coarsening.ipynb

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    "# Coarsening process in Cahn-Hilliard equation\n",
    "\n",
    "## The continuum hypothesis\n",
    "\n",
    "Let us consider a box of fluid with size $L_x\\times L_y\\times L_z$ (see figure below).\n",
    "The volume of the system is $V=L_x L_y L_z$.\n",
    "\n",
    "__Continuum hypothesis.__ \n",
    "We assume that we can divide the system into many many small volume elements $dV$'s, such that each volume element $dV$ contains a large number of molecules $\\sim10^{23}$, but $dV$ is still small compared to the overall system volume $V=L_xL_yL_z$ . \n",
    "We can then define the hydrodynamic variable $\\rho(\\mathbf{r},t)$, which is the mass density of the fluid.\n",
    "Physically, $\\rho(\\mathbf{r},t)$ is defined such that:\n",
    "\\begin{equation} \n",
    "\\rho(\\mathbf{r},t)\\,dV = \\text{mass of the fluid inside the volume element $dV$, located at $\\mathbf{r}$ and at time $t$}.\n",
    "\\end{equation}\n",
    "So if we want to find the total mass of the fluid, we integrate $\\rho(\\mathbf{r},t)$ over the whole system.\n",
    "In the figure below, the volume elements  $dV$'s are represented by pixels and the colour of this pixel represents the value of the fluid density $\\rho(\\mathbf{r},t)$ at that point.\n",
    "In fluids, there are also other hydrodynamic variables of interest such as pressure $p(\\mathbf{r},t)$ and fluid velocity $\\mathbf{u}(\\mathbf{r},t)$, but we will not worry about these in this Notebook.\n"
   ]
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     },
     "execution_count": 4,
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    }
   ],
   "source": [
    "from IPython.display import Image\n",
    "Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-coarsening/master/figures/density-field.png')"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "\n",
    "We consider the following problems involving phase separation:\n",
    "1. Two-phase, one-component systems, _e.g._, liquid water and water vapour phase separation\n",
    "2. One-phase, two-component systems, _e.g._, liquid water and liquid oil phase separation.\n",
    "\n",
    "For both cases, the phase-separation dynamics can be described by a single scalar order parameter $\\phi(\\mathbf{r},t)$ (which is sometimes also called the rescaled density).\n",
    "\n",
    "For Case 1, $\\phi(\\mathbf{r},t)$ is defined to be the rescaled density, relative to the critical density $\\rho_c$:\n",
    "\\begin{equation}\n",
    "\\phi(\\mathbf{r},t) = \\frac{\\rho(\\mathbf{r},t) - \\rho_c}{\\rho_c}.\n",
    "\\end{equation}\n",
    "In the above equation, $\\rho(\\mathbf{r},t)$ is the mass density of the fluid (which can be gas/liquid/both) at position $\\mathbf{r}$ and at time $t$,\n",
    "and $\\rho_c$ is the density of the fluid at the critical point.\n",
    "Basically, below the critical temperature, the liquid phase has density $\\rho(\\mathbf{r},t)\\simeq\\rho_l$ and the gas phase has density $\\rho(\\mathbf{r},t)\\simeq\\rho_g<\\rho_l$.\n",
    "As we approach the critical temperature, the density difference between the liquid and the gas phase goes to zero, _i.e._ $\\rho_l-\\rho_g\\rightarrow 0$.\n",
    "Finally beyond the critical point, there is no distinction between liquid and gas.\n",
    "Thus from the definition of $\\phi(\\mathbf{r},t)$ above, $\\phi(\\mathbf{r},t)>0$ corresponds to the liquid phase and $\\phi(\\mathbf{r},t)<0$ corresponds to the gas phase.\n",
    "\n",
    "For Case 2, $\\phi(\\mathbf{r},t)$ is defined to be the relative densities between the two molecules, say A and B:\n",
    "\\begin{equation}\n",
    "\\phi(\\mathbf{r},t) = \\frac{2\\rho(\\mathbf{r},t) - \\rho_A - \\rho_B}{\\rho_A - \\rho_B}.\n",
    "\\end{equation}\n",
    "In the above equation $\\rho_A$ is the density of pure A molecules and $\\rho_B$ is the density of the pure B molecules.\n",
    "Thus from the definition above, $\\phi(\\mathbf{r},t)\\simeq 1$ indicates that the fluid contains pure A molecules and $\\phi(\\mathbf{r},t)\\simeq-1$ indicates that the fluid contains pure B molecules.\n",
    "\n",
    "By conservation of mass and assuming that there is no chemical reaction $A\\leftrightarrow B$, it follows that the integral:\n",
    "\\begin{equation}\n",
    "\\phi_0 = \\frac{1}{V} \\underbrace{\\int_V dV \\, \\phi(\\mathbf{r},t)}_{\\text{total amount of fluid}} = \\text{constant}.\n",
    "\\end{equation}\n",
    "We call the constant $\\phi_0$ to be the global density, which is fixed by the initial condition.\n",
    "\n",
    "__Local equilibrium hypothesis.__ \n",
    "We assume each fluid element $dV$ to be in a state of thermodynamic equilibrium _locally_ with a given temperature, entropy, internal energy, pressure, chemical potential, _etc_.\n",
    "Note that this does not mean that the whole system is in a state of _global_ thermodynamic equilibrium because a fluid element $dV$ at $\\mathbf{r}$ might have a different pressure or chemical potential compared to the fluid element $dV$ at $\\mathbf{r}'$.\n",
    "However over time, if we leave the system to sit long enough, energy and mass can be exchanged between different fluid elements $dV$'s until a global thermodynamic equilibrium is reached.\n",
    "\n",
    "Since each fluid element $dV$ is in thermodynamic equilibrium locally, we can define the free energy of this fluid element to be:\n",
    "\\begin{equation}\n",
    "g(\\phi,\\nabla\\phi)\\,dV = \\text{free energy of the fluid element $dV$, located at $\\mathbf{r}$ and at time $t$}.\n",
    "\\end{equation}\n",
    "We call $g(\\phi,\\nabla\\phi)$ to be the free energy density.\n",
    "In general $g$ depends on the order parameter $\\phi$ and its gradient $\\nabla\\phi$.\n",
    "Landau and Ginzburg's idea is to Taylor expand $g$ around the critical point.\n",
    "At the critical point $\\phi=0$ and thus we can expand $g$ for small $\\phi$:\n",
    "\\begin{equation}\n",
    "g(\\phi,\\nabla\\phi) = \\underbrace{\\frac{\\alpha}{2}\\phi^2 + \\frac{\\beta}{4}\\phi^4}_{\\text{local}} + \\underbrace{\\frac{\\kappa}{2}|\\nabla\\phi|^2}_{\\text{semi-local}}.\n",
    "\\end{equation}\n",
    "Note that, we have ignored the cubic term $~\\phi^3$, because we can redefine $\\phi$ to eliminate this cubic term, making it unnecessary.\n",
    "We have also ignored the linear term $~\\phi$ because this does not contribute the dynamics, as we shall see below.\n",
    "Finally, $\\sim|\\nabla\\phi|^2$ is the smallest scalar term we can form using $\\nabla$ and $\\phi$. \n",
    "Note that the first two terms above are purely local as they only depend on $\\phi$.\n",
    "The last term in the equation above is semi-local, because it depends on the gradient $\\nabla\\phi$.\n",
    "This means to calculate the last term in the equation above, we need some information about the neighbouring fluid elements $dV$'s.\n",
    "\n",
    "__Coarse graining.__\n",
    "When we did the Landau expansion, _i.e._ Taylor expansion around $\\phi=0$ above, we get various phenomenological parameters such as $\\alpha$ and $\\beta$, which are unknown.\n",
    "In this section, we will investigate how to derive these parameters from microscopic models through _coarse-graining_.\n",
    "Let us consider some fluid, as shown in the figure on the left below.\n",
    "Yellow indicates the liquid phase ($\\phi\\simeq1$) while dark blue indicates the gas phase ($\\phi\\simeq-1$).\n",
    "\n",
    "As before, we can divide the system into many many volume elements $dV$'s.\n",
    "Let us just consider one such volume element $dV$, which is indicated by dark square in the figure below.\n",
