- 0.002882\n",
      "\t<m> = 0.978420 +/- 0.001281\n",
      "\tSimulation finished. Took 10.9956581593 sec.\n",
      "Simulating Ising model:\n",
      "\t20 by 20 lattice\n",
      "\tJ = 0.000000   (+ve means preferable )\n",
      "\th = 3.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t3000 total sweeps, 2000 of them burn sweeps\n",
      "\t\tSampling every 4 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.005153\n",
      "\t\tFraction proposals accepted during sampling regime = 0.005190\n",
      "\t<E> = -2.986140 +/- 0.001750\n",
      "\t<m> = 0.995380 +/- 0.000583\n",
      "\tSimulation finished. Took 11.0862588882 sec.\n"
     ]
    }
   ],
   "source": [
    "kT = 1.0\n",
    "# array of h's to loop over\n",
    "h = np.linspace(0.0, 3.0, 5)\n",
    "# pre-allocate arrays to store thermodyanmic quantities\n",
    "E = np.zeros((np.size(h), ))\n",
    "m = np.zeros((np.size(h), ))\n",
    "cv_ovr_k = np.zeros((np.size(h), ))\n",
    "chi = np.zeros((np.size(h), ))\n",
    "\n",
    "for i in range(np.size(h)):\n",
    "    model = IsingModel(20, 20, 0.0, h[i])\n",
    "    energy_samples, magnetization_samples = simulate(model, kT, 3000, 2000, 4)\n",
    "    \n",
    "    # record <E>, <m> at h[i]\n",
    "    E[i] = np.mean(energy_samples)\n",
    "    m[i] = np.mean(magnetization_samples)\n",
    "    # related to fluctuations. multiply by M * N for cv of SYSTEM\n",
    "    cv_ovr_k[i] = (np.mean(energy_samples ** 2) - np.mean(energy_samples) ** 2) / (kT ** 2)\n",
    "    chi[i] = (np.mean(magnetization_samples ** 2) - np.mean(magnetization_samples) ** 2) / kT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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E830VvE5x914ql759YfFiWLYMOnSApUvDjih8SiwR9N133zFt2jRmz55NkyZN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\nVrhM8xLKlLUS43T3bXur0O7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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8ee0ec8d50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot results, compare to analytical model\n",
    "plt.figure()\n",
    "plt.scatter(h, E, label='Simulation')  # plot simulated data\n",
    "plt.plot(h, -h * np.tanh(h / kT), label='Analytical solution')  # plot analytical sol'n\n",
    "plt.xlabel(\"h (in units of $kT$)\")\n",
    "plt.ylabel(r\"$\\langle e \\rangle$\")\n",
    "plt.legend(loc='lower left')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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R/Otf/yInJ6fY43v27EnPnj1p1aoVLVq0IDk5udAv37yGDh3KqlWruOKKK/K9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\nElVKLCIiElX/Px4DxMR8RBQgAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8ee015ee50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot results, compare to analytical model\n",
    "plt.figure()\n",
    "plt.scatter(h, m, label='Simulation')  # plot simulated data\n",
    "plt.plot(h, np.tanh(h / kT), label='Analytical solution')  # plot analytical sol'n\n",
    "plt.xlabel(\"h (in units of $kT$)\")\n",
    "plt.ylabel(r\"$\\langle m \\rangle$\")\n",
    "plt.legend(loc='lower right')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8ee276a410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot results, compare to analytical model\n",
    "plt.figure()\n",
    "plt.scatter(h, cv_ovr_k, label='Simulation')  # plot simulated data\n",
    "plt.plot(h, (h / kT) ** 2 * (1.0 - np.tanh(h / kT) ** 2) / (model.M * model.N), label='Analytical solution')  # plot analytical sol'n\n",
    "plt.xlabel(\"h (in units of $kT$)\")\n",
    "plt.ylabel(r\"$c_v$\")\n",
    "plt.legend(loc='upper right')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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FwK3RDcwsA3gEGACcDIw0sxPCk5cDw4C3o/qcCFwGnAhcCDxqZlZXKyEiIolL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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8ee2986f90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot results, compare to analytical model\n",
    "plt.figure()\n",
    "plt.scatter(h, chi, label='Simulation')  # plot simulated data\n",
    "plt.plot(h, (1.0 - np.tanh(h / kT) ** 2) / kT / (model.M * model.N), label='Analytical solution')  # plot analytical sol'n\n",
    "plt.xlabel(\"h (in units of $kT$)\")\n",
    "plt.ylabel(r\"$\\chi$\")\n",
    "plt.legend(loc='upper right')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 8.7 and 8.9 together"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot analytical solution to Ising Model in the limit of an infinitely large lattice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def exact_soln(model, kT):\n",
