monte_carlo.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Statistics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Monte Carlo sampling for evaluation of partition functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Metropolis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's implement the Metropolis algorithm for the 2D Ising model. For configurations, we'll use the same storage scheme as before."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "spin (generic function with 1 method)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spin(b::Bool) = Float64(2 * b - 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To allow us to study different topologies, we will use a separate function to define what we mean by neighbouring sites."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "neighbours_2D_PBC (generic function with 1 method)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "neighbours_2D_PBC(nx::Int, ny::Int, Lx::Int, Ly::Int) = ((mod1(nx+1, Lx), ny), (mod1(nx-1, Lx), ny), \n",
    "    (nx, mod1(ny+1, Ly)), (nx, mod1(ny-1, Ly)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can actually implement the Metropolis update. This takes the configuration, as well as the coordinates nx and ny of a spin to try to flip."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ising_metupd_2D_PBC! (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ising_metupd_2D_PBC!(c::BitArray{2}, nx::Int, ny::Int, h::Float64, Jx::Float64, Jy::Float64, β::Float64)\n",
    "    Ly,Lx = size(c)\n",
    "    \n",
    "    nb_x1, nb_x2, nb_y1, nb_y2 = neighbours_2D_PBC(nx, ny, Lx, Ly)\n",
    "    \n",
    "    b = c[ny,nx]\n",
    "    \n",
    "    s = spin(b)\n",
    "    sn_x = spin(c[nb_x1[2], nb_x1[1]]) + spin(c[nb_x2[2], nb_x2[1]])\n",
    "    sn_y = spin(c[nb_y1[2], nb_y1[1]]) + spin(c[nb_y2[2], nb_y2[1]])\n",
    "    \n",
    "    e = -(h + Jx * sn_x + Jy * sn_y)\n",
    "    \n",
    "    amp = exp(2β * s * e)\n",
    "    \n",
    "    accept = rand() < amp #If amp<1, this will set accept to true with probability amp. If amp > 1, accept is always false.\n",
    "    if accept\n",
    "        c[ny, nx] = !b\n",
    "    end\n",
    "    \n",
    "    accept\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see what kind of configurations we can generate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10×10 BitArray{2}:\n",
       "  true  false  false  false   true   true  false  false   true   true\n",
       " false   true  false   true   true  false  false   true   true  false\n",
       " false   true   true   true  false  false   true   true  false   true\n",
       " false  false   true  false   true  false  false  false  false  false\n",
       "  true   true   true  false  false   true  false  false   true  false\n",
       "  true   true  false   true  false  false   true  false  false   true\n",
       "  true  false  false   true  false   true  false  false   true  false\n",
       "  true   true   true  false  false   true   true  false  false  false\n",
       " false   true   true  false   true  false  false   true   true   true\n",
       " false   true  false   true  false  false   true   true   true  false"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = bitrand(10,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let h = 0.0, Jx = 1.0, Jy = 1.0, T = 2.269\n",
    "    Ly,Lx = size(c)\n",
    "    for j in 1:Lx*Ly\n",
    "        nx = rand(1:Lx)\n",
    "        ny = rand(1:Ly)\n",
    "        ising_metupd_2D_PBC!(c, nx, ny, h, Jx, Jy, 1/T)\n",
    "    end\n",
    "    heatmap(c)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Estimating a variable"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We wish to estimate the magnetisation. First, we define a function to evaluate it on a configuration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ising_mag_2D_PBC (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ising_mag_2D_PBC(c::BitArray{2})::Float64\n",
