Expansion del potencial del cubo.ipynb

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     "end_time": "2020-04-16T20:12:26.889681Z",
     "start_time": "2020-04-16T20:12:26.884753Z"
    }
   },
   "source": [
    "1.a\n",
    "\n",
    "Solución general para la ecuación de Laplace para un cubo con todas menos la cara $z=a$:\n",
    "\n",
    "$$\n",
    "\\Phi(x,y,z)=\\sum_{n_x,n_y} A_{n_x,n_y} \\sin(\\pi n_x/a x )\\sin(\\pi n_y/a y )\\frac{\\sinh(\\pi \\sqrt{n_x^2+n_y^2}/a z )}\n",
    "{\\sinh(\\pi \\sqrt{n_x^2+n_y^2})}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "con\n",
    "$$\n",
    "\\iint  \\sin(\\pi n_x/a x )\\sin(\\pi n_y/a y )  f(x,y)dx dy= \\sum_{n'_x,n'_y} A_{n'_x,n'_y} \\iint\\sin(\\pi n'_x/a x )\\sin(\\pi n'_y/a y )  \\sin(\\pi n_x/a x )\\sin(\\pi n_y/a y ) dxdy\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\iint  \\sin(\\pi n_x/a x )\\sin(\\pi n_y/a y )  f(x,y)dx dy=  A_{n_x,n_y} \\int\\sin^2(\\pi n_x/a x )dx \\int\\sin^2(\\pi n_y/a y )dy \n",
    "$$\n",
    "\n",
    "Para un potencial constante:\n",
    "$$\n",
    " A_{n_x,n_y}  =\\frac{4 \\times {\\rm V}}{a^2} \\iint  \\sin(\\pi n_x/a x )\\sin(\\pi n_y/a y ) dx dy\n",
    "$$\n",
    "\n",
    "luego,\n",
    "$$\n",
    " A_{n_x,n_y}  =\\frac{16}{\\pi^2 n_x n_y} \\frac{1-(-1)^{n_x}}{2}\\frac{1-(-1)^{n_y}}{2}\\times {\\rm V} \n",
    "$$\n",
    "De esta manera,\n",
    "\n",
    "\n",
    "$$\n",
    "\\Phi(x,y,z)=\\frac{16}{\\pi^2}\\sum_{n_x,n_y} \\frac{\\sin(\\pi (2 n_x+1)/a x )\\sin(\\pi (2 n_y+1)/a y )}{(2n_x+1)(2n_y+1)} \\frac{\\sinh(\\pi \\sqrt{(2 n_x+1)^2+(2 n_y+1)^2}/a z )}\n",
    "{\\sinh(\\pi \\sqrt{(2 n_y+1)^2+(2 n_y+1)^2}/a)} \\times [{\\rm V}]\n",
    "$$\n",
    "\n",
    "Una primera simplificación consiste en aproximar los $\\sinh$ por exponenciales:\n",
    "\n",
    "$$\n",
    "\\Phi(x,y,z)\\approx \\frac{16}{\\pi^2}\\sum_{n_x,n_y} \n",
    " \\frac{\\sin(\\pi (2 n_x+1)/a x )\\sin(\\pi (2 n_y+1)/a y )}{(2n_x+1)(2n_y+1)} \n",
    "e^{-\\pi \\sqrt{(2 n_x+1)^2+(2 n_y+1)^2}/a (1-z) } \\times [{\\rm V}]\n",
    "$$\n",
    "\n",
    "Notamos entonces que los términos con $n_x,n_y\\neq 0$ sólo contribuyen cerca de $z=a$. Podemos aproximar entonces \n",
    "\n",
    "$$\n",
    "\\Phi(x,y,z)\\approx \\frac{16}{\\pi^2}\n",
    " \\sin(\\pi /a x )\\sin(\\pi/a y )\n",
    "e^{\\pi \\sqrt{2}/a (z-a) } \\times [{\\rm V}]\n",
    "$$\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-16T21:12:48.661122Z",
     "start_time": "2020-04-16T21:12:48.649162Z"
    }
   },
   "source": [
    "1.b\n",
    "Si ahora dos caras opuestas están al mismo potencial, podemos implementar esa simetría directamente\n",
    "\n",
    "$$\n",
    "\\Phi(x,y,z)=\\sum_{n_x,n_y} A_{n_x,n_y} \\sin(\\pi n_x/a x )\\sin(\\pi n_y/a y )\\frac{\\cosh(\\pi \\sqrt{n_x^2+n_y^2} (z/a-1/2) )}\n",
    "{\\cosh(\\pi \\sqrt{n_x^2+n_y^2}/2)}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Una alternativa es simplemente superponer las soluciones de 1.1\n",
    "$$\n",
    "\\frac{16}{\\pi^2}\\sin(\\pi/a x)\\sin(\\pi/a y)e^{-\\pi \\sqrt{2}/2}(e^{\\pi \\sqrt{2}/a(z-a/2)}+e^{-\\pi \\sqrt{2}/a (z-a/2)})\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\frac{32}{\\pi^2}\\sin(\\pi/a x)\\sin(\\pi/a y)\\cosh(\\pi \\sqrt{2}/a(z-a/2))\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-16T20:49:15.707774Z",
     "start_time": "2020-04-16T20:49:15.422508Z"
    }
   },
   "outputs": [
    {
     "data": {
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     },
     "execution_count": 24,
     "metadata": {},
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    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "## import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "\n",
    "zs = np.linspace(0,1,100)\n",
    "\n",
    "nx=0\n",
    "ny=0\n",
    "\n",
    "plt.plot(zs,np.sinh(np.pi*np.sqrt((2*nx+1)**2+(2*ny+1)**2)*zs)/np.sinh(np.pi*np.sqrt((2*nx+1)**2+(2*ny+1)**2)))\n",
    "plt.plot(zs,np.exp(-np.sqrt((2*nx+1)**2+(2*ny+1)**2)*np.pi*(1.-zs)),label=\"nx=0,ny=0\")\n",
    "\n",
    "\n",
    "nx=1\n",
    "ny=0\n",
    "\n",
    "plt.plot(zs,np.sinh(np.pi*np.sqrt((2*nx+1)**2+(2*ny+1)**2)*zs)/np.sinh(np.pi*np.sqrt((2*nx+1)**2+(2*ny+1)**2)))\n",
    "plt.plot(zs,np.exp(-np.sqrt((2*nx+1)**2+(2*ny+1)**2)*np.pi*(1.-zs)),label=\"nx=1,ny=0\")\n",