    "Inside this volume element, we have a large number of molecules, which are indicated by the blue discs in the figure on the right below.\n",
    "Let us denote $N_p$ to be the number of particles/fluid molecules, _e.g._ $\\text{H}_2\\text{O}$, inside this volume element.\n",
    "(In the figure we have $N_p=36$ particles for illustration purposes, but in reality $N_p$ is very large!)\n",
    "Let us suppose that we can further divide the volume element $dV$ into $N$ lattice cells, as shown in the figure on the right below.\n",
    "(In the figure, $N=8\\times8=64$, but again in reality, $N$ is very large.)\n",
    "Each cell has a volume $a^3$, which is even smaller than $dV$, and $a$ is the typical size of the molecule.\n",
    "Now we assume that each fluid molecule has to fit inside one of the lattice cells and cannot be on the fence.\n",
    "Furthermore, each lattice cell cannot contain more than one fluid molecule due to hard-core repulsion."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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   "source": [
    "Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-coarsening/master/figures/coarse-graining.png')"
   ]
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   "cell_type": "markdown",
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    "We can then compute the free energy of this volume element.\n",
    "The free energy is given by:\n",
    "\\begin{equation}\n",
    "F = U - TS, \\quad\\text{where $U$ is the potential energy and $S$ is the entropy.}\n",
    "\\end{equation}\n",
    "First let us calculate $U$.\n",
    "Let us denote the interaction energy between two neighbouring molecules to be $-\\varepsilon$, where $\\varepsilon>0$.\n",
    "The interaction energy is negative because the molecules tend to attract each other.\n",
    "Now let us denote $z$ to be the number of nearest neighbour cells.\n",
    "In two-dimension, $z=4$, as shown by the red lines in the figure on the right below.\n",
    "Thus each fluid molecule has, on average, $z\\frac{N_p}{N}$ other neighbouring molecules.\n",
    "The interaction energy between the fluid molecules is then:\n",
    "\\begin{equation}\n",
    "U = -\\frac{\\varepsilon z}{2}N_p\\frac{N_p}{N} \\quad\\Rightarrow\\quad \n",
    "\\frac{U}{N} = -\\frac{\\varepsilon z}{2}\\frac{N_p}{N}\\frac{N_p}{N}.\n",
    "\\end{equation}\n",
    "Note that we have added a factor of $1/2$ to avoid double counting the bonds between two neighbouring molecules.\n",
    "Now to calculate the entropy, we use the Boltzmann formula:\n",
    "\\begin{equation}\n",
    "S = k_B \\ln\\Omega,\\quad\\text{where $\\Omega$ is the number of microstates.}\n",
    "\\end{equation}\n",
    "In our case $\\Omega$ is the number of arranging $N_p$ indistinguishable particles into $N$ cells:\n",
    "\\begin{equation}\n",
    "\\Omega = C^N_{N_p} = \\frac{N!}{(N-N_p)!N_p!}.\n",
    "\\end{equation}\n",
    "Thus the entropy is:\n",
    "\\begin{equation}\n",
    "S = k_B\\left[ \\ln N! - \\ln (N-N_p)! - \\ln N_p! \\right].\n",
    "\\end{equation}\n",
    "Next we can use the Stirling's approximation $\\ln N!\\simeq N\\ln N - N$ to get:\n",
    "\\begin{equation}\n",
    "\\frac{S}{N} = -k_B \\left[ \\left(1-\\frac{N_p}{N}\\right)\\ln\\left(1-\\frac{N_p}{N}\\right) + \\frac{N_p}{N}\\ln\\left(\\frac{N_p}{N}\\right) \\right]\n",
    "\\end{equation}\n",
    "Thus the free energy (per unit cell) of this volume element is:\n",
    "\\begin{align}\n",
    "\\frac{F}{N} &= \\frac{U}{N} - T\\frac{S}{N} \\\\\n",
    "&= -\\frac{\\varepsilon z}{2}\\frac{N_p}{N}\\frac{N_p}{N} +\n",
    "k_BT \\left[ \\left(1-\\frac{N_p}{N}\\right)\\ln\\left(1-\\frac{N_p}{N}\\right) + \\frac{N_p}{N}\\ln\\left(\\frac{N_p}{N}\\right) \\right].\n",
    "\\end{align}\n",
    "Note that the free energy per unit cell above is an intensive quantity, as expected.\n",
    "From the picture below, obviously if $N_p=N$ we get a pure liquid phase and if $N_p=0$ we get a pure gas phase.\n",
    "Thus at critical point, we expect $N_p=N/2$.\n",
    "This motivates us to expand $F/N$ around the critical point by writing:\n",
    "\\begin{equation}\n",
    "\\frac{N_p}{N} = \\frac{1}{2} + \\phi,\n",
    "\\end{equation}\n",
    "where $\\phi$ is small.\n",
    "The free energy per unit cell is then:\n",
    "\\begin{equation}\n",
    "\\frac{F}{N} = -\\frac{\\varepsilon z}{2}\\left(\\frac{1}{2}+\\phi\\right)\\left(\\frac{1}{2}+\\phi\\right) +\n",
    "k_BT \\left[ \\left(\\frac{1}{2}-\\phi\\right)\\ln\\left(\\frac{1}{2}-\\phi\\right) + \\left(\\frac{1}{2}+\\phi\\right)\\ln\\left(\\frac{1}{2}+\\phi\\right) \\right]\n",
    "\\end{equation}\n",
    "Next we expand the logarithm as power series up to order $\\phi^4$:\n",
    "\\begin{equation}\n",
    "\\ln\\left(\\frac{1}{2}\\pm\\phi\\right) = -\\ln2 \\pm 2\\phi - 2\\phi^2 \\pm \\frac{8}{3}\\phi^3 - 4\\phi^4 \\dots.\n",
    "\\end{equation}\n",
    "We can then expand $F/N$ up to order $\\phi^4$:\n",
    "\\begin{equation}\n",
    "\\frac{F}{N} = -\\frac{z\\varepsilon}{2}\\phi + \\left(2k_BT - \\frac{z\\varepsilon}{2}\\right)\\phi^2 + \\frac{4k_BT}{3}\\phi^4 + \\mathcal{O}(\\phi^6).\n",
    "\\end{equation}\n",
    "Note that we have ignored the constant terms in the equation above.\n",
    "Furthermore, we can also ignore the linear term $\\propto\\phi$ as this does not affect the $\\dot{\\phi}$-dynamics as we shall see below.\n",
    "To find the free energy density, we can then divide $F/N$ by the volume of the lattice cell $a^3$ to get:\n",
    "\\begin{equation}\n",
    "g(\\phi) = \\frac{F}{Na^3} = \\underbrace{\\left(\\frac{2k_BT}{a^3} - \\frac{z\\varepsilon}{2a^3}\\right)}_\\alpha\\phi^2 + \n",
    "\\underbrace{\\frac{4k_BT}{3a^3}}_\\beta\\phi^4 + \\mathcal{O}(\\phi^6).\n",
    "\\end{equation}\n",
    "If we compare with the Landau-Ginzburg free energy density $g(\\phi,\\nabla\\phi)$, we identify the coefficients of $\\phi^2$ and $\\phi^4$ to be:\n",
    "\\begin{equation}\n",
    "\\alpha = \\frac{2k_BT}{a^3} - \\frac{z\\varepsilon}{2a^3} \\quad\\text{and}\\quad \\beta = \\frac{4k_BT}{3a^3},\n",
    "\\end{equation}\n",
    "respectively.\n",
    "Note that $\\beta$ is always positive, whereas $\\alpha$ becomes negative when:\n",
    "\\begin{equation}\n",
    "T < \\frac{z\\varepsilon}{4k_B} = T_c.\n",
    "\\end{equation}\n",
    "We call the right hand side the critical temperature $T_c$, below which, phase separation can occur (for a given range of $\\phi_0$).\n",
    "If $T>T_c$, the coefficient of $\\phi^2$ in the free energy is always positive, and the system remains homogenously mixed, and there is no phase separation into liquid or gas.\n",
    "Finally, we should also note that we did not the get the gradient term $\\sim|\\nabla\\phi|^2$ through coarse-graining.\n",
    "This term has to be added phenomenologically with some unknown coefficient $\\kappa>0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Coarse-grained free energy\n",
    "\n",
    "The total coarse-grained free energy of the system can then be written as:\n",
    "\\begin{equation}\n",
    "\\mathcal{F}[\\phi] = \\int_V dV \\bigg[ \\underbrace{\\frac{\\alpha}{2}\\phi^2 + \\frac{\\beta}{4}\\phi^4}_{f(\\phi)} + \\frac{\\kappa}{2}|\\nabla\\phi|^2 \\bigg],\n",
    "\\end{equation}\n",
    "where $f(\\phi)$ is the local free energy density term\n",
    "and $\\frac{\\kappa}{2}|\\nabla\\phi|^2$ is a semi-local free energy density term.\n",
    "$\\beta$ and $\\kappa$ are positive constants.\n",
    "$\\alpha$ can be positive or negative.\n",