    "    Ly,Lx = size(c)\n",
    "    N = Lx * Ly\n",
    "    Nup = count(c) #counts the number of spins set to \"true\"\n",
    "    \n",
    "    (2Nup - N) / N\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function evaluate the magnetization and adds it to a preexisting results array. It does the same for $M^2$. We will use this for sampling expectation values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ising_meas_mag_2D_PBC! (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ising_meas_mag_2D_PBC!(meas::AbstractMatrix{Float64}, nsweep::Int, c::BitArray{2})\n",
    "    m = ising_mag_2D_PBC(c)\n",
    "    \n",
    "    #write to the results array\n",
    "    meas[1,nsweep] = meas[1,nsweep] + m\n",
    "    meas[2,nsweep] = meas[2,nsweep] + m^2\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next let's define a function to gather measurements over a number of sweeps. We will take a \"measfunc\" argument that specifies a function that carried out the measurement. This way we can use this Monte-Carlo sampling routine for any measurement we want."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ising_MC_2D_PBC (generic function with 1 method)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ising_MC_2D_PBC(c::BitArray{2}, h::Float64, Jx::Float64, Jy::Float64, β::Float64, Nsweeps::Int; \n",
    "                         keep_cs::Bool=false, Nmeas::Int=2, measfunc=ising_meas_mag_2D_PBC!)\n",
    "    Ly, Lx = size(c)\n",
    "    N = Lx * Ly\n",
    "    \n",
    "    cs = BitArray{2}[]\n",
    "    \n",
    "    ms = zeros((Nmeas, Nsweeps))\n",
    "    \n",
    "    Racc = zeros(Nsweeps)\n",
    "    \n",
    "    for i in 1:Nsweeps\n",
    "        acc = 0\n",
    "        for j in 1:N\n",
    "            nx = rand(1:Lx)\n",
    "            ny = rand(1:Ly)\n",
    "            acc += ising_metupd_2D_PBC!(c, nx, ny, h, Jx, Jy, β)\n",
    "        end\n",
    "        measfunc(ms, i, c)\n",
    "        Racc[i] = acc / N\n",
    "        if keep_cs\n",
    "            push!(cs, copy(c))\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    #rmul!(ms, 1/N)\n",
    "    \n",
    "    keep_cs ? (ms, Racc, cs) : ms, Racc\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "time_ising_MC (generic function with 1 method)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "time_ising_MC(c, nswp) = @time ising_MC_2D_PBC(c, 0.0, 1.0, 1.0, 1/2.269, nswp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.013299 seconds (4 allocations: 23.797 KiB)\n"
     ]
    }
   ],
   "source": [
    "let c = bitrand(10,10), nswp = 1000\n",
    "    time_ising_MC(c, nswp)\n",
    "end;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Getting results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To avoid autocorrelation effects, which lead to underestimation of the error, and also to exploit the central limit theorem, we use \"binning\". Binning is the division of measurements into blocks of $l_{\\text{bin}}$ sequential measurements. The new samples are the averages within the blocks. For sufficiently large blocks, the mean of a block approximately obeys gaussian statistics, and neighbouring can be treated as independent."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "binmeas (generic function with 1 method)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function binmeas(meas::Vector{Float64}, lbin::Int)\n",
    "    nbins, leftover = divrem(length(meas), lbin)\n",
    "    to_bin = view(meas, leftover+1:length(meas)) # trim from start to help thermalization\n",
    "    to_bin = reshape(to_bin, (lbin, nbins))\n",
    "    binned = vec(mean(to_bin, dims=1)) # take means over bins\n",