    "\n",
    "\n",
    "\n",
    "nx=1\n",
    "ny=1\n",
    "\n",
    "plt.plot(zs,np.sinh(np.pi*np.sqrt((2*nx+1)**2+(2*ny+1)**2)*zs)/np.sinh(np.pi*np.sqrt((2*nx+1)**2+(2*ny+1)**2)))\n",
    "plt.plot(zs,np.exp(-np.sqrt((2*nx+1)**2+(2*ny+1)**2)*np.pi*(1.-zs)),label=\"nx=1,ny=1\")\n",
    "\n",
    "\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\Phi(x,a/2,z)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1.4 \n",
    "$$\n",
    "\\phi(x,y,z)=\\frac{32}{\\pi^2}\\sin(\\pi/a x)\\sin(\\pi/a y)\\exp(\\pi \\sqrt{2}/a(z-a))\n",
    "$$\n",
    "\n",
    "$$\n",
    "V(x,y,z)=V(z=a)\\phi(x,y,z)+V(z=0)\\phi(x,y,a-z)+V(x=0)\\phi(z,y,a-x)+V(x=a)\\phi(z,y,x)...\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2020-04-16T22:05:38.809435Z",
     "start_time": "2020-04-16T22:05:38.150285Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Encabezado: cargar librerías\n",
    "%matplotlib inline\n",
    "import numpy as np                # Librería para funciones matemáticas\n",
    "import matplotlib.pyplot as plt   # Librería para graficos\n",
    "import matplotlib.cm as cm        # Módulo para controlar mapas de colores\n",
    "from mpl_toolkits.mplot3d import Axes3D   # Módulo 3D\n",
    "\n",
    "\n",
    "fig,ax=plt.subplots(figsize=(5,5))\n",
    "\n",
    "x = np.linspace(.0,1.,100)\n",
    "z = np.linspace(0.,1,100)\n",
    "\n",
    "# Construye a partir del par de parámetros, una grilla con los valores\n",
    "X, Z = np.meshgrid(x, z)\n",
    "V = 1*sum([\n",
    "          16/np.pi**2 * np.sin(np.pi *(1+2*nx)*X)*np.sin(  np.pi/2 * (2*ny+1))*np.exp(np.pi*np.sqrt( (1+2*nx)**2+(1+2*ny)**2 )*(Z-1))/((2*nx+1)*(2*ny+1)) \n",
    "         for nx in range(10) for ny in range(10)])- \\\n",
    "    1*sum([\n",
    "          16/np.pi**2 * np.sin(np.pi *(1+2*nx)*Z)*np.sin(  np.pi/2 * (2*ny+1))*np.exp(np.pi*np.sqrt( (1+2*nx)**2+(1+2*ny)**2 )*(X-1))/((2*nx+1)*(2*ny+1)) \n",
    "         for nx in range(10) for ny in range(10)]) + \\\n",
    "    0*sum([\n",
    "          16/np.pi**2 * np.sin(np.pi *(1+2*nx)*X)*np.sin(  np.pi/2 * (2*ny+1))*np.exp(-np.pi*np.sqrt( (1+2*nx)**2+(1+2*ny)**2 )*Z)/((2*nx+1)*(2*ny+1)) \n",
    "         for nx in range(10) for ny in range(10)])+ \\\n",
    "    0*sum([\n",
    "          16/np.pi**2 * np.sin(np.pi *(1+2*nx)*Z)*np.sin(  np.pi/2 * (2*ny+1))*np.exp(-np.pi*np.sqrt( (1+2*nx)**2+(1+2*ny)**2 )*X)/((2*nx+1)*(2*ny+1)) \n",
    "         for nx in range(10) for ny in range(10)])\n",
    "         \n",
    "c = ax.pcolor(X,Z,V,cmap=cm.coolwarm)   # c =ax.color(X,Z,V,cmap=cm.coolwarm)\n",
    "fig.colorbar(c,ax=ax)\n",
    "ax.set_xlabel(\"x\")\n",
    "ax.set_ylabel(\"z\")\n",
    "\n",
    "plt.show()\n",
    "\n",
    "\n"
   ]
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      "Help on method colorbar in module matplotlib.figure:\n",
      "\n",
      "colorbar(mappable, cax=None, ax=None, use_gridspec=True, **kw) method of matplotlib.figure.Figure instance\n",
      "    Create a colorbar for a ScalarMappable instance, *mappable*.\n",
      "    \n",
      "    Documentation for the pyplot thin wrapper:\n",
      "    \n",
      "    \n",
      "    Add a colorbar to a plot.\n",
      "    \n",
      "    Function signatures for the :mod:`~matplotlib.pyplot` interface; all\n",
      "    but the first are also method signatures for the\n",
      "    :meth:`~matplotlib.figure.Figure.colorbar` method::\n",
      "    \n",
      "      colorbar(**kwargs)\n",
      "      colorbar(mappable, **kwargs)\n",
      "      colorbar(mappable, cax=cax, **kwargs)\n",
      "      colorbar(mappable, ax=ax, **kwargs)\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    mappable\n",
      "        The `matplotlib.cm.ScalarMappable` (i.e., `~matplotlib.image.Image`,\n",
      "        `~matplotlib.contour.ContourSet`, etc.) described by this colorbar.\n",
      "        This argument is mandatory for the `.Figure.colorbar` method but optional\n",
      "        for the `.pyplot.colorbar` function, which sets the default to the current\n",
      "        image.\n",
      "    \n",
      "        Note that one can create a `ScalarMappable` \"on-the-fly\" to generate\n",