    "    \"\"\"\n",
    "    Return average energy\n",
    "    \n",
    "    Parameters:\n",
    "        model: an instance of the IsingModel class.\n",
    "        kT: temperature ($k_b T$)\n",
    "    returns dictionary of exact values\n",
    "        e.g:\n",
    "            soln = exact_soln(model, kt)\n",
    "            soln['cv']\n",
    "        returns cv.\n",
    "    \"\"\"  \n",
    "    num_sites = model.M * model.N\n",
    "    J_theirs = model.J / kT\n",
    "    K = 2.0 * np.sinh(2.0 * J_theirs) / np.cosh(2.0 * J_theirs) ** 2\n",
    "    K1 = ellipk(K ** 2)\n",
    "    K_prime = 2.0 * np.tanh(2.0 * J_theirs) ** 2 - 1.0\n",
    "    E1 = ellipe(K ** 2)\n",
    "    \n",
    "    z = np.exp(-2.0 * J_theirs)\n",
    "    \n",
    "    avg_E = -J_theirs / np.tanh(2.0 * J_theirs) * (1.0 + 2.0 / np.pi * K_prime * K1)\n",
    "    cv = 2.0 / np.pi * (J_theirs / np.tanh(2.0 * J_theirs)) ** 2 *\\\n",
    "        (2.0 * K1 - 2.0 * E1 - (1.0 - K_prime) * (np.pi/2.0 + K_prime * K1))\n",
    "    avg_m = (1.0 + z**2)**0.25 * (1.0 - 6.0 * z ** 2 + z ** 4) ** (1.0/8.0) / np.sqrt(1.0-z**2)\n",
    "    \n",
    "    return {'m': avg_m, 'cv': cv, 'e': avg_E}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example of how this function works:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\t20 by 20 lattice\n",
      "\tJ = 1.000000   (+ve means preferable )\n",
      "\th = 0.000000\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{'cv': 1.4142776913433117, 'e': -0.7029496109538429, 'm': 0.78475513138392128}"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model = IsingModel(20, 20, 1.0, 0.0)\n",
    "model.print_params()\n",
    "soln = exact_soln(model, 2.2)\n",
    "soln"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.7029496109538429"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "soln['e']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Explore parameter space; compare simulation to analytical model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When $h=0$, an Ising model simulation is characterized by $M$, $N$, and $J/k_B T$. Equivalently, we can fix $kT$ and vary $J$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "kT = 1.0\n",
    "J = np.linspace(0.01, 0.8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Store analytical solution in these pre-allocated arrays."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "cvs = np.zeros((np.size(J), ))\n",
    "avg_es = np.zeros((np.size(J), ))\n",
    "avg_ms = np.zeros((np.size(J), ))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Store simulated solutions here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "cvs_sim = np.zeros((np.size(J), ))\n",
    "avg_es_sim = np.zeros((np.size(J), ))\n",
    "avg_ms_sim = np.zeros((np.size(J), ))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.010000   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.985082\n",
      "\t\tFraction proposals accepted during sampling regime = 0.985048\n",
      "\t<E> = -0.000183 +/- 0.000058\n",
      "\t<m> = 0.000295 +/- 0.004264\n",
      "\tSimulation finished. Took 26.8854730129 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.026122   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.960974\n",
      "\t\tFraction proposals accepted during sampling regime = 0.960618\n",
      "\t<E> = -0.001381 +/- 0.000149\n",
      "\t<m> = -0.002109 +/- 0.004178\n",
      "\tSimulation finished. Took 26.0055580139 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.042245   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.936751\n",
      "\t\tFraction proposals accepted during sampling regime = 0.936434\n",
      "\t<E> = -0.003719 +/- 0.000228\n",
      "\t<m> = -0.000634 +/- 0.004408\n",
      "\tSimulation finished. Took 34.9737138748 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.058367   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.912159\n",
      "\t\tFraction proposals accepted during sampling regime = 0.912476\n",
      "\t<E> = -0.006756 +/- 0.000328\n",
      "\t<m> = -0.002049 +/- 0.004557\n",
      "\tSimulation finished. Took 35.149805069 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.074490   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.887419\n",
      "\t\tFraction proposals accepted during sampling regime = 0.887910\n",
      "\t<E> = -0.010977 +/- 0.000423\n",