    "If $\\alpha<0$, then the system will favour phase separation into the liquid $\\phi\\simeq\\sqrt{\\frac{-\\alpha}{\\beta}}$ and vapour phase $\\phi\\simeq-\\sqrt{\\frac{-\\alpha}{\\beta}}$.\n",
    "On the other hand if $\\alpha>0$, the system will remain in a homogenous phase with $\\phi=\\phi_0$ everywhere.\n",
    "The reason for this has something to do with the shape of the local (or bulk) free energy density $f(\\phi)$, as you can see below.\n",
    "For $\\alpha<0$, the local (or bulk) free energy density $f(\\phi)$ has two minima at $\\phi=\\pm\\sqrt{\\frac{-\\alpha}{\\beta}}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
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",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-coarsening/master/figures/Landau-energy.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The _global_ equilibrium state is given by the minimum of the total free energy $\\mathcal{F}[\\phi]$, _i.e_,\n",
    "\\begin{equation}\n",
    "\\frac{\\delta \\mathcal{F}}{\\delta\\phi} = 0,\n",
    "\\end{equation}\n",
    "subject to the constraint\n",
    "\\begin{equation}\n",
    "\\phi_0 = \\frac{1}{V} \\int_V dV \\, \\phi(\\mathbf{r},t) = \\text{constant}.\n",
    "\\end{equation}\n",
    "There are two equilibrium states:\n",
    "1. Homogenous state, where the density is constant everywhere $\\phi(\\mathbf{r},t)=\\phi_0=$ constant.\n",
    "2. Droplet state, where the system is separated into two domains with density $\\phi\\simeq+\\sqrt{\\frac{-\\alpha}{\\beta}}$ and $\\phi\\simeq-\\sqrt{\\frac{-\\alpha}{\\beta}}$ in each domain.\n",
    "\n",
    "The control parameters in our system are $\\alpha$ and $\\phi_0$.\n",
    "Depending on the values of $\\alpha$ and $\\phi_0$, the equilibrium state of the system can either be the homogenous state or the droplet state."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-coarsening/master/figures/equilibrium-phase-diagram.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Functional derivative\n",
    "\n",
    "Let us consider some functional $\\mathcal{F}[\\phi(\\mathbf{r})]$, which in general, can be written as:\n",
    "\\begin{equation}\n",
    "\\mathcal{F}[\\phi] = \\int_V g(\\phi,\\partial_\\alpha\\phi)\\,dV.\n",
    "\\end{equation}\n",
    "In the above equation, $g(\\phi,\\partial_\\alpha\\phi)$ is some function of $\\phi$ and its derivative $\\partial_\\alpha\\phi$.\n",
    "Here the index $\\alpha=x,y,z$ represents the Cartesian coordinates.\n",
    "We also use the following notation for partial derivatives $\\partial_x=\\frac{\\partial}{\\partial x}$, $\\partial_y=\\frac{\\partial}{\\partial y}$, and $\\partial_z=\\frac{\\partial}{\\partial z}$.\n",
    "In the case of our energy, $g$ is given by:\n",
    "\\begin{equation}\n",
    "g(\\phi,\\partial_\\alpha\\phi) = \\frac{\\alpha}{2}\\phi^2 + \\frac{\\beta}{4}\\phi^4 + \\frac{\\kappa}{2}(\\partial_\\alpha\\phi)(\\partial_\\alpha\\phi).\n",
    "\\end{equation}\n",
    "Here, we have used the repeated index notation, _i.e_ summation over that index is implied so the last term in the above equation is implied to be:\n",
    "\\begin{equation}\n",
    "(\\partial_\\alpha\\phi)(\\partial_\\alpha\\phi) =\n",
    "(\\partial_x\\phi)^2 + (\\partial_y\\phi)^2 + (\\partial_z\\phi)^2 = |\\nabla\\phi|^2\n",
    "\\end{equation}\n",
    "\n",
    "Now let us consider some density field $\\phi(\\mathbf{r})$.\n",
    "Let's suppose that we add some perturbation to the density field such that \n",
    "$\\phi(\\mathbf{r})\\rightarrow\\phi(\\mathbf{r}) + \\delta\\phi(\\mathbf{r})$,\n",
    "where $\\delta\\phi$ is a small perturbation.\n",
    "And we also suppose that $\\delta\\phi$ vanishes at the boundary (_i.e._ the surface of the box).\n",
    "We want to calculate the change in the total energy due to this perturbation:\n",
    "\\begin{align}\n",
    "\\delta \\mathcal{F} &= \\mathcal{F}[\\phi+\\delta\\phi] - \\mathcal{F}[\\phi] \\\\\n",
    "&= \\int_V g(\\phi+\\delta\\phi,\\partial_\\alpha\\phi+\\partial_\\alpha\\delta\\phi)\\,dV - \\int_V g(\\phi,\\partial_\\alpha\\phi)\\,dV\n",
    "\\end{align}\n",
    "Next we take Taylor series for the first term to get:\n",
    "\\begin{align}\n",
    "\\delta \\mathcal{F}\n",
    "&= \\int_V g(\\phi+\\delta\\phi,\\partial_\\alpha\\phi+\\partial_\\alpha\\delta\\phi)\\,dV - \n",
    "\\int_V g(\\phi,\\partial_\\alpha\\phi)\\,dV \\\\\n",
    "&= \\int_V \\left\\{ g(\\phi,\\partial_\\alpha\\phi) + \\frac{\\partial g}{\\partial\\phi}\\delta\\phi + \n",
    "\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\partial_\\alpha(\\delta\\phi) \\right\\} \\,dV - \n",
    "\\int_V g(\\phi,\\partial_\\alpha\\phi)\\,dV \\\\\n",
    "&= \\int_V \\left\\{ \\frac{\\partial g}{\\partial\\phi}\\delta\\phi + \\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\partial_\\alpha(\\delta\\phi) \\right\\}dV\n",
    "\\end{align}\n",
    "Using product rule (or integration by parts), we can write:\n",
    "\\begin{equation}\n",
    "\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\partial_\\alpha(\\delta\\phi)\n",
    "= \\partial_\\alpha\\left(\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\delta\\phi\\right) - \\partial_\\alpha\\left(\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\right)\\delta\\phi\n",
    "\\end{equation}\n",
    "Thus, $\\delta \\mathcal{F}$ becomes:\n",
    "\\begin{align}\n",
    "\\delta \\mathcal{F} = \\int_V \\left\\{ \\frac{\\partial g}{\\partial\\phi}\\delta\\phi \\right\\}dV + \\int_V \\partial_\\alpha\\left(\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\delta\\phi\\right)dV - \\int_V \\left\\{ \\partial_\\alpha\\left(\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\right)\\delta\\phi \\right\\}dV\n",
    "\\end{align}\n",
    "For the second term, we can use divergence theorem:\n",
    "\\begin{equation}\n",
    "\\int_V \\partial_\\alpha J_\\alpha\\,dV = \\oint_S J_\\alpha\\,dS_\\alpha,\n",
    "\\end{equation}\n",
    "where $S$ is the surface bounding the volume $V$.\n",
    "So we end up with:\n",
    "\\begin{align}\n",
    "\\delta \\mathcal{F} &= \\int_V \\left\\{ \\frac{\\partial g}{\\partial\\phi}\\delta\\phi \\right\\}dV + \\underbrace{\\oint_S \\left(\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\delta\\phi\\right)dS_\\alpha}_{=0} - \\int_V \\left\\{ \\partial_\\alpha\\left(\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\right)\\delta\\phi \\right\\}dV \\\\\n",
    "&= \\int_V \\bigg\\{ \\underbrace{\\frac{\\partial g}{\\partial\\phi} - \\partial_\\alpha\\left(\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\right)}_{\\delta \\mathcal{F}/\\delta\\phi(\\mathbf{r})} \\bigg\\}\\delta\\phi\\,dV \n",
    "\\end{align}\n",
    "The surface integral above vanishes since $\\delta\\phi$ vanishes at the boundary.\n",
    "The functional derivative of $\\mathcal{F}$ with respect to $\\phi(\\mathbf{r})$ is defined to be the terms inside the curly brackets:\n",
    "\\begin{equation}\n",
    "\\frac{\\delta \\mathcal{F}}{\\delta\\phi(\\mathbf{r})} = \\frac{\\partial g}{\\partial\\phi} - \\partial_\\alpha\\left(\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\right).\n",
    "\\end{equation}\n",
    "The functional derivative is like a generalization of the regular derivative.\n",
    "When the total energy is minimum (or maximum), the functional derivative of that energy must be equal to zero:\n",
    "\\begin{equation}\n",
    "\\frac{\\delta \\mathcal{F}}{\\delta\\phi} = 0.\n",
    "\\end{equation}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dynamics\n",
    "\n",
    "The chemical potential $\\mu(\\mathbf{r},t)$ is defined to be the energy cost of adding a particle locally at $\\mathbf{r}$. \n",