      "        colorbars not attached to a previously drawn artist, e.g. ::\n",
      "    \n",
      "            fig.colorbar(cm.ScalarMappable(norm=norm, cmap=cmap), ax=ax)\n",
      "    \n",
      "    cax : :class:`~matplotlib.axes.Axes` object, optional\n",
      "        Axes into which the colorbar will be drawn.\n",
      "    \n",
      "    ax : :class:`~matplotlib.axes.Axes`, list of Axes, optional\n",
      "        Parent axes from which space for a new colorbar axes will be stolen.\n",
      "        If a list of axes is given they will all be resized to make room for the\n",
      "        colorbar axes.\n",
      "    \n",
      "    use_gridspec : bool, optional\n",
      "        If *cax* is ``None``, a new *cax* is created as an instance of Axes.  If\n",
      "        *ax* is an instance of Subplot and *use_gridspec* is ``True``, *cax* is\n",
      "        created as an instance of Subplot using the :mod:`~.gridspec` module.\n",
      "    \n",
      "    Returns\n",
      "    -------\n",
      "    colorbar : `~matplotlib.colorbar.Colorbar`\n",
      "        See also its base class, `~matplotlib.colorbar.ColorbarBase`.\n",
      "    \n",
      "    Notes\n",
      "    -----\n",
      "    Additional keyword arguments are of two kinds:\n",
      "    \n",
      "      axes properties:\n",
      "    \n",
      "    \n",
      "        ============= ====================================================\n",
      "        Property      Description\n",
      "        ============= ====================================================\n",
      "        *orientation* vertical or horizontal\n",
      "        *fraction*    0.15; fraction of original axes to use for colorbar\n",
      "        *pad*         0.05 if vertical, 0.15 if horizontal; fraction\n",
      "                      of original axes between colorbar and new image axes\n",
      "        *shrink*      1.0; fraction by which to multiply the size of the colorbar\n",
      "        *aspect*      20; ratio of long to short dimensions\n",
      "        *anchor*      (0.0, 0.5) if vertical; (0.5, 1.0) if horizontal;\n",
      "                      the anchor point of the colorbar axes\n",
      "        *panchor*     (1.0, 0.5) if vertical; (0.5, 0.0) if horizontal;\n",
      "                      the anchor point of the colorbar parent axes. If\n",
      "                      False, the parent axes' anchor will be unchanged\n",
      "        ============= ====================================================\n",
      "    \n",
      "    \n",
      "      colorbar properties:\n",
      "    \n",
      "    \n",
      "        ============  ====================================================\n",
      "        Property      Description\n",
      "        ============  ====================================================\n",
      "        *extend*      {'neither', 'both', 'min', 'max'}\n",
      "                      If not 'neither', make pointed end(s) for out-of-\n",
      "                      range values.  These are set for a given colormap\n",
      "                      using the colormap set_under and set_over methods.\n",
      "        *extendfrac*  {*None*, 'auto', length, lengths}\n",
      "                      If set to *None*, both the minimum and maximum\n",
      "                      triangular colorbar extensions with have a length of\n",
      "                      5% of the interior colorbar length (this is the\n",
      "                      default setting). If set to 'auto', makes the\n",
      "                      triangular colorbar extensions the same lengths as\n",
      "                      the interior boxes (when *spacing* is set to\n",
      "                      'uniform') or the same lengths as the respective\n",
      "                      adjacent interior boxes (when *spacing* is set to\n",
      "                      'proportional'). If a scalar, indicates the length\n",
      "                      of both the minimum and maximum triangular colorbar\n",
      "                      extensions as a fraction of the interior colorbar\n",
      "                      length. A two-element sequence of fractions may also\n",