      "\t<m> = 0.002309 +/- 0.004950\n",
      "\tSimulation finished. Took 35.2088198662 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.090612   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.863412\n",
      "\t\tFraction proposals accepted during sampling regime = 0.863325\n",
      "\t<E> = -0.016597 +/- 0.000519\n",
      "\t<m> = 0.003264 +/- 0.005113\n",
      "\tSimulation finished. Took 35.1798188686 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.106735   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.838973\n",
      "\t\tFraction proposals accepted during sampling regime = 0.838650\n",
      "\t<E> = -0.023298 +/- 0.000647\n",
      "\t<m> = -0.004080 +/- 0.005192\n",
      "\tSimulation finished. Took 35.2942621708 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.122857   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.813769\n",
      "\t\tFraction proposals accepted during sampling regime = 0.813362\n",
      "\t<E> = -0.030584 +/- 0.000729\n",
      "\t<m> = -0.003021 +/- 0.004932\n",
      "\tSimulation finished. Took 35.3450169563 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.138980   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.789295\n",
      "\t\tFraction proposals accepted during sampling regime = 0.789346\n",
      "\t<E> = -0.039935 +/- 0.000912\n",
      "\t<m> = 0.002934 +/- 0.006047\n",
      "\tSimulation finished. Took 35.4257879257 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.155102   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.763644\n",
      "\t\tFraction proposals accepted during sampling regime = 0.763969\n",
      "\t<E> = -0.049953 +/- 0.000910\n",
      "\t<m> = -0.001536 +/- 0.006077\n",
      "\tSimulation finished. Took 35.4322469234 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.171224   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.738987\n",
      "\t\tFraction proposals accepted during sampling regime = 0.738220\n",
      "\t<E> = -0.061626 +/- 0.000997\n",
      "\t<m> = -0.004514 +/- 0.006253\n",
      "\tSimulation finished. Took 35.3428740501 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.187347   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.713087\n",
      "\t\tFraction proposals accepted during sampling regime = 0.713250\n",
      "\t<E> = -0.074041 +/- 0.001179\n",
      "\t<m> = 0.001302 +/- 0.006313\n",
      "\tSimulation finished. Took 35.4189300537 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.203469   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.686584\n",
      "\t\tFraction proposals accepted during sampling regime = 0.686817\n",
      "\t<E> = -0.089488 +/- 0.001280\n",
      "\t<m> = 0.003168 +/- 0.007064\n",
      "\tSimulation finished. Took 35.6282119751 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.219592   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.659368\n",
      "\t\tFraction proposals accepted during sampling regime = 0.660836\n",
      "\t<E> = -0.104954 +/- 0.001405\n",
      "\t<m> = 0.000677 +/- 0.007670\n",
      "\tSimulation finished. Took 35.6793830395 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.235714   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.634008\n",
      "\t\tFraction proposals accepted during sampling regime = 0.633906\n",
      "\t<E> = -0.121557 +/- 0.001611\n",
      "\t<m> = 0.001241 +/- 0.007985\n",
      "\tSimulation finished. Took 31.9926559925 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.251837   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.607047\n",
      "\t\tFraction proposals accepted during sampling regime = 0.605488\n",
      "\t<E> = -0.141711 +/- 0.001539\n",
      "\t<m> = -0.007352 +/- 0.007944\n",
      "\tSimulation finished. Took 30.0467078686 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.267959   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.579002\n",
      "\t\tFraction proposals accepted during sampling regime = 0.578862\n",
      "\t<E> = -0.162725 +/- 0.001685\n",
      "\t<m> = 0.004106 +/- 0.009088\n",
      "\tSimulation finished. Took 32.9923009872 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.284082   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.549985\n",
      "\t\tFraction proposals accepted during sampling regime = 0.550617\n",
      "\t<E> = -0.186710 +/- 0.001905\n",
      "\t<m> = -0.000243 +/- 0.009665\n",
      "\tSimulation finished. Took 29.1815109253 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.300204   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.521233\n",