    "Mathematically, it can be written as:\n",
    "\\begin{equation}\n",
    "\\mu = \\frac{\\delta \\mathcal{F}}{\\delta\\phi}.\n",
    "\\end{equation}\n",
    "Using the formula for functional derivative above, we can calculate:\n",
    "\\begin{equation}\n",
    "\\mu = \\frac{\\partial g}{\\partial\\phi} - \\partial_\\alpha\\left(\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\right) \n",
    "= \\alpha\\phi + \\beta\\phi^3 - \\kappa\\underbrace{\\partial_\\alpha\\partial_\\alpha\\phi}_{\\nabla^2\\phi}\n",
    "\\end{equation}\n",
    "Note that:\n",
    "\\begin{equation}\n",
    "\\partial_\\alpha\\partial_\\alpha = \\partial_x^2 + \\partial_y^2 + \\partial_z^2 = \\nabla^2 \n",
    "\\end{equation}\n",
    "\n",
    "Since the total amount of fluid in the system is conserved, the dynamics of $\\phi(\\mathbf{r},t)$ must follow the continuity equation:\n",
    "\\begin{equation}\n",
    "\\frac{\\partial\\phi}{\\partial t} + \\nabla\\cdot\\mathbf{J} = 0,\n",
    "\\end{equation}\n",
    "where $\\mathbf{J}$ is the current.\n",
    "To see this, we can integrate the continuity equation over the whole box to get:\n",
    "\\begin{align}\n",
    "\\frac{d}{dt}\\int_V\\phi\\,dV + \\int_V \\nabla\\cdot\\mathbf{J}\\,dV &= 0 \\\\\n",
    "\\frac{d}{dt}\\int_V\\phi\\,dV + \\underbrace{\\oint_S \\mathbf{J}\\cdot\\hat{\\mathbf{n}}\\,dS}_{=0} &= 0,\n",
    "\\end{align}\n",
    "where we have used the divergence theorem on the second term.\n",
    "$S$ is the surface covering the boundary of the box.\n",
    "Now since we have periodic boundary condition on each side of the box, the second term in the equation above vanishes\n",
    "so we end up with:\n",
    "\\begin{equation}\n",
    "\\frac{d}{dt}\\int_V\\phi\\,dV  = 0 \n",
    "\\quad\\Rightarrow\\quad \n",
    "\\int_V\\phi\\,dV  = \\text{constant},\n",
    "\\end{equation}\n",
    "which implies the total amount of fluid is conserved.\n",
    "\n",
    "Finally, the current is given by:\n",
    "\\begin{equation}\n",
    "\\mathbf{J} = -M\\nabla\\mu,\n",
    "\\end{equation}\n",
    "_i.e._, particles diffuse from regions with high chemical potential to regions with low chemical potential.\n",
    "$M>0$ is called the mobility.\n",
    "Thus the dynamics can also be written as:\n",
    "\\begin{equation}\n",
    "\\frac{\\partial\\phi}{\\partial t} = M\\nabla^2\\mu.\n",
    "\\end{equation}\n",
    "Now let us consider the rate of change of the total free energy $\\mathcal{F}$:\n",
    "\\begin{align}\n",
    "\\frac{d\\mathcal{F}}{dt} = \\frac{d}{dt} \\int_V g(\\phi,\\partial_\\alpha\\phi)\\,dV \n",
    "=  \\int_V \\frac{\\partial}{\\partial t}g(\\phi,\\partial_\\alpha\\phi)\\,dV.\n",
    "\\end{align}\n",
    "Next using chain rule, we get:\n",
    "\\begin{align}\n",
    "\\frac{d\\mathcal{F}}{dt} &= \\int_V \\left\\{ \\frac{\\partial g}{\\partial\\phi}\\frac{\\partial\\phi}{\\partial t} + \\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\partial_\\alpha\\left(\\frac{\\partial\\phi}{\\partial t}\\right) \\right\\} dV \\\\\n",
    "&= \\int_V \\bigg\\{ \\underbrace{\\frac{\\partial g}{\\partial\\phi} - \\partial_\\alpha\\left(\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\right)}_{=\\mu} \\bigg\\}\\underbrace{\\frac{\\partial\\phi}{\\partial t}}_{=M\\nabla^2\\mu} \\,dV,\n",
    "\\end{align}\n",
    "where we have used integration by parts again and the surface integral term vanishes (similar to previous section).\n",
    "Finally, we substitute the dynamics to get:\n",
    "\\begin{align}\n",
    "\\frac{d\\mathcal{F}}{dt} = M\\int_V \\mu\\nabla^2\\mu\\,dV = -M\\int_V \\nabla\\mu\\cdot\\nabla\\mu\\,dV \\le 0,\n",
    "\\end{align}\n",
    "where we have used the integration by parts again.\n",
    "Thus we have shown that $\\frac{d\\mathcal{F}}{dt}$ is always negative as long as the dynamics follows $\\dot{\\phi}=\\nabla^2\\mu$ and $M\\ge0$.\n",
    "In other words the dynamics $\\dot{\\phi}=\\nabla^2\\mu$ will guarantee that the total energy always decreases with time until the minimum (equilibrium state) is reached."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear stability of the homogenous state\n",
    "\n",
    "The dynamics for $\\phi(\\mathbf{r},t)$ can be now written as:\n",
    "\\begin{equation}\n",
    "\\frac{\\partial\\phi}{\\partial t} = M\\nabla^2\\left[\\alpha\\phi + \\beta\\phi^3 - \\kappa\\nabla^2\\phi \\right]\n",
    "\\end{equation}\n",
    "Suppose we write the density field as follows:\n",
    "\\begin{equation}\n",
    "\\phi(\\mathbf{r},t) = \\phi_0 + \\delta\\phi(\\mathbf{r},t), \n",
    "\\end{equation}\n",
    "where $\\delta\\phi(\\mathbf{r},t)$ is a small fluctuation around the homogenous state $\\phi=\\phi_0$.\n",
    "We are interested in whether the fluctuations will grow or decay as the time $t$ increases.\n",
    "To do this, we substitute $\\phi=\\phi_0+\\delta\\phi$ into the dynamics to get:\n",
    "\\begin{equation}\n",
    "\\frac{\\partial\\delta\\phi}{\\partial t} = M\\nabla^2\\left[(\\alpha + 3\\beta\\phi_0^2)\\delta\\phi - \\kappa\\nabla^2\\delta\\phi \\right].\n",
    "\\end{equation}\n",
    "Note that we have neglected small terms of order $\\sim\\delta\\phi^2$ and higher.\n",
    "Now we write $\\delta\\phi(\\mathbf{r},t)$ as a Fourier series:\n",
    "\\begin{equation}\n",
    "\\delta\\phi(\\mathbf{r},t) = \\sum_{\\mathbf{q}} \\delta\\tilde\\phi_{\\mathbf{q}}(t)e^{i\\mathbf{q}\\cdot\\mathbf{r}}.\n",
    "\\end{equation}\n",
    "$\\mathbf{q}=(q_x,q_y,q_z)^T$ is the wavevector and each component of $\\mathbf{q}$ is discrete:\n",
    "\\begin{align}\n",
    "q_x &= 0, \\pm\\frac{2\\pi}{L_x}, \\pm\\frac{4\\pi}{L_x}, \\dots \\\\ \n",
    "q_y &= 0, \\pm\\frac{2\\pi}{L_y}, \\pm\\frac{4\\pi}{L_y}, \\dots \\\\ \n",
    "q_z &= 0, \\pm\\frac{2\\pi}{L_z}, \\pm\\frac{4\\pi}{L_z}, \\dots \\\\ \n",
    "\\end{align}\n",
    "The summation in the Fourier series indicates summing over all discrete values of $\\mathbf{q}$'s.\n",
    "The Fourier series can be thought as decomposing a function $\\phi(x)$, which is defined on $x\\in[0,L]$, \n",
    "into standing waves $1,e^{\\pm i\\frac{2\\pi}{L}x},e^{\\pm i\\frac{4\\pi}{L}x},\\dots$, with $\\tilde{\\phi}_q$ as the amplitude for each standing wave.\n",
    "Substituting the Fourier series into the dynamics, a set of first order ODE:\n",
    "\\begin{equation}\n",
    "\\frac{d\\delta\\tilde\\phi_\\mathbf{q}}{dt} = -M\\left[(\\alpha + 3\\beta\\phi_0^2)q^2  + \\kappa q^4 \\right] \\delta\\tilde\\phi_{\\mathbf{q}},\\quad\\text{for each }\\mathbf{q}.\n",
    "\\end{equation}\n",
    "The solution is:\n",
    "\\begin{equation}\n",
    "\\delta\\tilde\\phi_{\\mathbf{q}}(t) = A_\\mathbf{q} e^{r(q)t},\n",
    "\\end{equation}\n",
    "where\n",
    "\\begin{equation}\n",
    "r(q) = M[(-\\alpha - 3\\beta\\phi_0^2)q^2 - \\kappa q^4]\n",
    "\\end{equation}\n",
    "is the growth rate constant (here $q=|\\mathbf{q}|$).\n",
    "If $r(q)>0$ then the fluctuation $\\delta\\tilde\\phi_{\\mathbf{q}}$ will grow exponentially.\n",
    "If $r(q)<0$ then the fluctuation $\\delta\\tilde\\phi_{\\mathbf{q}}$ will decay exponentially to zero.\n",
    "Now let's consider case by case.\n",
    "\n",
    "#### Case I: $\\alpha>0$\n",
    "\n",
    "In this case, the coefficients of $q^2$ and $q^4$ in $r(q)$ are always negative.\n",
    "\\begin{equation}\n",
    "r(q) = M[\\underbrace{(-\\alpha - 3\\beta\\phi_0^2)}_{<0}q^2 \\underbrace{- \\kappa}_{<0} q^4].\n",
    "\\end{equation}\n",
    "Therefore the growth rate is always negative\n",
    "So the fluctuations decay to zero exponentially for all wavevector $\\mathbf{q}$.\n",
