PoissonSolver.ipynb

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   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "from scipy import special\n",
    "%config InlineBackend.figure_formats = ['svg']"
   ]
  },
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   "source": [
    "nxyz"
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   "source": [
    "Nx = 48\n",
    "Ny = 48\n",
    "Nz = 120\n",
    "nxyz = Nx*Ny*Nz\n",
    "dx = 1\n",
    "dy = 1\n",
    "dz = 1\n",
    "dV = dx*dy*dz\n",
    "e2 = 197.3269631 / 137.035999679; # Squared charge of proton"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "b6fea287-a7c2-4b47-b838-043e34ba0a85",
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   "source": [
    "xs = np.arange(0,Nx*dx,dx); xs-=(Nx//2)*dx\n",
    "ys = np.arange(0,Ny*dy,dy); ys-=(Ny//2)*dy\n",
    "zs = np.arange(0,Nz*dz,dz); zs-=(Nz//2)*dz\n",
    "X,Y,Z = np.meshgrid(xs,ys,zs,indexing=\"ij\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6e19736a-749d-46d4-8314-518dd01d56af",
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   "source": [
    "rho[xsteps,ysteps,zsteps]"
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   "source": [
    "def solvePoisson(rho):\n",
    "    \n",
    "    D = np.sqrt(3)*np.max([Nx*dx,Ny*dy,Nz*dz])\n",
    "    \n",
    "    e2 = 197.3269631 / 137.035999679; # Squared charge of proton\n",
    "    N3 = 3*np.max([Nx,Ny,Nz])\n",
    "\n",
    "    rho3 = np.zeros([N3,N3,N3])\n",
    "    rho3[0:Nx,0:Ny,0:Nz]=rho\n",
    "\n",
    "    kx = (2*np.pi/(N3*dx))*np.concatenate([np.arange(0,N3//2),np.arange(-N3//2,0)])\n",
    "    ky = (2*np.pi/(N3*dy))*np.concatenate([np.arange(0,N3//2),np.arange(-N3//2,0)])\n",
    "    kz = (2*np.pi/(N3*dz))*np.concatenate([np.arange(0,N3//2),np.arange(-N3//2,0)])\n",
    "    Kx,Ky,Kz = np.meshgrid(kx,ky,kz,indexing=\"ij\")\n",
    "    K2 = Kx**2+Ky**2+Kz**2\n",
    "\n",
    "    coul_ker = 4*np.pi*e2*(1-np.cos(D*np.sqrt(K2)))/(K2+np.finfo(np.double).eps)\n",
    "    coul_ker[0,0,0] = 2*e2*np.pi*D**2\n",
    "\n",
    "    phi3 = np.real(np.fft.ifftn(coul_ker*np.fft.fftn(rho3)))\n",
    "    phi = phi3[0:Nx,0:Ny,0:Nz]\n",
    "    \n",
    "    return phi"
   ]
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   "execution_count": 5,
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   "source": [
    "sig = 1.87; # Standard deviation of charge distribution\n",
    "numimg = 2; # Number of image charges\n",
    "\n",
    "xoff2, yoff2, zoff2 = [0,0,0]#[0,(Ny//8)*dy,(Nz//6)*dz]#\n",
    "R2 = np.sqrt((X-xoff2)**2+(Y-yoff2)**2+(Z-zoff2)**2)\n",
    "\n",
    "gauss = np.zeros([Nx,Ny,Nz])\n",
    "for imagex in range(-numimg,numimg+1):\n",
    "    for imagey in range(-numimg,numimg+1):\n",
    "        for imagez in range(-numimg,numimg+1):\n",
    "            gauss += np.exp(-((X-xoff2+imagex*(Nx*dx))**2+\n",
    "                              (Y-yoff2+imagey*(Ny*dy))**2+\n",
    "                              (Z-zoff2+imagez*(Nz*dz))**2)/(2.*sig**2));\n",
    "gauss *= 1/((2*np.pi*sig**2)**(3/2))\n",
    "phigauss = solvePoisson(gauss)\n",
    "phigaussalyt = 1*e2/(R2 + np.finfo(np.double).eps) * special.erf(R2/(np.sqrt(2)*sig)) # Analytical solution\n",
    "phigaussalyt[Nx//2,Ny//2,Nz//2] = e2*np.sqrt(2/np.pi)/sig\n",
    "engauss = 0.217223 # Compute in Mathematica"
   ]
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   "source": [
    "Coulomb energy:\n",
    "\n",
    "$$E_{coul} = \\frac{1}{2}\\int \\rho(\\vec{r}) \\Phi(\\vec{r}) d^3r$$"
   ]
  },
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     "text": [
      "Total charge = 1.0000000000000002\n",
      "\n",
      "Accuracy of Coulomb Energy: dE/E = -1.3495899696069829e-06\n",
      "\n",
      "Accuracy of Coulomb Potential: dV/V = 3.7390242954208074e-11\n",
      "\n"
     ]
    },
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   "source": [
    "print(f\"Total charge = {gauss.sum()*dV}\")\n",
    "print()\n",
    "print(f\"Accuracy of Coulomb Energy: dE/E = {(1/2*np.sum(gauss*phigauss)*dV-engauss)/engauss}\")\n",
    "print()\n",
    "print(f\"Accuracy of Coulomb Potential: dV/V = {np.max( np.abs(phigauss-phigaussalyt) )/np.max( np.abs(phigauss) )}\")\n",
    "print()\n",
    "\n",
    "plt.contourf(phigauss[Nx//2,:,:],extent=[zs[0],zs[-1],ys[0],ys[-1]])\n",
    "plt.colorbar()\n",
    "plt.title(\"Slice Through Center of\\n Coulomb Potential\")\n",
    "plt.xlabel(\"z (fm)\")\n",
    "plt.ylabel(\"y (fm)\")\n",
    "plt.show()\n",
    "\n",
    "plt.plot(xs,-1+(R2*phigauss)[:,Ny//2+yoff2//dy,Nz//2+zoff2//dz]/e2,\"b\",label=\"x-axis\")\n",
    "#plt.show()\n",
    "plt.plot(ys,-1+(R2*phigauss)[Nx//2,:,Nz//2+zoff2//dz]/e2,\"r\",label=\"y-axis\")\n",
    "#plt.show()\n",
    "plt.plot(zs,-1+(R2*phigauss)[Nx//2,Ny//2+yoff2//dy,:]/e2,\"k\",label=\"z-axis\")\n",
    "plt.ylim([-1e-10,1e-10])\n",
    "plt.xlabel(\"x,y,z (fm)\")\n",
    "plt.ylabel(\"$-1 + r \\Phi/e^2$\")\n",
    "plt.title(\"Deviation from 1/r Potential\\nAlong 3 Cardinal Axes\\nThrough Center of Dist\")\n",
    "#plt.vlines(xoff2,-1e-10,1e-10,\"gray\")\n",
    "#plt.vlines(yoff2,-1e-10,1e-10,\"gray\")\n",
    "#plt.vlines(zoff2,-1e-10,1e-10,\"gray\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
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   "cell_type": "code",
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   "id": "20f45edd-9925-4e7a-9be6-de122940ae75",
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   "source": []
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