      "\t\tFraction proposals accepted during sampling regime = 0.522471\n",
      "\t<E> = -0.212577 +/- 0.002103\n",
      "\t<m> = -0.005825 +/- 0.010674\n",
      "\tSimulation finished. Took 24.1588242054 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.316327   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.493320\n",
      "\t\tFraction proposals accepted during sampling regime = 0.492067\n",
      "\t<E> = -0.239936 +/- 0.002482\n",
      "\t<m> = 0.010373 +/- 0.011720\n",
      "\tSimulation finished. Took 22.5794479847 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.332449   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.464092\n",
      "\t\tFraction proposals accepted during sampling regime = 0.462605\n",
      "\t<E> = -0.273203 +/- 0.002600\n",
      "\t<m> = 0.008125 +/- 0.014581\n",
      "\tSimulation finished. Took 26.8091981411 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.348571   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.434126\n",
      "\t\tFraction proposals accepted during sampling regime = 0.432287\n",
      "\t<E> = -0.303324 +/- 0.002898\n",
      "\t<m> = -0.035825 +/- 0.014839\n",
      "\tSimulation finished. Took 32.4453659058 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.364694   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.399204\n",
      "\t\tFraction proposals accepted during sampling regime = 0.398525\n",
      "\t<E> = -0.342420 +/- 0.003156\n",
      "\t<m> = 0.011259 +/- 0.018308\n",
      "\tSimulation finished. Took 32.9885690212 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.380816   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.366789\n",
      "\t\tFraction proposals accepted during sampling regime = 0.365217\n",
      "\t<E> = -0.383990 +/- 0.003320\n",
      "\t<m> = 0.028759 +/- 0.021584\n",
      "\tSimulation finished. Took 37.0623738766 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.396939   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.328279\n",
      "\t\tFraction proposals accepted during sampling regime = 0.333078\n",
      "\t<E> = -0.429280 +/- 0.003732\n",
      "\t<m> = -0.074922 +/- 0.026707\n",
      "\tSimulation finished. Took 37.917703867 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.413061   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.296337\n",
      "\t\tFraction proposals accepted during sampling regime = 0.293644\n",
      "\t<E> = -0.485419 +/- 0.004236\n",
      "\t<m> = 0.057378 +/- 0.034419\n",
      "\tSimulation finished. Took 37.4019610882 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.429184   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.233734\n",
      "\t\tFraction proposals accepted during sampling regime = 0.230568\n",
      "\t<E> = -0.572342 +/- 0.005215\n",
      "\t<m> = -0.078099 +/- 0.055989\n",
      "\tSimulation finished. Took 37.3161380291 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.445306   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.181449\n",
      "\t\tFraction proposals accepted during sampling regime = 0.165842\n",
      "\t<E> = -0.668871 +/- 0.004434\n",
      "\t<m> = -0.770451 +/- 0.008960\n",
      "\tSimulation finished. Took 36.6339850426 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.461429   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.137343\n",
      "\t\tFraction proposals accepted during sampling regime = 0.133647\n",
      "\t<E> = -0.734184 +/- 0.004424\n",
      "\t<m> = 0.823932 +/- 0.008124\n",
      "\tSimulation finished. Took 37.1369731426 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.477551   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.113102\n",
      "\t\tFraction proposals accepted during sampling regime = 0.104629\n",
      "\t<E> = -0.796548 +/- 0.003699\n",
      "\t<m> = -0.874115 +/- 0.004736\n",
      "\tSimulation finished. Took 37.1866159439 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.493673   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.090118\n",
      "\t\tFraction proposals accepted during sampling regime = 0.087914\n",
      "\t<E> = -0.848527 +/- 0.003422\n",
      "\t<m> = 0.899366 +/- 0.003551\n",
      "\tSimulation finished. Took 37.0187978745 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.509796   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.075960\n",
      "\t\tFraction proposals accepted during sampling regime = 0.069270\n",