    "\n",
    "#### Case IIa: $\\alpha>0$ and $-\\sqrt{\\frac{-\\alpha}{3\\beta}}<\\phi_0<\\sqrt{\\frac{-\\alpha}{3\\beta}}$\n",
    "\n",
    "In this case, the coefficient of $q^2$ is positive whereas the coefficient of $q^4$ is negative.\n",
    "\\begin{equation}\n",
    "r(q) = M[\\underbrace{(-\\alpha - 3\\beta\\phi_0^2)}_{>0} q^2 \\underbrace{-\\kappa}_{<0} q^4].\n",
    "\\end{equation}\n",
    "We can plot the growth rate $r(q)$ as a function of $q=|\\mathbf{q}|$.\n",
    "The growth rate is positive for some range of $q$. \n",
    "This indicates the fluctuations will grow exponentially with time for some values of $\\mathbf{q}$.\n",
    "We can also define a characteristic wavevector $q^*$ such that the growth rate is maximum.\n",
    "This correponds to the initial growth lengthscale $\\lambda^*=\\frac{2\\pi}{q^*}$.\n",
    "This instability is illustrated in the numerical simulation at the bottom of this notebook.\n",
    "\n",
    "#### Case IIb: $\\alpha>0$ and $|\\phi_0|>\\sqrt{\\frac{-\\alpha}{3\\beta}}$\n",
    "\n",
    "In this case, both coefficients of $q^2$ and $q^4$ are negative in the growth rate.\n",
    "Thus all fluctuations decay to zero for all $\\mathbf{q}$, similar to case I.\n",
    "However from the previous section, we also learnt that if $\\sqrt{\\frac{-\\alpha}{3\\beta}}<\\phi_0<\\sqrt{\\frac{-\\alpha}{\\beta}}$ or\n",
    "$-\\sqrt{\\frac{-\\alpha}{\\beta}}<\\phi_0<\\sqrt{\\frac{-\\alpha}{3\\beta}}$, the actual equilibrium state (or lowest energy state) of the system is actually a droplet state.\n",
    "However, if we initialize the system from a homogenous state $\\phi(\\mathbf{r},t=0)=\\phi_0+\\text{ small noise}$, the system will actually remain in the homogenous state, even though the homogenous state is not the lowest energy state.\n",
    "We call the homogenous state to be a metastable state in this regime.\n",
    "In order for the system to jump into the actual equilibrium state, we need to add a rather large perturbation in the initial condition.\n",
    "For instance we may introduce a nucleus (_i.e._ a tiny droplet) into the system at the initial time.\n",
    "If the size of the nucleus is larger than the critical size, the nucleus will then grow until the equilibrium state of a macroscopic droplet is reached. \n",
    "This process is called nucleation process, which is quite distinct from Case IIa."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-coarsening/master/figures/stability-phase-diagram.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numerical simulation\n",
    "\n",
    "Let us consider a two-dimensional system. In computer simulation the space $\\mathbf{r}$ is discretized into lattice with step size $\\Delta x$ along the $x$-axis and $\\Delta y$ along the $y$-axis.\n",
    "(Ideally $\\Delta x$ and $\\Delta y$ have to be small.)\n",
    "The coordinates $x$ and $y$ then become:\n",
    "\\begin{align}\n",
    "x &\\rightarrow i\\Delta x, \\quad \\text{where } i = 0,1,2,\\dots,N_x-1 \\\\\n",
    "y &\\rightarrow j\\Delta y, \\quad \\text{where } j = 0,1,2,\\dots,N_y-1.\n",
    "\\end{align}\n",
    "$N_x\\in\\mathbb{N}$ and $N_y\\in\\mathbb{N}$ are the number of lattice points along $x$ and along $y$ respectively.\n",
    "The system size is now $L_x\\times L_y$, where $L_x = N_x \\Delta x$ and $L_y = N_y \\Delta y$. \n",
    "Similarly the time $t$ is also discretized into:\n",
    "\\begin{equation}\n",
    "t \\rightarrow n\\Delta t, \\quad \\text{where } n = 0,1,2,\\dots,N_t-1\n",
    "\\end{equation}\n",
    "where $N_t\\Delta t$ is total length of time that we run the simulation for.\n",
    "The density field then becomes:\n",
    "\\begin{equation}\n",
    "\\phi(\\mathbf{r},t) \\rightarrow \\phi^n_{ij}.\n",
    "\\end{equation}\n",
    "\n",
    "Now we want to solve:\n",
    "\\begin{equation}\n",
    "\\frac{\\partial\\phi}{\\partial t} = M\\nabla^2\\mu, \\quad \\text{where} \\quad \n",
    "\\mu = \\alpha\\phi + \\beta\\phi^3 - \\kappa\\nabla^2\\phi.\n",
    "\\end{equation}\n",
    "First we can write the time derivative as:\n",
    "\\begin{equation}\n",
    "\\frac{\\partial\\phi}{\\partial t} \\simeq \\frac{\\phi_{ij}^{n+1} - \\phi_{ij}^n}{\\Delta t} + \\mathcal{O}(\\Delta t)\n",
    "\\end{equation}\n",
    "So the dynamics becomes:\n",
    "\\begin{equation}\n",
    "\\phi_{ij}^{n+1} = \\phi_{ij}^n + \\Delta t M\\nabla^2 \\mu_{ij}^n.\n",
    "\\end{equation}\n",
    "So for a given initial condition $\\phi_{ij}^0$ we can find $\\phi$ for subsequent timesteps: $\\phi_{ij}^1, \\phi_{ij}^2, \\dots$. \n",
    "Next we need to calculate the spatial derivatives of $\\phi$ and $\\mu$ in the lattice, _i.e._:\n",
    "\\begin{align}\n",
    "\\frac{\\partial\\phi}{\\partial x} &\\simeq \\frac{\\phi_{i+1,j} - \\phi_{i-1,j}}{2\\Delta x} + \\mathcal{O}(\\Delta x^2) \\\\\n",
    "\\frac{\\partial\\phi}{\\partial y} &\\simeq \\frac{\\phi_{i,j+1} - \\phi_{i,j-1}}{2\\Delta y} + \\mathcal{O}(\\Delta y^2)\\\\\n",
    "\\frac{\\partial^2\\phi}{\\partial x^2} &\\simeq \\frac{\\phi_{i+1,j} - 2\\phi_{ij} + \\phi_{i-1,j}}{\\Delta x^2} + \\mathcal{O}(\\Delta x) \\\\\n",
    "\\frac{\\partial^2\\phi}{\\partial y^2} &\\simeq \\frac{\\phi_{i,j+1} - 2\\phi_{ij} + \\phi_{i,j-1}}{\\Delta y^2} + \\mathcal{O}(\\Delta y) \\\\\n",
    "\\end{align}\n",
    "The Laplacian is then given by:\n",
    "\\begin{align}\n",
    "\\nabla^2\\phi = \\frac{\\partial^2\\phi}{\\partial x^2} + \\frac{\\partial^2\\phi}{\\partial y^2} \n",
    "= \\frac{\\phi_{i+1,j} - 2\\phi_{ij} + \\phi_{i-1,j}}{\\Delta x^2} +\n",
    "  \\frac{\\phi_{i,j+1} - 2\\phi_{ij} + \\phi_{i,j-1}}{\\Delta y^2} + \\mathcal{O}(\\Delta x) + \\mathcal{O}(\\Delta y).\n",
    "\\end{align}\n",
    "In Python $\\phi_{ij}$ is represented as an array:\n",
    "\\begin{align}\n",
    "\\phi &=\n",
    "\\begin{pmatrix}\n",
    "\\phi_{00} & \\phi_{01} & \\ldots & \\phi_{0,N_y-1} \\\\\n",
    "\\phi_{10} & \\phi_{11} &  & \\phi_{1,N_y-1} \\\\\n",
    "\\vdots &  & \\ddots & \\vdots \\\\\n",
    "\\phi_{N_x-1,0} & \\phi_{N_x-1,1} & \\ldots & \\phi_{N_x-1,N_y-1} \\\\\n",
    "\\end{pmatrix} \\downarrow i\\text{-direction} \\\\\n",
    "&\\quad\\quad\\quad\\quad\\quad\\longrightarrow j\\text{-direction}\n",
    "\\end{align}\n",
    "Note that the $x$- and $y$-axis (or $i$- and $j$-direction) are transposed.\n",
    "The array $\\phi_{i+1,j}$ is then equivalent to shifting every elements inside the array $\\phi_{ij}$ upwards.\n",
    "This is represented by the `np.roll` function in Python:\n",
    "\n",
    "`phi_i_plus_1 = np.roll(phi, -1, axis=0)`\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "\n",
    "# lattice parameters\n",
    "dx = 1.0  \n",
    "Nx, Ny = 64, 64\n",
    "dt = 0.01 \n",
    "Nt = 1000000\n",
    "T = int(Nt*dt)  # total length of time \n",
    "\n",
    "# free energy parameters\n",
    "M, alpha, beta, kappa = 1.0, -1.0, 1.0, 1.0\n",
    "phi0 = 0.0\n",
    "\n",
    "# array of cartesian coordinates (needed for plotting)\n",
    "x = np.arange(0, Nx)*dx\n",
    "y = np.arange(0, Ny)*dx\n",
    "y, x = np.meshgrid(y, x) \n",
    "\n",
    "# create a phi-matrix of size Nx by Ny (similarly for mu-matrix)\n",
    "phi = np.zeros((Nx, Ny))  # phi at instantaneous time\n",
    "phi_t = np.zeros((Nx, Ny, T))  # phi as a function of time t\n",
    "mu = np.zeros((Nx, Ny))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# method to calculate the laplacian\n",
    "def laplacian(phi):\n",
    "    # axis=0 --> roll along x-direction\n",
    "    # axis=1 --> roll along y-direction\n",
    "    laplacianphi = (np.roll(phi,+1,axis=0) - 2.0*phi + np.roll(phi,-1,axis=0))/(dx*dx) \\\n",