      "                      be given, indicating the lengths of the minimum and\n",
      "                      maximum colorbar extensions respectively as a\n",
      "                      fraction of the interior colorbar length.\n",
      "        *extendrect*  bool\n",
      "                      If *False* the minimum and maximum colorbar extensions\n",
      "                      will be triangular (the default). If *True* the\n",
      "                      extensions will be rectangular.\n",
      "        *spacing*     {'uniform', 'proportional'}\n",
      "                      Uniform spacing gives each discrete color the same\n",
      "                      space; proportional makes the space proportional to\n",
      "                      the data interval.\n",
      "        *ticks*       *None* or list of ticks or Locator\n",
      "                      If None, ticks are determined automatically from the\n",
      "                      input.\n",
      "        *format*      None or str or Formatter\n",
      "                      If None, the\n",
      "                      :class:`~matplotlib.ticker.ScalarFormatter` is used.\n",
      "                      If a format string is given, e.g., '%.3f', that is\n",
      "                      used. An alternative\n",
      "                      :class:`~matplotlib.ticker.Formatter` object may be\n",
      "                      given instead.\n",
      "        *drawedges*   bool\n",
      "                      Whether to draw lines at color boundaries.\n",
      "        *label*       str\n",
      "                      The label on the colorbar's long axis.\n",
      "        ============  ====================================================\n",
      "    \n",
      "        The following will probably be useful only in the context of\n",
      "        indexed colors (that is, when the mappable has norm=NoNorm()),\n",
      "        or other unusual circumstances.\n",
      "    \n",
      "        ============   ===================================================\n",
      "        Property       Description\n",
      "        ============   ===================================================\n",
      "        *boundaries*   None or a sequence\n",
      "        *values*       None or a sequence which must be of length 1 less\n",
      "                       than the sequence of *boundaries*. For each region\n",
      "                       delimited by adjacent entries in *boundaries*, the\n",
      "                       color mapped to the corresponding value in values\n",
      "                       will be used.\n",
      "        ============   ===================================================\n",
      "    \n",
      "    \n",
      "    \n",
      "    If *mappable* is a :class:`~matplotlib.contours.ContourSet`, its *extend*\n",
      "    kwarg is included automatically.\n",
      "    \n",
      "    The *shrink* kwarg provides a simple way to scale the colorbar with respect\n",
      "    to the axes. Note that if *cax* is specified, it determines the size of the\n",
      "    colorbar and *shrink* and *aspect* kwargs are ignored.\n",
      "    \n",
      "    For more precise control, you can manually specify the positions of\n",
      "    the axes objects in which the mappable and the colorbar are drawn.  In\n",
      "    this case, do not use any of the axes properties kwargs.\n",
      "    \n",
      "    It is known that some vector graphics viewers (svg and pdf) renders white gaps\n",
      "    between segments of the colorbar.  This is due to bugs in the viewers, not\n",
      "    Matplotlib.  As a workaround, the colorbar can be rendered with overlapping\n",
      "    segments::\n",
      "    \n",
      "        cbar = colorbar()\n",
      "        cbar.solids.set_edgecolor(\"face\")\n",
      "        draw()\n",
      "    \n",
      "    However this has negative consequences in other circumstances, e.g. with\n",
      "    semi-transparent images (alpha < 1) and colorbar extensions; therefore, this\n",
      "    workaround is not used by default (see issue #1188).\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(fig.colorbar)"
   ]
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