      "\t<E> = -0.904613 +/- 0.003080\n",
      "\t<m> = 0.924505 +/- 0.002759\n",
      "\tSimulation finished. Took 37.0243210793 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.525918   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.063725\n",
      "\t\tFraction proposals accepted during sampling regime = 0.058883\n",
      "\t<E> = -0.947712 +/- 0.003138\n",
      "\t<m> = 0.935365 +/- 0.002584\n",
      "\tSimulation finished. Took 37.097520113 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.542041   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.052122\n",
      "\t\tFraction proposals accepted during sampling regime = 0.047570\n",
      "\t<E> = -0.998804 +/- 0.002637\n",
      "\t<m> = 0.950174 +/- 0.002025\n",
      "\tSimulation finished. Took 37.2304561138 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.558163   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.043307\n",
      "\t\tFraction proposals accepted during sampling regime = 0.039907\n",
      "\t<E> = -1.040723 +/- 0.002753\n",
      "\t<m> = -0.958802 +/- 0.001832\n",
      "\tSimulation finished. Took 37.2567398548 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.574286   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.042280\n",
      "\t\tFraction proposals accepted during sampling regime = 0.033796\n",
      "\t<E> = -1.082010 +/- 0.002437\n",
      "\t<m> = 0.965156 +/- 0.001551\n",
      "\tSimulation finished. Took 32.7042148113 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.590408   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.030587\n",
      "\t\tFraction proposals accepted during sampling regime = 0.028492\n",
      "\t<E> = -1.122237 +/- 0.002179\n",
      "\t<m> = -0.970877 +/- 0.001276\n",
      "\tSimulation finished. Took 30.3476259708 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.606531   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.031069\n",
      "\t\tFraction proposals accepted during sampling regime = 0.024153\n",
      "\t<E> = -1.159853 +/- 0.002189\n",
      "\t<m> = -0.974844 +/- 0.001144\n",
      "\tSimulation finished. Took 37.2795369625 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.622653   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.021633\n",
      "\t\tFraction proposals accepted during sampling regime = 0.020516\n",
      "\t<E> = -1.200153 +/- 0.002030\n",
      "\t<m> = -0.979236 +/- 0.001064\n",
      "\tSimulation finished. Took 37.2885651588 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.638776   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.034272\n",
      "\t\tFraction proposals accepted during sampling regime = 0.017310\n",
      "\t<E> = -1.237184 +/- 0.002044\n",
      "\t<m> = -0.982118 +/- 0.001032\n",
      "\tSimulation finished. Took 37.2259430885 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.654898   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.016981\n",
      "\t\tFraction proposals accepted during sampling regime = 0.014937\n",
      "\t<E> = -1.273970 +/- 0.001804\n",
      "\t<m> = 0.984844 +/- 0.000824\n",
      "\tSimulation finished. Took 37.182790041 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.671020   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.015109\n",
      "\t\tFraction proposals accepted during sampling regime = 0.012753\n",
      "\t<E> = -1.311530 +/- 0.001760\n",
      "\t<m> = -0.987483 +/- 0.000787\n",
      "\tSimulation finished. Took 37.1761710644 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.687143   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.012236\n",
      "\t\tFraction proposals accepted during sampling regime = 0.011034\n",
      "\t<E> = -1.345619 +/- 0.001686\n",
      "\t<m> = -0.988490 +/- 0.000762\n",
      "\tSimulation finished. Took 37.2568011284 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.703265   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.018523\n",
      "\t\tFraction proposals accepted during sampling regime = 0.009511\n",
      "\t<E> = -1.381404 +/- 0.001673\n",
      "\t<m> = 0.990182 +/- 0.000708\n",
      "\tSimulation finished. Took 37.5289170742 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.719388   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.009117\n",
      "\t\tFraction proposals accepted during sampling regime = 0.008354\n",
      "\t<E> = -1.416145 +/- 0.001627\n",
      "\t<m> = 0.991450 +/- 0.000655\n",