    "    binned, leftover\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given a vector containing a sampling of `length(vector)` iid random variables, computes a sampling of the mean and estimates the standard deviation on the mean."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "measval (generic function with 1 method)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function measval(meas::Vector{Float64})\n",
    "    m = mean(meas)\n",
    "    m_stdev = √(varm(meas, m) / length(meas)) # stdev of the mean of iid random variables\n",
    "    m, m_stdev\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Test"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try out metropolis sampling for measuring the magnetization at the critical temperature."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  2.120742 seconds (6 allocations: 4.578 MiB, 0.74% gc time)\n",
      "mean(Racc) = 0.07890504999999999\n"
     ]
    },
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       "<g clip-path=\"url(#clip3301)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 396.835, 754.86)\" x=\"396.835\" y=\"754.86\">5.0×10</text>\n",
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       "<g clip-path=\"url(#clip3301)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:38px; text-anchor:start;\" transform=\"rotate(0, 545.376, 727.449)\" x=\"545.376\" y=\"727.449\">4</text>\n",
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       "<g clip-path=\"url(#clip3301)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:38px; text-anchor:start;\" transform=\"rotate(0, 750.274, 727.449)\" x=\"750.274\" y=\"727.449\">5</text>\n",
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       "<g clip-path=\"url(#clip3301)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:38px; text-anchor:start;\" transform=\"rotate(0, 955.172, 727.449)\" x=\"955.172\" y=\"727.449\">5</text>\n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:38px; text-anchor:start;\" transform=\"rotate(0, 1160.07, 727.449)\" x=\"1160.07\" y=\"727.449\">5</text>\n",
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       "  687.905,974.386 689.959,969.455 692.013,990.644 694.067,973.568 696.122,1001.62 698.176,971.511 700.23,975.14 702.284,1037.06 704.338,988.739 706.392,978.051 \n",
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     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let Lx = 10, Ly = 10, h = +0.0, Jx = 1.0, Jy = 1.0, T = 2.0, Nsweep = 200000, Nburn = 0\n",
    "    \n",
    "    #Choose bin sizes\n",
    "    lbins = Int[100,500,1000,5000,10000,20000,40000]\n",
    "    \n",
    "    N = Lx * Ly\n",
    "    c = falses(Ly,Lx)\n",
    "    @time meas, Racc = ising_MC_2D_PBC(c, h, Jx, Jy, 1.0/T, Nsweep, Nmeas=2, measfunc=ising_meas_mag_2D_PBC!)\n",
    "    @show mean(Racc)\n",
    "    \n",
    "    meas = meas'\n",
    "    ms = meas[:,1]\n",
    "    m2s = meas[:,2]\n",
    "    \n",
    "    p_m_raw = plot(\n",
    "        title=\"Magnetization of binned samples\",\n",
    "        xlabel=\"MC time\", ylabel=\"m\",\n",
    "        ylim=(-1,+1)\n",
    "    )\n",
    "    \n",
    "    ms = ms[Nburn+1:end] #burn-in\n",
    "    \n",
    "    avs = zeros(length(lbins))\n",
    "    stds = zeros(length(lbins))\n",
    "    \n",
    "    for (j, lbin) in enumerate(lbins)\n",
    "        msb, offset = binmeas(ms, lbin)\n",
    "        if length(msb) <= 1000 # only plot binned data with a reasonable number of points\n",
    "            plot!(range(Nburn+offset+1, Nsweep, length=length(msb)), msb, label=\"$lbin\")\n",
    "        end\n",
    "        avs[j], stds[j] = measval(msb)\n",
    "    end\n",
    "    \n",
    "    p_m = plot(\n",
    "        lbins, avs, yerror=stds, \n",
    "        xscale=:log10,\n",
    "        ylim=(-1,+1),\n",
    "        xlabel=\"bin length (sweeps)\", ylabel=\"<m>\"\n",
    "    )\n",
    "\n",
    " \n",
    "    p_m2_raw = plot(\n",
    "        ylim=(0,+1),\n",
    "        xlabel=\"MC time\", ylabel=\"m²\"\n",
    "    )\n",
    "    \n",
    "    m2s = m2s[Nburn+1:end] #burn-in\n",
    "    \n",
    "    avs = zeros(length(lbins))\n",
    "    stds = zeros(length(lbins))\n",
    "    \n",
    "    for (j, lbin) in enumerate(lbins)\n",
    "        m2sb, offset = binmeas(m2s, lbin)\n",
    "        if length(m2sb) <= 1000\n",
    "            plot!(range(Nburn+offset+1, Nsweep, length=length(m2sb)), m2sb, label=\"$lbin\")\n",
    "        end\n",
    "        avs[j], stds[j] = measval(m2sb)\n",
    "    end\n",
    "    \n",
    "    \n",
    "    p_m2 = plot(\n",
    "        lbins, avs, yerror=stds,\n",
    "        xscale=:log10,\n",
    "        ylim=(0,1),\n",
    "        xlabel=\"bin length (sweeps)\", ylabel=\"<m²>\"\n",
    "    )\n",
    "    \n",
    "    plot(p_m_raw, p_m, p_m2_raw, p_m2, layout=4, legend=false, titlefontsize=8, guidefontsize=8)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see lots of jumps in the magnetization. Also, that the error bars, especially for the magnetization, grow slowly with the bin size, suggests a lot of autocorrelation. Let's have a look at the sampling procedure..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Ergodicity\n",
    "\n",
    "Let us evaluate the Metropolis algorithm. After burn-in, it should take us through a selection of the most probable configurations. Let's check that on a small system. We're using the same functions we used in the partition function notebook to get the true probability of each configuration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ising_weight_2D_PBC (generic function with 1 method)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ising_weight_2D_PBC(c::BitArray{2}, h::Float64, Jx::Float64, Jy::Float64, β::Float64)::Float64\n",
    "    Ly,Lx = size(c)\n",
    "\n",
    "    en = 0.0\n",
    "    @inbounds @fastmath for k in 1:Lx, j in 1:Ly\n",
    "        s = spin(c[j,k])\n",
    "        en -= h * s\n",
    "        en -= Jx * s * spin(c[j,mod1(k+1, Lx)])\n",
    "        en -= Jy * s * spin(c[mod1(j+1, Ly),k])\n",
    "    end\n",
    "    \n",
    "    exp(-en * β)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ising_pd_2D_PBC (generic function with 1 method)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ising_pd_2D_PBC(Lx::Int, Ly::Int, h::Float64, Jx::Float64, Jy::Float64, β::Float64)\n",
    "    N = Lx*Ly\n",
    "    @assert N ≤ 64 \"Up to 64 spins!\" #Although it would take a *long* time, our implementation would do 64!\n",
    "    c = falses(Ly,Lx)\n",
    "    Ws = zeros(2^N)\n",
    "    Ms = zeros(2^N)\n",
    "    for j in 0:2^N-1\n",
    "        c.chunks[1] = j\n",
    "        W = ising_weight_2D_PBC(c, h, Jx, Jy, β)\n",
    "        M = ising_mag_2D_PBC(c)\n",
    "        Ws[j+1] = W\n",
    "        Ms[j+1] = M\n",
    "    end\n",
    "    Z = sum(Ws)\n",
    "    \n",
    "    P = Ws ./ Z #normalize the weights to probabilities\n",
    "    \n",
    "    P, Ms, Z\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's see how well Metropolis did at the critical temperature. We will use the raw samples to estimate the probability of a correlation and compare it to the exact result. This will show us autocorrelation issues."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.463820 seconds (9 allocations: 24.000 MiB, 19.32% gc time)\n",
      "acc / (Nsamp * N) = 0.1456875\n",
      "sum(P) = 1.0000000000000007\n"
     ]
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     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let Lx = 4, Ly = 5, T = 2.269, Jx = 1.0, Jy = 1.0, h = 0.0, Nburn = 100, Nsamp = 800\n",
    "    N = Lx * Ly\n",
    "    @time P, Ms, Z = ising_pd_2D_PBC(Lx,Ly,h,Jx,Jy,1/T)\n",
    "    \n",
    "    visits = zeros(Int, length(P))\n",
