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    "<center><h2>Ergodic Theorem Replication </h2></center>\n",
    "<p>\n",
    "A demonstration  of the  Ergodic theorem. The program  continuously generates 16 random    particles\n",
    "in a box and counts the number  in the right and left halfs.  The distributions of numbers are plotted. \n",
    "See too the program  \"ParticlesinBox\" for time evolution of the histogram."
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    "\"\"\" From \"COMPUTATIONAL PHYSICS\" & \"COMPUTER PROBLEMS in PHYSICS\"\n",
    "    by RH Landau, MJ Paez, and CC Bordeianu (deceased).\n",
    "    Copyright R Landau, Oregon State Unv, MJ Paez, Univ Antioquia, \n",
    "    C Bordeianu, Univ Bucharest, 2021. \n",
    "    Please respect copyright & acknowledge our work.\"\"\"\n",
    "\n",
    "# ErgodicTheorem.ipynb: Demostration of the ergodic theorem\n",
    "\n",
    "from vpython import *\n",
    "import random\n",
    "import numpy as np\n",
    "\n",
    "L = 1                                    # side square\n",
    "scene  =  canvas(width = 400,height = 400,range = (1.3)  )\n",
    "ndist  =  graph(x = 500, ymax  =  50,\n",
    "             width = 400, height = 300, xtitle = 'Nr', ytitle = 'N')\n",
    "bars = gvbars(delta = 0.8,color = color.red)\n",
    "Natom = 16                                # number of atoms\n",
    "Nr = 0                                # number particles right side\n",
    "dt = 1e-6                             # time step\n",
    "dN = [0]*(Natom)\n",
    "ar = 0.03                                # radius of atom\n",
    "deltaN = 1                               # for histogram\n",
    "Atom = []\n",
    "t = 1\n",
    "dnew = [0]*(Natom)\n",
    "\n",
    "for jj in range (0,200):\n",
    "   Nr = 0\n",
    "   positions = []                           # position of atoms\n",
    "   curve(pos = [(-L,-L,0),(L,-L,0),(L,L,0),(-L,L,0),(-L,-L,0)]) #limits figure\n",
    "   curve(pos = [(0,-L,0),(0,L,0)],color = color.yellow)\n",
    "   inside = label(pos = vec(0.4,1.1,0),text = 'Particles here = ',box = 0)\n",
    "   inside2 = label(pos = vec(0.8,1.1,0),box = 0)    \n",
    "   for i in range (Natom):              # initial positions and velocities\n",
    "      rate(100)\n",
    "      x = 2.*(L-ar)*random.random()-L+ar   # positons  \n",
    "      y = 2.*(L-ar)*random.random()-L+ar   # border forbidden\n",
    "      Atom = Atom+[sphere(pos = vec(x,y,0),radius = ar,color = color.green) ]# add atoms\n",
    "      positions.append((x,y,0))            # add positions to list\n",
    "      pos = np.array(positions)               # array with positions\n",
    "      ddp = pos[i]\n",
    "      if ddp[0] >= 0 and ddp[0] <= L: Nr += 1        # count initial right atoms        \n",
    "      inside2.text = '%4s'%Nr    \n",
    "   dN[Nr]+ = 1                             # for the histogram\n",
    "   for j in arange(0,16): bars.plot(pos = (j,dN[j]))\n",
    "   for obj in scene.objects:      # to start new walk\n",
    "        if (obj is curve or obj is sphere or obj is label) :     continue\n",
    "        else: obj.visible  =  0  # clear curve to make new one\n",
    "   inside2.text = '%4s'%Nr"
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