      "\tSimulation finished. Took 37.5666739941 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.735510   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.028644\n",
      "\t\tFraction proposals accepted during sampling regime = 0.007135\n",
      "\t<E> = -1.451126 +/- 0.001486\n",
      "\t<m> = 0.992743 +/- 0.000583\n",
      "\tSimulation finished. Took 37.6182549 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.751633   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.015874\n",
      "\t\tFraction proposals accepted during sampling regime = 0.006097\n",
      "\t<E> = -1.486367 +/- 0.001379\n",
      "\t<m> = 0.994002 +/- 0.000515\n",
      "\tSimulation finished. Took 37.4880900383 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.767755   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.006237\n",
      "\t\tFraction proposals accepted during sampling regime = 0.005276\n",
      "\t<E> = -1.518995 +/- 0.001282\n",
      "\t<m> = 0.994262 +/- 0.000469\n",
      "\tSimulation finished. Took 37.5401530266 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.783878   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.006018\n",
      "\t\tFraction proposals accepted during sampling regime = 0.004648\n",
      "\t<E> = -1.552976 +/- 0.001300\n",
      "\t<m> = -0.995000 +/- 0.000462\n",
      "\tSimulation finished. Took 32.3083651066 sec.\n",
      "Simulating Ising model:\n",
      "\t24 by 24 lattice\n",
      "\tJ = 0.800000   (+ve means preferable )\n",
      "\th = 0.000000\n",
      "\tTemperature (kT) = 1.000000\n",
      "\t7000 total sweeps, 3000 of them burn sweeps\n",
      "\t\tSampling every 10 sweeps\n",
      "\t\tFraction proposals accepted during burn sweeps = 0.005736\n",
      "\t\tFraction proposals accepted during sampling regime = 0.003977\n",
      "\t<E> = -1.588056 +/- 0.001174\n",
      "\t<m> = -0.996033 +/- 0.000414\n",
      "\tSimulation finished. Took 34.4844589233 sec.\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python2.7/dist-packages/ipykernel/__main__.py:25: RuntimeWarning: invalid value encountered in double_scalars\n"
     ]
    }
   ],
   "source": [
    "for i in range(np.size(J)):\n",
    "    # construct model with this J.\n",
    "    model = IsingModel(24, 24, J[i], 0.0)\n",
    "    \n",
    "    # get and store analytical sol'n\n",
    "    soln = exact_soln(model, kT)\n",
    "    \n",
    "    cvs[i] = soln['cv']\n",
    "    avg_es[i] = soln['e']\n",
    "    avg_ms[i] = soln['m']\n",
    "    \n",
    "    # Simulate, record results\n",
    "    energy_samples, magnetization_samples = simulate(model, kT, 7000, 3000, 10)\n",
    "    \n",
    "    avg_es_sim[i] = np.mean(energy_samples)\n",
    "    avg_ms_sim[i] = np.mean(magnetization_samples)\n",
    "    cvs_sim[i] = (np.mean(energy_samples ** 2) - np.mean(energy_samples) ** 2) / (\n",
    "        kT ** 2) * model.M * model.N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot results. Points: simulation, Lines: analytical soln"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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JXNZsFYlIYKNGjaCoaDQwCViA2Slcd93NYZclOWpfg6Otu/8XcIC71wKbmrEmEQmorKyM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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8ee4b7a610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8ee27850d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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mVOAydz8tNYZxm7sfk+G1POqfBxq/FGO2/fzZjmnoC19EtlfsL19qZoOB20me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\nIoH+P/6SP1u7oTSdAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8ee2a7ff90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "critical_J = kT * np.log(1.0 + np.sqrt(2.0))/2.0\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(J, avg_es)\n",
    "plt.scatter(J, avg_es_sim)\n",
    "plt.xlabel('J')\n",
    "plt.ylabel(r'$\\langle e \\rangle$')\n",
    "plt.savefig('avgE.png', format='png', dpi=200)\n",
    "plt.show()\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(J, avg_ms, color='b')\n",
    "plt.plot(J, -avg_ms, color='b')\n",
    "plt.plot([np.min(J), critical_J], [0.0, 0.0], color='b')\n",
    "plt.scatter(J, avg_ms_sim)\n",
    "plt.xlabel('J')\n",
    "plt.ylabel(r'$\\langle m \\rangle$')\n",
    "plt.show()\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(J, cvs)\n",
    "plt.scatter(J, cvs_sim)\n",
    "plt.xlabel('J')\n",
    "plt.ylabel(r'$c_v$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}