    "    c = bitrand(Ly,Lx)\n",
    "    #burn in \n",
    "    for j in 1:Nburn\n",
    "        nx = rand(1:Lx)\n",
    "        ny = rand(1:Ly)\n",
    "        ising_metupd_2D_PBC!(c, nx, ny, h, Jx, Jy, 1.0/T)\n",
    "    end\n",
    "    acc = 0\n",
    "    for j in 1:Nsamp\n",
    "        for k in 1:N\n",
    "            nx = rand(1:Lx)\n",
    "            ny = rand(1:Ly)\n",
    "            acc += ising_metupd_2D_PBC!(c, nx, ny, h, Jx, Jy, 1.0/T)\n",
    "        end\n",
    "        visits[c.chunks[1] + 1] += 1\n",
    "    end\n",
    "    @show acc / (Nsamp * N)\n",
    "    F = visits / Nsamp\n",
    "    \n",
    "    srt = sortperm(P, rev=true)\n",
    "    P = P[srt]\n",
    "    Ms = Ms[srt]\n",
    "    F = F[srt]\n",
    "    \n",
    "    @show sum(P)\n",
    "    Ws = P * Z\n",
    "    Es = -log.(Ws) * T\n",
    "\n",
    "    Ms_unique = sort(unique(Ms))\n",
    "    PM = zeros(length(Ms_unique))\n",
    "    FM = zeros(length(Ms_unique))\n",
    "    for j in 1:length(Ms_unique)\n",
    "        flt = Ms .== Ms_unique[j]\n",
    "        PM[j] = sum(P[flt])\n",
    "        FM[j] = sum(F[flt])\n",
    "    end\n",
    "\n",
    "    bar(Ms_unique, FM, label=\"MC estimate\", xlabel=\"M/N\", ylabel=\"p(M)\")\n",
    "    plot!(Ms_unique, PM, label=\"Exact\")\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It seems the Metropolis algorithm breaks the spin-flip symmetry. This makes sense, since if we start magnetized it is very improbable to flip enough spins to cross over to the opposite magnetization, if we only flip one at a time. In other words, we expect strong autocorrelation problems, as we witnessed above. We would need to sample for a very long time to fix this. Alternatively, we can change our update algorithm."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cluster updates\n",
    "\n",
    "To improve the ergodicity near the critical point, we can try a different update scheme, flipping multiple spins at once. This is the Wolff algorithm (DOI:10.1103/PhysRevLett.62.361). It adds a number of sites from an existing magnetic domain while respecting detailed balance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Tuple{Int64,Int64}"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Coord = Tuple{Int,Int}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ising_clustupd_2D_PBC! (generic function with 1 method)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ising_clustupd_2D_PBC!(c::BitArray{2}, to_visit::Vector{Coord}, nx1::Int, ny1::Int, β::Float64)\n",
    "    Ly,Lx = size(c)\n",
    "    \n",
    "    c_size = 1\n",
    "\n",
    "    resize!(to_visit, 0)\n",
    "    c[ny1, nx1] = !c[ny1, nx1] # flip first site\n",
    "    push!(to_visit, (nx1,ny1)) # schedule neighbours for a visit\n",
    "    \n",
    "    while !isempty(to_visit)\n",
    "        # this site was already added to the cluster and flipped\n",
    "        (nx,ny) = pop!(to_visit)\n",
    "        b = c[ny,nx]\n",
    "        \n",
    "        # now determine which of its neighbours to add\n",
    "        for (nb_x, nb_y) in neighbours_2D_PBC(nx, ny, Lx, Ly)\n",
    "            nb_b = c[nb_y,nb_x]\n",
    "            \n",
    "            # If neighbour is anti-aligned, add to cluster with prob 1-exp(-2β).\n",
    "            # Aligned neighbours are either already in the cluster, or do not belong\n",
    "            # to the original domain.\n",
    "            if xor(b, nb_b) && rand() < 1.0 - exp(-2β)\n",
    "                c[nb_y,nb_x] = !nb_b\n",
    "                c_size += 1\n",
    "                # Schedule this site's neighbours for a visit\n",
    "                push!(to_visit, (nb_x, nb_y))\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    c_size\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "test_clusters (generic function with 2 methods)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function test_clusters(Lx,Ly,itr, T=2.269)\n",
    "    c = bitrand(Ly,Lx)\n",
    "    to_visit = Coord[]\n",
    "    #burn in \n",
    "    for j in 1:itr\n",
    "        nx = rand(1:Lx)\n",
    "        ny = rand(1:Ly)\n",
    "        @time ising_clustupd_2D_PBC!(c, to_visit, nx, ny, 1.0/T)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.000005 seconds (1 allocation: 80 bytes)\n",
      "  0.000003 seconds\n",
      "  0.000002 seconds\n",
      "  0.000003 seconds\n",
      "  0.000002 seconds\n",
      "  0.000012 seconds (2 allocations: 416 bytes)\n",
      "  0.000002 seconds\n",
      "  0.000003 seconds\n",
      "  0.000012 seconds\n",
      "  0.000012 seconds (1 allocation: 544 bytes)\n"
     ]
    }
   ],
   "source": [
    "test_clusters(10,10,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Occasional allocations happen when the to_visit vector needs to be substantially resized."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.387193 seconds (9 allocations: 24.000 MiB, 1.38% gc time)\n",
      "  0.003129 seconds (1.13 k allocations: 17.719 KiB)\n",
      "c_frac / N = 0.756750000000004\n",
      "sum(P) = 1.0000000000000007\n"
     ]
    },
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   ],
   "source": [
    "let Lx = 4, Ly = 5, T = 2.269, Nburn = 100, Nsamp = 800, Nupd = 1\n",
    "    N = Lx * Ly\n",
    "    @time P, Ms, Z = ising_pd_2D_PBC(Lx,Ly,0.0,1.0,1.0,1/T)\n",
    "    \n",
    "    visits = zeros(Int, length(P))\n",
    "    c = bitrand(Ly,Lx)\n",
    "    to_visit = Coord[]\n",
    "    #burn in \n",
    "    for j in 1:Nburn\n",
    "        nx = rand(1:Lx)\n",
    "        ny = rand(1:Ly)\n",
    "        ising_clustupd_2D_PBC!(c, to_visit, nx, ny, 1.0/T)\n",
    "    end\n",
    "    c_frac = 0\n",
    "    @time for j in 1:Nsamp\n",
    "        for k in 1:Nupd\n",
    "            nx = rand(1:Lx)\n",
    "            ny = rand(1:Ly)\n",
    "            c_frac += ising_clustupd_2D_PBC!(c, to_visit, nx, ny, 1.0/T) / (Nsamp * Nupd)\n",
    "        end\n",
    "        visits[c.chunks[1] + 1] += 1\n",
    "    end\n",
    "    @show c_frac / N\n",
    "    F = visits / Nsamp\n",
    "    \n",
    "    srt = sortperm(P, rev=true)\n",
    "    P = P[srt]\n",
    "    Ms = Ms[srt]\n",
    "    F = F[srt]\n",
    "    \n",
    "    @show sum(P)\n",
    "    Ws = P * Z\n",
    "    Es = -log.(Ws) * T\n",
    "\n",
    "    Ms_unique = sort(unique(Ms))\n",
    "    PM = zeros(length(Ms_unique))\n",
    "    FM = zeros(length(Ms_unique))\n",
    "    for j in 1:length(Ms_unique)\n",
    "        flt = Ms .== Ms_unique[j]\n",
    "        PM[j] = sum(P[flt])\n",
    "        FM[j] = sum(F[flt])\n",
    "    end\n",
    "\n",
    "    bar(Ms_unique, FM, label=\"MC estimate\", xlabel=\"M/N\", ylabel=\"p(M)\")\n",
    "    plot!(Ms_unique, PM, label=\"Exact\")\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is much more symmetric than the metropolis version!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ising_MC_cluster_2D_PBC (generic function with 1 method)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ising_MC_cluster_2D_PBC(c::BitArray{2}, β::Float64, Nsweeps::Int; \n",
    "                           keep_cs::Bool=false, Nmeas::Int=2, measfunc=ising_meas_mag_2D_PBC!)\n",
    "    Ly, Lx = size(c)\n",
    "    N = Lx * Ly\n",
    "    \n",
    "    cs = BitArray{2}[]\n",
    "    \n",
    "    ms = zeros((Nmeas, Nsweeps))\n",
    "    \n",
    "    c_fracs = zeros(Nsweeps)\n",
    "    \n",
    "    to_visit = Coord[]\n",
    "    \n",
    "    for i in 1:Nsweeps\n",
    "        c_frac = 0.0\n",
    "        Nupd = 0\n",
    "        while c_frac < 1.0 #Note: Not obviously the most sensible choice for a sweep.\n",
    "            nx = rand(1:Lx)\n",
    "            ny = rand(1:Ly)\n",
    "            c_frac += ising_clustupd_2D_PBC!(c, to_visit, nx, ny, β)\n",
    "            measfunc(ms, i, c)\n",
    "            Nupd += 1\n",
    "        end\n",
    "        for k in 1:Nmeas\n",
    "            ms[k,i] /= Nupd\n",
    "        end\n",
    "        \n",
    "        c_fracs[i] = c_frac / Nupd\n",
    "        if keep_cs\n",
    "            push!(cs, copy(c))\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    keep_cs ? (ms, c_fracs, cs) : ms, c_fracs\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "test_cluster_MC (generic function with 2 methods)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test_cluster_MC(c, Nsweep, T=2.269) = @time ising_MC_cluster_2D_PBC(c, 1.0/T, Nsweep, Nmeas=2, measfunc=ising_meas_mag_2D_PBC!)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.060073 seconds (46.27 k allocations: 2.311 MiB)\n"
     ]
    }
   ],
   "source": [
    "let c = bitrand(4,4), Nsweep = 200\n",
    "    test_cluster_MC(c, Nsweep)\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean(c_fracs) = 61.223136363636364\n"
     ]
    },
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      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let Lx = 10, Ly = 10, T = 2.269, Nsweep = 22000, Nburn = 2000\n",
    "    \n",
    "    lbins = Int[10, 50, 100, 200, 400, 1000, 2000]\n",
    "    \n",
    "    N = Lx * Ly\n",
    "    c = bitrand(Ly,Lx)\n",
    "    meas, c_fracs = ising_MC_cluster_2D_PBC(c, 1.0/T, Nsweep, Nmeas=2, measfunc=ising_meas_mag_2D_PBC!)\n",
    "    @show mean(c_fracs)\n",
    "    \n",
    "    meas = meas'\n",
    "    ms = meas[:,1]\n",
    "    m2s = meas[:,2]\n",
    "    \n",
    "    p_m_raw = plot(\n",
    "        title=\"Magnetization of (binned) samples\",\n",
    "        xlabel=\"MC time\", ylabel=\"m\",\n",
    "        ylim=(-1,+1)\n",
    "    )\n",
    "    \n",
    "    ms = ms[Nburn+1:end] #burn-in\n",
    "    \n",
    "    avs = zeros(length(lbins))\n",
    "    stds = zeros(length(lbins))\n",
    "    \n",
    "    for (j, lbin) in enumerate(lbins)\n",
    "        msb, offset = binmeas(ms, lbin)\n",
    "        if length(msb) <= 1000 # only plot binned data with a reasonable number of points\n",
    "            plot!(range(Nburn+offset+1, Nsweep, length=length(msb)), msb, label=\"$lbin\")\n",
    "        end\n",
    "        avs[j], stds[j] = measval(msb)\n",
    "    end\n",
    "    \n",
    "    p_m = plot(\n",
    "        lbins, avs, yerror=stds, \n",
    "        xscale=:log10,\n",
    "        ylim=(-0.1,+0.1),\n",
    "        xlabel=\"bin length (sweeps)\", ylabel=\"<m>\"\n",
    "    )\n",
    "\n",
    " \n",
    "    p_m2_raw = plot(\n",
    "        ylim=(0,+1),\n",
    "        xlabel=\"MC time\", ylabel=\"m²\"\n",
    "    )\n",
    "    \n",
    "    m2s = m2s[Nburn+1:end] #burn-in\n",
    "    \n",
    "    avs = zeros(length(lbins))\n",
    "    stds = zeros(length(lbins))\n",
    "    \n",
    "    for (j, lbin) in enumerate(lbins)\n",
    "        m2sb, offset = binmeas(m2s, lbin)\n",
    "        if length(m2sb) <= 1000\n",
    "            plot!(range(Nburn+offset+1, Nsweep, length=length(m2sb)), m2sb, label=\"$lbin\")\n",
    "        end\n",
    "        avs[j], stds[j] = measval(m2sb)\n",
    "    end\n",
    "    \n",
    "    \n",
    "    p_m2 = plot(\n",
    "        lbins, avs, yerror=stds,\n",
    "        xscale=:log10,\n",
    "        ylim=(0,1),\n",
    "        xlabel=\"bin length (sweeps)\", ylabel=\"<m²>\"\n",
    "    )\n",
    "    \n",
    "    plot(p_m_raw, p_m, p_m2_raw, p_m2, layout=4, legend=false, titlefontsize=8, guidefontsize=8)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "We observe a lack of jumps in the magnetization, much better convergence of the error bars with bin size, and much smaller errors!"
   ]
  }
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