    "                 + (np.roll(phi,+1,axis=1) - 2.0*phi + np.roll(phi,-1,axis=1))/(dx*dx)\n",
    "\n",
    "    return laplacianphi\n",
    "\n",
    "# create animation\n",
    "def animate(phi_t):\n",
    "    # initialize figure and movie objects\n",
    "    fig, ax = plt.subplots(figsize=(4,4)) \n",
    "\n",
    "    # set label\n",
    "    ax.set_xlabel('$x$')\n",
    "    ax.set_ylabel('$y$')\n",
    "\n",
    "    # set x range and y range\n",
    "    ax.set_xlim([0, Nx*dx])\n",
    "    ax.set_ylim([0, Ny*dx])\n",
    "\n",
    "    # set tick interval\n",
    "    ax.tick_params(axis='both')\n",
    "    ax.set_xticks(np.arange(0, Nx, 10)*dx)\n",
    "    ax.set_yticks(np.arange(0, Ny, 10)*dx)\n",
    "\n",
    "    # set aspect ratio\n",
    "    ax.set_aspect('equal')\n",
    "\n",
    "    # create colormap of phi\n",
    "    colormap = ax.pcolormesh(x, y, phi_t[:,:,0], shading='auto', vmin=-1.2, vmax=1.2)\n",
    "    plt.colorbar(colormap)\n",
    "\n",
    "    def animate(t):\n",
    "        # set title\n",
    "        ax.set_title(f't = {t}')\n",
    "        colormap.set_array(phi_t[:,:,t].flatten())  # update data\n",
    "\n",
    "    # interval = time between frames in miliseconds\n",
    "    ani = animation.FuncAnimation(fig, animate, interval=10, frames=range(0,T,10))  \n",
    "    ani.save(\"movie.mp4\")\n",
    "\n",
    "# plot phi at 4 different times\n",
    "def plot(phi_t, t1, t2, t3, t4):\n",
    "    # initialize figure and movie objects\n",
    "    fig, (ax1, ax2, ax3, ax4) = plt.subplots(1,4,figsize=(8,2)) \n",
    "\n",
    "    ax1.set_title(f't = {t1}')\n",
    "    ax1.set_aspect('equal')\n",
    "    colormap = ax1.pcolormesh(x, y, phi_t[:,:,t1], \\\n",
    "                              shading='auto', vmin=-1.2, vmax=1.2)\n",
    "\n",
    "    ax2.set_title(f't = {t2}')\n",
    "    ax2.set_aspect('equal')\n",
    "    colormap = ax2.pcolormesh(x, y, phi_t[:,:,t2], \\\n",
    "                              shading='auto', vmin=-1.2, vmax=1.2)\n",
    "\n",
    "    ax3.set_title(f't = {t3}')\n",
    "    ax3.set_aspect('equal')\n",
    "    colormap = ax3.pcolormesh(x, y, phi_t[:,:,t3], \\\n",
    "                              shading='auto', vmin=-1.2, vmax=1.2)\n",
    "\n",
    "    ax4.set_title(f't = {t4}')\n",
    "    ax4.set_aspect('equal')\n",
    "    colormap = ax4.pcolormesh(x, y, phi_t[:,:,t4], \\\n",
    "                              shading='auto', vmin=-1.2, vmax=1.2)\n",
    "\n",
    "    plt.show()\n",
    "\n",
    "# update phi\n",
    "def update(phi):\n",
    "    # calculate mu\n",
    "    mu = alpha*phi + beta*phi*phi*phi - kappa*laplacian(phi)\n",
    "\n",
    "    # update phi\n",
    "    phi = phi + dt*M*laplacian(mu)\n",
    "    \n",
    "    return phi\n",
    "\n",
    "# method to run the simulation from a uniform phi + a little bit of noise\n",
    "def run():\n",
    "    # initialize the system with uniform state + a little bit of noise\n",
    "    phi = np.ones((Nx, Ny))*phi0 + np.random.normal(0.0, 0.001, (Nx, Ny))\n",
    "    \n",
    "    # update the \\phi_{ij}^n for all n\n",
    "    for n in range(0, Nt, 1):\n",
    "            \n",
    "        # save current phi\n",
    "        if (n % int(1/dt) == 0):\n",
    "            phi_t[:,:,int(n*dt)] = phi\n",
    "            \n",
    "        phi = update(phi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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8t8qpw2vU7+PFdSqp2lZ7/h7r+k4H+9Y5YRhXtamkWp5R35CSvOosk1Zn+ucAQLqCr427JMKlWi6fUnpUN03qVFFnj1yZNJz3ZpKOTKs/zL4xlUNDJ/uC8swBnXXqHPLqXnNA29tx1PGNOeksJXCePWcTEMCq2TTg/hu1qFJE7gVw75x9X3TKnwPwuYZU5pAwQ2MYSw8RoGBZEAzDiJZ4r8wOQ7IMjaDcIUp0ZoOO63fGyd+cHvPkTNsxTfK1bK/KjOk+x+1+uScvjq5QmbF/hXZ7Z5c7dfQ4CcS6PBnR3ebsy6mMak9re3IpJ3+zv9+drXK7yO6HK190ZFLBn0nK632MTutM0siElqfGPBmVGtZz2487gaicuYSBE/ps2/x4vplRJ1PBlCuRnJkkN2hV0lZhJ4RGjNG0kmQZGsNYglgWBMMwokfilx24XpJtaNzuedF1CnPiufrd+oyT8Cs9qgNruRP6CLraPHlRbFeZku/S47Md+r7ZLp3Fynd4AaFGVLHglE78oKCqDc4CaZQmjaqFGnGXUzmTWEj5t+Lmz8voJBC6dJkR+sa9i2THnVkwp5yeUFmXcpzsSulPynJhV5JIgMmkJpD0UJ7JNjSGsQQQ0GadYkmlnk5ZL0d/HdxwkqW0rJLSf2o2o+V2pyxp7fWIv18ydI4757r7U5XLtaBzT6Uy887K8lnn/pwB89KgbVnPxFkSUL7fHWiX8r9Ac3oudTyTeZPAHphJJ8MwIsdmnQzDiBQRMzTNRyq769ekTONW6Yf68qrsXzrjyCw6R9xype6+c1zKzq3d1Jr4j4Buf7qaxKkkgVxaOahbj0ziPL9ole67Wr0xllQ2vW0YRuQUi2ZoDMOIEAFNOjUdOtqjHhkV5tzSZAxDyKxKVOmSN/0jEmMJUFW2lKRRPfK0GpWko1uuJiPdOmL2DOPVmvpp2uQ8yTTJfyP5ff/1CpIPkHzB/2vZ4QyjEv5gcJgtrjTTC+hDAHY6ry33tmGERUJuMaUp0onkBgBvBPAZAP/D31137u2mMd+ZrULtU2rCELZ/vu2bb30LJVVFDrnldMnp0XV0dD6eZc6SFWRWwXFodB0SZys4J1ZbSuGWYyaj4txbCUOzxmj+GsBHAPQ4++rOvd2e7ql0imEsagTJn3VqRqbKNwE4KiKPzef9IrJdRK4QkSty6c4Gt84wEoAAEIbbYkozejRXAXiLnyOmHcAykl/HPHJvn8F8Z6DiTLPvw62vkTKqmlxy1pFJzlkv1u4tcc9361L32WX68ZzpdjJt5krSSS+bdlJ6Z8f1nrIjupYtM+6dlJqoI4CXey8tlFBJX+sUeY9GRD4uIhtEZBBeeocficjvw3JvG0Z4bDB43tyCOnNvGxHQ6MHgVIWBWrcX064hRfO9GsBncpUXwGdsg/ZynLTfyK904gl1zvpV6DcrP6XXTZ3QcudBDQzUc8Db33lIj2dOaSAfN1Rpxd5NywaI4z11HYamGhoR+TG82SXLvW0Y9RDj3koYkucZbBhLDQEk4bNOi8fQLMaB4aRQaeC3ilyaXa4hUEfOU+l08iLvfbktmrzv5asPB+WNnZqVdXlW86+XGMvrtYam1Mn8qeOaIPDwb7z9y3foub27nbCux5xbqiSjWjoia4bGMIyoMelkGEbkmKExEkeUyw586SQZnT0q9KhUGR3U8vHL9duz6ZIhAMBrVu4K9l3QfiQo96VVLmUrrPWYhdY32qny7OLug0H54d5NAIDHOwadd6rfznIn7nL2hLN0oRRLuVVLFEoOewnGDI1hJICkDzuaoTGMJGA9mhhiM1DRU22JQWkVdofONE2tVKe5U1v03MGLDwTl69fsAAC8pO1QsK8vNRGUc04GvRTPlCpuTN0eqhOeK7m60/7+i/R9T05sCsptIyqjeiYdB8HSqu/ZKiu9m0CFW04Ui9PQGMZiIubLC8JghsYwYk+8V2aHwQzNUqEZAa6gWT4LnSqdxtbqjBC2jAXFV63cHZRLkqk/rcezTvLxNNzymT/vBUe+ZR2ZlRMtn5/zZrHGluvM194L1LlvdH9/UO44ojKqtNrbzXDa9Bko69EYhhE5CR9qNENjLBx3MNgPuZnv0h7NxBo9vHmV+vm7fjLLUt5ArduLydURG7VSL8drm16jk17PZGPueLDv4n5d5vDzdSuC8uwu/Wpkh70eGamxbZrKIvCjaWZwcsMw5gkl3FbzOuR1JJ8juYvkGQkB6PE3/vFfk7y8Ee03Q2MYSaABga9IpgF8HsD1ALYCeAfJrXNOux7AZn+7CcAXGtH8mtKJ5A8BfFhEnmxEhUaCqZX8DZrFIN+pv2Ezy1UOndN1Mij3pdVPpp2e30o6wsGIki9OT3oy2Hdux3BQ/lm/+s7MOtJPsv5gdtmgb5WB4XizDcAuEdkNACTvhJeN5BnnnBsBfE1EBMAvSPaVQu4upOIwPZqPAPgrkl/xY/sahtFkGiSd1gPY77we8vfVe07d1DQ0IvK4iLwOwPcB3EfyZpIdtd5nGEaDEHg9qDAbMEDyUWe7yblSpS7pXPMU5py6CTXrRJIAnoOn1/4cwB+R/LiI/MNCG2AkFJ5dRhUzerzYqTM/yzK6PCBbYRan6uzRPHGvV/B/8t3V391uezpUOhVyKp2QjsGMT/jHclxErqhybAjAOc7rDQAOzuOcuqnZoyH5EIADAP4KXhfqPfAyTG4juT3E+9tJ/orkkyR3kPyUv99ybxtGSBoknR4BsJnkeSRz8LKS3DPnnHsA/IE/+3QlgNMLHZ8BwvVo3gdghz845PJBkjsrvWEO0wBeJyJjJLMAHiL5zwB+F17u7Vv8abaPIS4pcQ0jbjSgoycieZIfAPADAGkAt4nIDpLv849/EcC9AG4AsAvABID3LrzmEIZGRJ4+y+E3hni/ACj5lWf9TRDn3NtGY0mIs1msJ48a1DYRuReeMXH3fdEpC4D3N6Y2ZUF+NKVpslqQTJN8Al42ygdE5JeYk3sbQNXc26WBrZnCRKVTDGNRQwFYZKgtrjRlCYKIFABcRrIPwN0kL6njvdsBbAeA3rY1cf7NWVpU+/n396ec2C2pSf09G3GyFUwVdcC14PuoFJxJj0YMDLvXK8WsccN+ns5rPvf8hLbHTbGLQgw+djFowkJoqmewiJyCJ5Gug597GwDmnXvbMJYIjVqC0CoiNzQkV/o9Gfj+N9cAeBaWe9swwmO5t2uyFsDt/jqLFIBvicj3ST4My72dLNxYK6nKLvj0w15mJtRXpe2ExnZ5cUxjvpzq6QrK/X7IzfIYNOFXb4dhRjzJdKqg9e6fVK+KtJOzOzvmZEGY9cvu/Tdz5DjmvZUwRG5oROTXAF5WYb/l3jaMsJihMQwjamiBr4wljysjfOmUHdPlBZ2HVTrtOrwyKD+/XCNi9Wc8V6uyZQlucoVaoTydkwvO0OOsaHlUvBmvvTMq3546quuEOw/oNXIj2g4W4pB7O9mYoTGMJJBwG2eGxjDijg0GG4nBTaQXYUYE5j2ZkZpUj7fuQyqdxp9XB7mfrTg/KPdmPK/vrerPV5ZArhQYC9BV2C5uArnS7BKgcgkAXpj2pNpDxy/Q9rzQF5RXDznybEzrC2adqkmnqHNvA9ajMQyjCZihMQwjSgibdYonlm87esqc95z9vrzglM7atB2bDsp9z6usOdy1Oij/ky+Hxlap1NnSoalQyuMLuwuRPGZEP8qnCirP9s0MBOWHTnhSbecOjevUv1MlV+cRDYKVmnLqKBTL7q3p2BiNYRhNwQyNkTiiHBguetcuDQoDQHpMewo9+9z6tPdydNLzZ/nWlmXBvgvXaIK5Td2a9K0UDjTt6InTeQ1jPTShywqeOaa9pqld3rUHnHBty17U3lbmtLaT044/T6WeTDMGgF3M0BiGETUmnQzDiBaB5d42jIqruov6zXBlSOakDuouyzt5tsc8GTV2UFdWP79mU1B+amCj1tHlX88NKDepg8zZYf1Yd+p4MtYc8OrrOFpNLqnvjNv+OCw9sB6NYRjRY4bGMIyosR5NXDDfmfnR6BmokoxyL1VFRqULmgO709/fNqzLFZa9qB/PfJdKo0Kbd46bXCE9o9/E7IQj1cbcsucbQ8dHhjNOcK1acqnZM00uZmgMw4iUmIfpDIMZGsOIOYRJp5qQPAfA1wCsgTdJt11EbiW5AsA/AhgEsAfA20TkZNTtMZpEtSUKroxyUrKw5Og3qTM/mdMqlySjF5HUmRKP7nWd2SwUCmfuL1SRSK1cnV2DpBuaZqRbyQP4sIhcBOBKAO8nuRVeCtwHRWQzgAf914ZhVCLhWRAiNzQickhEHvfLowB2AlgPLyXu7f5ptwP4najbYhiJJeGGpqljNCQH4WVEOCMlLsmKKXHPStxmmiIMKFWTuD0Llxorvd0y3XgIznqpsmSvqRqpX6ulRSmVwzjgxUAuBdjq7fCQ7AbwbQB/LCIjZLg8wSRvAnATALSne6JroGHEGItHEwKSWXhG5g4R+Y6/+wjJtX5vpmpK3Mhzb9fTC6n1SxrGeNY6px5397Jz05XPidMvM1A7CV21+3efW60v3XyXDMTtWbnEuGlhaEZKXAL4MoCdIvJZ55ClxDWMkCQ993YzejRXAXgXgKdIPuHv+wSAW2ApcQ2jNjEf6A1DM1LiPoQ5Y3kOjUuJO9+BWLf7Xuqes8K+OeeW+XKUiu4+531Sdo062uauDnDlQPHM0JIsVBkALVaQJNXKzZYOleqrJk8buYI6zhKpGglssot5BhtGzDHPYMMwmgKT2AtzSLahCSOXUhXkkCNxJOPM1vhu7mXu7lk9Xsw55bL9KX+f1lHMVC6LOzlUQSXQWUzsTmmm8nJGOTWjJ5SXC2eUgyRoQGW3fKCya36zpVXCv1CRYGM0hmE0g2ZIp7DrD0nuATAKoAAgLyJX1Lp2C11ZDcMITXOWINSz/vC1InJZGCMDJL1HU2nGaG7Zl0aS01sttmu50JENyvlub/9Mt9pftzzbrZfNdzllP19ZvkP/08U2lSGSc1YWZ1w9NPeGABSctue1zGk9OTXllTMTeh+ZcX1bbkzL2TGvTbnRorNPpVNmXFdLp6acfNMzef9vFS3XytmqJUiTBoNvBHC1X74dwI8BfLQRF7YejWEkgeb0aMrWHwKotv5QANxP8jF/iVBNkt2jMYylgNS11mmA5KPO6+3+Mh4AAMkfwosNNZdP1tGiq0TkoL8Q+gGSz4rIT8/2huQZmlSV2aOs3oq40qi7DQAw06sSabJfp36m+vV6kyu9v7MDKiHa+jQ9yIpuLS/vcNKGZL30HT1ZTd3RkXKukXbi1jrTSukKi3YKTicz70xRTRS0/eN5757G/L8AMDylmRqHJzT39LHTftmJxdt+VMsdx7TceUzb1nbSa39m1ImvWy3WbiVJZXKqYdTpR3P8bOMmInJN1XrIsOsPD/p/j5K8G8A2AGc1NCadDCMJiITbFkbN9Ycku0j2lMoA3gDg6VoXTlaPhgBy+stebHMGcnv1131qQH+lx9Z5vYLx9XqZwnrteaxZdTooX7JsGACwvkP39WdHg3JPSt/XmdJf93a/95KlE+Hf6a2knZ+jVB0pB4vO70DBCfk/K/6gtdPjmRZ9FqMFp3fjj1ofmu4N9u0d09zUe0+sCMqnDmhPqGufl9Ct+4DT4zmi9+wmXks54Tc1hoxzn0u9d1Nr1X8ImjQYXHH9Icl1AL4kIjcAWA3gbj/MSwbAN0TkvloXTpahMYylSJMc9kTkBCqsP/Sl0g1+eTeAS+u9thkaw0gArsd4EkmWoSFR6FVZMLWyPSiPnKu3MnK+mv+2wREAwKWrjwT7NnfrGNeG3HBQXuE7o3SlNDezK4eycAZynQHQtP9zU48sWgjFCkNrBbjSSiXVjC+zpjpVWp1ephLp4IDKqBc26GzmU+evBQAc2tMX7Fu2S5937x69XvsRlVHpMa9MxydnycqokmSqkLWhXmxRpWEY0SJobJiMFmCGxjASgPVomohkUhg5X33/T25RuVC4UH3wX77hQFB+6TKvvLHteLCvL60+MJ1UmZTzhXCqbMaoshxKt3Q57dkFe6FCXOLSTBUA9Kd1jcLqjM6wXdCu8vKlPd5ze2pAp+seWXtuUD4yoIHilz+nUqx7nycTsic1r3a5jHJzXSf821MJd4Yp7S9/ac9WObkOEv6oEmVoDGMpYoGvDMOIHhELfFULkrcBeBOAoyJyib9vXnm3C+0pHN2mD3ztRdrVv2rV7qB8YcehoLwy4806lc0k1Zg9qkYqIT8rKec+ir6jX5qzFc9tT+v+ZY5DYl/ak6Krsyqtzu3QGbqf9Z4flPcv06Uz+XZvVrBvl0qI7AmVqtQqsGhkVAW5BKhkmu3vnPuO+knw4wGaswThqwCum7PP8m4bRh0kPd1KM3Jv/xTA8JzdN8LybhtGOARejy/MFlNaNUYzr7zb+U7gosv2Bq9fM/BCUD6vTZ3w3FmVkkzKVXGtTIocmi8178857MqrkqNie1b39aR0Jml5Vmf5fpJTPfRUdqNfUue+PqeO7Al9H6fctvkSNsZfljKqRRFo069USTKd3KLPYt4k5LFUI/aDwW7u7fSKvtY2xjBaRNJ/D1tlaELFvQDKc28vv3CVvGHVM8GxTTl9mzuQWb6K2l8ekPT/VERUfS7+7i7qiu10Ws/NtumAe3al01u82PvzdH5Q3zejK+t7nTg2aSfpHWdKPaeE/J/cBIFOLKR8ny6RGRn07nv40oUvVEr6rFOr4tFY3m3DCEvYMJ4xtkXNmN7+JryAxwMkhwDcDMu7bRih8Rz2YmxFQtCM3NvvqHKo7rzb3ZmpMrnUl1L/DHew12TSwqn0DNvhZExIqf+R+z95zYDnRzK2VQNmDY1tCMq5UR0Y7ZpWictSwKw4Z1dwB4CdxIPFLpWGE2v0/oa3en9fsnVo4XU3JzBAZMR+MNgwDOvRGIYRNRJvH5kwJMrQtDFvcqmFuM845ywfcGf8Nrd5y0JetVLd7r99oa70Pn1MYxfnTqnMyJUS1k26GiEm/9MKAazchISzy1U6nd6k5/Rc5Pmpvnpg14KbkPSPd6IMjWEsWUw6GYYRKfUlkIsliTI0aRZNLsUE99m7s1GlZQpb2tWh77fXaUqXn5yvCcy7D2lAqOyIV6YzExW7Gai0yqJip86qja/R+xjbos/i+jV7AAAXtGmUgXljPRrDMCIn2XYmWYaGEOvFxJBKvZv+jC5sfWmP+pE8Nbg2KI/9pj8odx72eghZN+3uVEwGhv3lBpJV35nZHidJ4Qb1r9lwzomgfFGX16srxfZZUBOKydZOiTI0hrEkEZjDnmEY0UKIOew1k4VnMDaiprRa3l31vS57Kihf1K8Dow9v0EHimd3egGrmpH4k6azCL3P/b8bAcIV4M67vzHSflifW6wTFq5YfDsrrsl502k7nWcwbMzSGYUSOGRrDMCLFxmgMo5zSDFQWKnvcEKCbOjWR38NrdOnC1HJvOULHIZ3ZwZQTLqmVsy6+jCo60mlqhbatbc1oUD6v81hQLmXeqJaEsB5s1skwjIgRk06GYUSMwAyNYVTCTcbX7mRXcBPSDSxXp74pP9Zu0ckikK6QQ7xpuDGB/VmnYrvKumldhI61vSNBeWVGZVTpvhuSpz3ZyskMjWEkAfOjMQwjWgRAIdldmpYaGpLXAbgVQBrAl0Tklla2x4gGd8V9T1pnmlZ3qcx4oWclAKDQpvIkk1b5wjIZ1eRfd78dbttml+nh1Z0qATtTTnqahrUz+YPBrUq3ApJpAJ8HcD2ArQDeQXJrq9pjGLFGJNwWU1pmaABsA7BLRHaLyAyAO+Hl5DYWASmKbigGWztng60vNxls+S4g3wUUsww2pFK6NRtSN78NxRyDLd8lwdabnQy2LPPBFty38yzmTRMMDcm3ktxBskjyirOcdx3J50juIvmxMNdupaFZD2C/83rI32cYhovAW98VZlsYTwP4XQA/rXbCfJVIK8doKs1dnvGk3NzbAKa3bdz7dKStKmcAwPGaZ1l986zv0TNO+E1kdTWAJ53yPVr8e6e+v695keBjvzF8xQJI9IPBIrITmDsedgaBEvHPLSmRZ872plYamiEA5zivNwA4OPckN/c2yUdFpGqXrtFYfcmtb1HdW32zTgMkXQu+3f8ONYpKSuQVtd7USkPzCIDNJM8DcADA2wG8s4XtMYz4En785fjZDB7JHwJYU+HQJ0XkuyGuH0qJzKVlhkZE8iQ/AOAH8Ka3bxORHa1qj2HEmgbNKInINQu8RCglMpeW+tGIyL0A7q3jLY3sAlp9i7u+RXRvsZq6npcSocTnBgzDqEBvdpX8u4G3hjr3vsN/99h8x4pI/gcAfwtgJYBTAJ4QkWtJroPnUHuDf94NAP4aqkQ+U+vatgTBMJJAE+LRiMjdAO6usP8ggBuc1/UqkZb60YRmPg5C86jjNpJHST7t7FtB8gGSL/h/lzeornNI/gvJnb6D1Icirq+d5K9IPunX96ko63PqTZP8N5Lfj7o+kntIPkXyidKsS8T19ZG8i+Sz/v/xldHVF9KHJg5J9qoQe0PTxKUKXwVw3Zx9HwPwoIhsBvCg/7oR5AF8WEQuAnAlgPf79xRVfdMAXicilwK4DMB1JK+MsL4SHwKw03kddX2vFZHLHOkQZX23ArhPRC4EcCm8+4ymPgFEiqG2uBJ7Q4MmLVUQkZ8CGJ6z+0YAt/vl2wH8ToPqOiQij/vlUXgf0vUR1iciUlr5l/U3iao+ACC5AcAbAXzJ2R1ZfVWIpD6SywC8GsCXAUBEZkTkVFT1AbAeTRNo5VKF1SJyCPCMA4BVja6A5CCAlwH4ZZT1+TLmCQBHATwgIpHWB2+w8CMoD9kUZX0C4H6Sj/ne5FHWtwnAMQBf8aXhl0h2RVifLapsAvNyEEoCJLsBfBvAH4vISK3zF4KIFETkMnh+D9tIXhJVXSTfBOCoiDwWVR0VuEpELocnsd9P8tUR1pUBcDmAL4jIywCMo/EyUBHxBoPDbDElCYZmXg5CDeIIybUA4P892qgLk8zCMzJ3iMh3oq6vhN/F/zG88aio6rsKwFtI7oEndV9H8usR1leaGYGIHIU3c7ItwvqGAAz5vUIAuAue4Ynu/gqFUFtcSYKhCRyESObgOQjdU+M9jeIeAO/2y+8GEMZFuyb0Vq19GcBOEflsE+pbSbLPL3cAuAbAs1HVJyIfF5ENIjII7//1IxH5/ajqI9lFsqdUBvAGeCuRo7q/wwD2k3yJv+v18BYVRlJf4LCXYOkUez+aZi1VIPlNAFfDW5Q2BOBmALcA+BbJPwSwD0A4r6naXAXgXQCe8sdNAOATEda3FsDt/gxeCsC3ROT7JB+OqL5qRHV/qwHc7dlvZAB8Q0TuI/lIRPUBwAcB3OH/+O0G8F74z7bh9ZXCRCQY8ww2jJjTm+qXK3NzPS8qc//0N+btGRwlse/RGMZSRwBIwns0ZmgMI+5IcwJfRYkZGsNIAHGeUQqDjdEYRswheR+8UKFhOC4i4QZ0mogZGsMwIicJfjSGYSQcMzSGYUSOGRoDAODHx/n3fvnPSf5Nq9tkLB5s1skocTOAT5NcBW81+Vta3B5jEWGDwUYAyZ8A6AZwtR8nxzAagkknAwBA8qXw1kRNm5ExGo0ZGqMU0uAOeBHixkle2+ImGYsMMzRLHJKdAL4DL4bxTgB/BuB/tbRRxqLDxmgMw4gc69EYhhE5ZmgMw4gcMzSGYUSOGRrDMCLHDI1hGJFjhsYwjMgxQ2MYRuSYoTEMI3L+Px50wlE5TrTDAAAAAElFTkSuQmCC",
      "text/plain": [
       "<Figure size 288x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 576x144 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# run simulation for phi0 = -0.2\n",
    "phi0 = -0.2\n",
    "run()\n",
    "#animate(phi_t)\n",
    "plot(phi_t, 10, 100, 1000, 9990)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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l+y+SaPeTEzKoSSX3i52TMWTQuEQhzxgkTnK4wox1Zc4c+NVWzHS/tmJk//DFEgH/mrP2AADO7xDfGT2oqQeAmx2pHZZSCMf2+WqbptGwDMCkk2EYMWCzToZhRAqzdTSxQsRYlhF37ks6ZWmNsU3iPv/zgo5qd/uXKmmROyImd2pMmd/T3iBxuehvoHUkVarMQ6eisGeEFeTd9U8tlwFyLZdOXiYD4y8/VwbaX7HEreS4MtN4qEEtVJJDxTKDoWH8WtJlkgxWkjJRXVMzzmvT24ZhRE6pZB2NYRgRwiCTTnFCmGm+a7+OK3tkBgrnSfEXmU0AgGK7zET1PtcVlPMH5RakR9yq9TShZqJKyuytJKnKUYvMKieFZqMlgJ+tQK3xy+1KLnVJOMbEClc+uUneH1GzS684d09Q/r2+54Ly+ja3zm1nSmbomp3hQMskXxpNs8r1zWohLrV/2ttfDDHrpBPapT0Jk1UzZjrhm95fTl7N5wxWi4j2uoltcp6I0kT0GBF933vdR0QPENFz3l/LDmcY5fAGg8NsrUqcXkAfArBTvbbc24YRFg65tSixSCciWgfgjQA+A+B/eLtrzr0NzDTftVm/PiurDaZ7xMTtPM8d88uujcG+w329QbnnOXFe6xp0szHtJ+W8qXGRGTRVIXTBl1daLukunCuY3FSmn6/oeKfX9nXlUl454fXIrNvoGkn6NrzRnYO3yGzdtWeJzHx5t8w0+XIJkOjsZjiblZNIADDB0s6xkmv/SFFFkBdlpuxEQb6n4YKTg6eL6vrVDFRWLVDVkZbvb0nGSeM+lSu8Jz0elLVDon62RK6r2cqYZVQrWythiGuM5m8BfBRAt9pXc+7tlWsSNaRkGE2BkfxZpzgyVb4JwBFm3l7P55l5GzNfwcxX9C5LV/+AYSw0GABTuK1FicNEuBrAW7wcMTkAS4joG6gj9/ZsKsqoNpFRfsqLZQNiLj/aIxknnlm9OiiPvuhM8q59YtLnj6oZilMqrchpkVG+pKJCGTkFoKKV7XfzSiJx+kyJBADFvJqBWeLad7pf3tdpUU6rdX7P3uBiw16xTKKwz+84GJSXZyQViL6HvmSqVyJouTShZo98iQQARwtLgvLeyWUAgN1jYti+eKovKB8/JdKpOOwkE01IHVTU91DkXqlD2p/pdjJqWY88C2f3iFw8p1MewbPa5Rny79FMOaUWV4tBRiU91ilyi4aZP8HM65h5AC69w0+Y+Y9gubcNIzw2GFw3t6LG3Ntzoa2bNMkdT2eGAAD5vPwarW4bCsrnLpFfsafXOevmhcP9wb6TB2VwsuOw+Ke0iwsP2kbcL1rmtKp3Wn3rxQpPQNr9Cpcy8mtc6JDyVKf8Dkz2yscmVrjzFVUu7PWr5Jf5wqVisZyXd2Hrq1QoQW9aftG170gz/GSm4awsPdB7vCB+Sy9OisXy2CmxLJ8+4O59aa8MAOcPyL3oOyH3sG3UldNTYknoBAU6MqHYJi+mut13eXqp1PHIWvmuHztrbVC+aI3cw8t7nDV4drssa6rvofbtaravkaO1p67DEGtHw8w/g5tdstzbhlELLWythMGmcQyj1WGAEz7rtCA7mplJvNygXVYl8dKDejq51+YON3C6v0+clPcOyIDkvtHeoHxkROTAiWFPUo3J7UxNykBtqsJKjr5XfbFd/Vx1ySBje7e0s79bTPWLup1uW9+hcmErLbc8MxKUfX8Ybd43e9EqXy4BIpkOTfcG+54al4TpDx3cFJRP7loWlHt2u3+krv3q+k+qwfdxKQf+TJUG3PXIY0pecJsXupAXWTf5vJRH1sl3+tjmc4Ly3nPdtbxmzfPBvovysnKAHlDXQebNlVHW0RiGETUmnQzDiBzraFobXxpoa1rPSmXTYt76LuirvJkqYOaauSO92j1elQtuFmOsoMIAlHt8oVTe0TCTcnV3pEUWdKVFLvVkxCV+pqu8c6XPKwlYKcral0nN9vXQfjI6str3jXlsdEOw74E9Kpz+CXEOX7VL2pQ/6K41c2oi2Dcjil5HzvuhHpVCOzQqzIO88I7UsLQ3MyTSqf2EzCrqhIQnh5YDAP7lYvl+pwfkvJd3SRiHfnb856zhe+877CWYBd/RGMZCIEyf2spYR2MYScAsmuQxc1ZKym2e5MgpKdOdElN+BWR2odyiTNNqn16UqVjhIfFN67RqT7bSQkyqnb400vt0lHUc6/lOKbk0VJTwgB3jzuntwb2ybnNqu8ilZTvk+vL7RQ6mRr37PKkXHSsjl+po9RmkVPjItLrfKjp/iQoxyUy46zumQiZ+kj43KHcMSJtzeSn3eTOabU3oI8jGaAzDiJQWDy8Ig3U0htHytHZkdhgWZEejJUU5KqXpKDdDpeWLlg46erdcKpAw9VVrZ9xpQaqhr3OcZQZm37Q4Nf7q6Nnu2Kd6gn0rnpF7mB8Ux8PUsMyqYdqTHJVyoTczfFlXUVKLmakR15RqR96rexmJQ9+RXG9Q/nXnQFDuXyPOkv7KAWlS11kvZtEYhhE5NusUH2E7dW1BlLMaGrV4ztw/N9oqioN6r68clfxlhtQymztG1wTlwedddPbK5+Tb6jioBn1HZHA9sGIAsWTiWHilUh0zrCmVLG/EWXL5g3L9vb+TAfA9K1YG5V09x4Kyn3wvR7KcaF0sAD+aOBcnNwyjTojDbVXPQ3QDET1LRLuJ6IyEAOT4O+/9J4no8ma03zoaw0gCTVj4iojSAL4A4A0AtgB4BxFtmXXYGwBs9rabAXyxGc2vKp2I6McAPszMTzSjwiRQTXqEoRZ50oz6okIPAGvfocPTMtj75DGRTl3PO3nRNSgSKT2s5NKUkhFaqrTCWpW6Dbptnq9NSl1H1wGVweF5GRh/cr3ci/M73eJZvSp8ZJ7ZCmA3M78AAER0F1w2kmfUMTcB+DozM4BfE1Gvv+RuIxWH+W/4KIDPEdFXvbV9DcOImSZJp7UA9qnXg96+Wo+pmaodDTM/ysyvBfB9APcT0S1E1FHtc4ZhNAkGUKJwG9BPRI+o7WZ1pnIjyrO7pzDH1EyoWSciIgDPwum1vwbwp0T0CWb+x0YbUAv1rJvqS5i45Ukry6Fa0BJwXGUw2DchvjPHDoqMWn3APZOZYYkmx5SaXdLypBXkUiV027xQCFLXkT2pZNR+idQ/fEjuxf5VbgG19VlZz7n+9oQ+8hgzX1HhvUEA69XrdQAO1HFMzVS1aIjolwD2A/gcnAn1HrgMk1uJaFuIz+eI6LdE9AQR7SCiT3v7Lfe2YYSkSdLpYQCbiehsImqDy0py76xj7gXwx97s01UATjU6PgOEs2jeB2CHNzik+SAR7Sz3gVlMAngtM48SURbAL4noBwD+AC739q3eNNvHESIlrmEsSppg/DFzgYg+AOCHANIAbmfmHUT0Pu/9LwG4D8CNAHYDGAfw3sZrDtHRMPPTc7z9xhCfZwD+wrxZb2PUmXtbO5DNlyt+EqllFgyQ+1xSkl0nfxs83RuU2w/KDEzuuJMXdLrKolVJwm9zQUV9qyjz3HEVAX5IZNTguDPSx/Jy3+qmSbeNme+D60z0vi+pMgN4f3NqExryo/GnyapBRGkiehwuG+UDzPwbzMq9DaBi7m1/YOvUiUK5QwxjQUMMUIlCba1KLCEIzFwEcCkR9QK4h4gurOGz2wBsA4DNF3Uk8Ocw2eh1dUZKsrzlkXEJMGxXY52ZMc+iKSgrppUHfWtBXQdNqcR7Yypzwwmx7g6fdvdovCRWTv11N36K+SRWz2BmHoKTSDfAy70NAPXm3jaMxUKzQhDmi8g7GiJa7lky8PxvrgOwC5Z72zDC04QQhPkkDum0GsAdXpxFCsC3mfn7RPRvqDn3dv0atFpE90Kl1kHguT4/WRJZMHxaZFRWlphBypcUpRjWlYmLoM3q+VPXl5oho+SQU9490nnI66u/ta2VMETe0TDzkwAuK7Pfcm8bRlisozEMI2qSboQnrqPR0cS2xkV0+Pe50sJXk9Py6OSm1GxMocwCVkn0nSmHvg4uc80AUio4fcK7R/q+LVYS19EYxqIk4X21dTSG0erYYHC8JPxeJ54ZM1j6y7AvJoCUpPJXGyg1OPPnTtb4KeaTRHU0hrFosY7GMIwoIdisU6wQKidVM6JBR8hnVc7qdFr2lzLKkY1aN7CvYVLlr5NVWd8L/x41nG7HxmgMw4gF62jiJOF3O0GUsxx1GuB8ThxGCmoF6VLW+YykUmoAVFsCxQXyHWqLpk38ZPS96M655UzbU2ptnnpJ+G1LWEdjGIsTk06GYUQLw3Jvx8kCHmZsWdq0XEpJZoP+vIQp7+3pD8rFDvdIZdJKOlGFweKkRHL7bdZtz4hcKnZIeUqSIGB9h7tHneq+1d2EhNyqSiSqozGMRYt1NIZhRI1ZNMaCJqV+SrV0WpM/FZR/t0IGEKZ63CPVdkweLZ0xINGDDWmRSNwm1zfZKwtbTawQn5m13j3K65DuerGOxjCMSGnxZTrDYB2NYbQ4BJNOVSGi9QC+DmAVnN28jZlvI6I+AP8EYADAHgBvY+aTc5+LLWlcTPj3Oa1mWvTsybrcUFDOrhkPymOrXIqRjsOSNC0zoXNvq++vqMqtNgOlZ5j8GTQlnYpdcn1jq2SGrWP1cFBel3OPs806xbNIXQHAh5n5AgBXAXg/EW2BS4H7IDNvBvCg99owjHIkPAtC5B0NMx9k5ke98giAnQDWwqXEvcM77A4Avx91WwwjsSS8o4l1jIaIBuAyIpyREpeIyqbENeYXHfOUUzE769okPeV5KyT3384NTjp1HZR0LJ1jMutCxQoZLEtl9sWNlks6Pivj/k1KKof26ZVSHhmQNl+s7sWaNiedctTgrJNFb4eHiLoA/DOAP2PmYQq5nAAR3QzgZgBYucbGro3Fia1HEwIiysJ1Mncy83e83YeJaLVnzVRMiatzb59/ca7ufn0xJY3T+NfdaCI5AMhCrJG+zGhQvrhnf1B+bvNyAMDQ4W753Gg+KLcpi4ZOq5MXvFAH/TXFYd1UsWIAgDtc7uypFXIdQxtlYDh3jvgUXbjkQFDuSbtB8obXowFaWhaFIY6UuATgKwB2MvNn1VuWEtcwQpL03NtxWDRXA3gXgKeI6HFv3ycB3IqaU+IaxiKkxQd6wxBHStxfonLgtaXEbXG031Kb+ha1b8imnKjeK9e9BAD4+UXnBfuyp2XgdKla+Cp7TCLA6bR38oJEi8/wsylHGGlVbiywwpKc5eQSAEwtdwPcQ+fIvtGLZID3urV7gvI5ucNBeUnKacOmLD9rHY1hGFFinsGGYcQCJTyt8ILvaBbrbNNs9H2odwZKy6gciU/N8oy43V++xEmn4XPFj+ax6YGgzCT7e5+XdrQdczM0qXHlczKtZVQ5/5sal0LzZZIKJUBW/gVKeZFGk8vVDJMnmU5eJte89Zw9QfnSrr1BuS8ts3H+bFPDYTM2RmMYRhzEIZ3Cxh8S0R4AIwCKAArMfEW1c8cR62QYRqPEE4JQS/zhtcx8aZhOBligFo3JpblphoyaEZqgZNT67HEAwKv7ngv2pS6UY7fnNgTl6S6RKj0vukex44jMZqVHpUwTIqOo4MmoGmed2Fvnl3Py2Be6VVjBCimf2ij3Zexlrh1XKbn0qqW7g/J6FY6hZ+OamewwpsHgmwBc45XvAPAzAB9rxonNojGMJBCPRTMj/hBApfhDBvAjItruhQhVZUFaNIaxoOCaYp36iegR9XqbF8YDACCiH8OtDTWbT9XQoquZ+YAXCP0AEe1i5ofm+oB1NEZdVHLk605NAAA2tB0N9mX7RPYs2yJOer9eKjLq4NpeAEDXS5LqsfOQSJm2IZl1ypz2ZnOm1X+fnv5VDnmlrBjthbyTTpO9Mus0tlLeHz1bztezUeTQDav2AAAu7hoM9q3NyhhpnsrLpWYt0lajH82xucZNmPm6ivUQhY0/POD9PUJE9wDYCmDOjsakk2EkAeZwW2NUjT8kok4i6vbLAF4P4OlqJ14wFk1UA8DNiHqOg3qvv9n+Nb5105uW5T119HJ3z0RQ9pe6BIAd/WsAALs2LQ/2HTiiIsCPSqaBtlOunJFTzZAWJXUZRZUL20/uVlgug9d9K6UNr1om4QNbug5KO73BXu0jU2nQN6qlZmMaDC4bf0hEawB8mZlvBLASwD3eMi8ZAN9k5vurnXjBdDSGsWCJyWGPmY+jTPyhJ5Vu9MovALik1nNbR2MYCaAZS9rMJ4nuaJohl+qVC6Uqw1vFGtzjw/hbVDPJq11HmHvVXBkln9cyQ+fy7k7Jyle+PLm0Z0mw79BaSWR9aELKJyadHhqfFj+cotJL6ZRcRz4rIQ39OTcQvSonIROr22TRqv7MSFDuTcugtd9+LQHjkEsaC6o0DCNaGK2XjqZGrKMxjARgFk3M1CuXapECWhZpCaTPMY30GftnfI7LS6eiOiYdJGkrb4bra02r/b4JX0lylTPlK11/pfvZ6FrDug36DPpas2mRIr48WabWIt6ofHEmulR+65IrT7P6DsrcV2Cm3PGzOOiQiXwFWVdOJs1r8kLraAzDiBJb+MowjOhhtoWvqkFEtwN4E4AjzHyht6/mvNuOcDe7mrlfTRpNsDLTVXm8pJKGFXNn7B8vySyIb94DQEGZ+BrfFG9PicmeT4spn0/JjInv2g+IzGiG2V9NUjVjJkpTTVLptmvX/lKZWbxiCMf29Iz6znx+ZsrTeGeSaiLZ/UwsIQhfA3DDrH2Wd9swaiDp6VbiyL39EIATs3bfBMu7bRjhYLig0TBbizJfYzRNz7tdzazXs0R6tmKCRe4MFd06sUcLEmNzaKo3KO8bXxqUD47LMSfH3OfGx0VaFaekDi4os189C5TxZE1W5EJ7h8yI9HaKQ9uqTnEmW9cxBABYo2KFVqp1e5vhbObfzxnSoskySuO3Q59Vt7mag2Q9dcVFU+LwWrcPCUXLDwbPyL29tvxYh2EsdFpZFoVhvjqaUOteALNzb7dzmF9S/es35VkvelD3RLErKA9O9QXlXaNuPaBdJ8TAOnpEXN8zh8T6yR2X+to8Y2PZuDwN6Skp6zgV/cCUvLtfyqqB45yEG492iTv+jr6VQfmxlW7gd8kqsXLOWyY+J+d3HwrKG9qPAQCWK/d6Pciq15Kp5n9TybqpdHwzabnB2RA0c0WBpM86zdcaCJZ32zDCEnYZzxbui+KY3v4W3ILH/UQ0COAWWN5twwiNc9hr4V4kBHHk3n5HhbfqyLtdOSJay6UJlssaLjkpMji1LNj35OjaoPzo4XVy7Iu9AIDOvXKulYfUav8nZKA2MyZ+K+lJp41oWmkklWN6xkOiy94K/ZxWK/Wr5GalnJQLebmmyaWuPL5SBqcfXd8blJ88e3VQvnT1Afe3R5KcbWgT3bcsLZIqp3xxapFRmkZDF5JILFk3kqccZ9Dyg8GGYZhFYxhG1HBr+8iEIdEdTSW5dLwoPi6/m3AzSf96bKPs270mKHc/K59b/ZKzTzuOirt/ZkQlMZsU6UQFZcsWvXKpgn1b6SHxVuufIQhVwrN0Sq4vkxEZ1X7Utblzv8yCLXlJfHiG98r1//aczQCAl84VmfXqVZL87GX5/UFZ59D2I5yzKL+0WzUZFfesVNzEnaTQprcNw4gek06GYURKbQnkWpLEdTSV5NKRgji3PTW+Pig/eOBcAMCpp/qDfSt2ya9D9z6RRtkTzuU/dVokEqZlJiaQSADAulznr021h0fJKJqW66ZJF9WdGpfo7sywcsI7KZHlHcecpDp5UpwQ771QOQUOiOS6vOuloLzGT5CmdF0lGVULSZRUjcqkplybWTSGYUROsvuZZHU0jMqDvtqKuX/vBUF58jE3CLriaflV6twvwYrpYSnThGfJFJQVU+mXJJZZAF2H+lX1U74WxcIgZXm1TSkfn3FXzo6JFXNCBYT+qHC+1LBJzJet3a7uVdkhOVeFJUfD+NdUo9zn4rZymj3A28z2U6WJhoSQqI7GMBYlDHPYMwwjWghsDntxwkxBSAEgPjIA8OP95wVlXy4BwPInnLzI75Nc0OkRkUtQvjEoeVKk1Z2jgvZVCG1Q7U97A9h5JbNSKnr9GDqD8k8zm4Ny1yY30OxnDgBmZi2oFPVdbh2beolq4DhuH5imYB2NYRiRYx2NYRiRYmM08cKgGVHYvzi6KSgPPy379QyTL5nSp0Q6zfCNKSnfkFaXTHMxo+3qmqa8JTJHVBT6AXl7WUpk1JGczEb9vMPd26VnybKguZz47fSm5X7G7QHTytKnnKzTy8jWi806GYYRMWzSyTCMiGFYRxMnBaRmLFr1/G5Z4Gn5TvkiOgeVQ54/w7QQ5VIl9DX5lvy0zB6lRmXKKHdQzPqleZmBOrzUhWw8smRDsK+/Xy2SpWajKjnyLXb8cJnpCgkEazxZoklUR2MYixXzozEMI1oYMwN6E8i8djREdAOA2wCkAXyZmW+d6/ipUgbbD0lMk160qnuvRC+nT5VxyFvocqkS/rXqyRAlI9OjKqf3fjHxe37nkuI9uUoWCdvcJVlx+jKjQTmXVrLM+9uM+KekUMmB0M/rrpMU1kfyB4PnLTafiNIAvgDgDQC2AHgHEW2Zr/YYRkvDHG5rUebTotkKYDczvwAARHQXXE7uZyp9YKKUxcgeSei25iX5pcwOqSjsqQSGFURNRT8bldnhlFg33fvcr/DoC/lg347VMvi+KSfWzZKUGnz3BoYX66CwXi/Jt3T8VMsNEUMnQkRvBfAXAC4AsJWZH6lwXE1KBJhHiwbAWgD71OtBb59hGBqG+6EIszXG0wD+AMBDlQ6oV4nMp0VTLknTGXdK594GMHniv33kaf+9PdG0S9MP4Fj01bRgfU96f78nu3art+9rdn3Np3XupeI98thvmOu4mfDMFR0jgpl3AgBR5fxpqEOJAPPb0QwCWK9erwNwYPZBOvc2ET3CzFfE0zyrL8n1Lahrq23WqZ+ItOTZ5v0PNYtySuTKah+az47mYQCbiehsAPsBvB3AO+exPYbRuoQfozk2V4dHRD8GsKrMW59i5u+GOH8oJTKbeetomLlARB8A8EO4QaXbmXnHfLXHMFqaJg0GM/N1DZ4ilBKZzbz60TDzfahN7jfTBLT6FnZ9C+jaWmrqui4lQtw6F2AYRhl6siv4Vf1vDXXs/Yf+YXu9Y0VE9O8B/D2A5QCGADzOzNcT0Rq4aewbveNuBPC3ECXymWrnthAEw0gCMaxHw8z3ALinzP4DAG5Ur2tVIvPqRxMaIrqBiJ4lot1E9PGI6ridiI4Q0dNqXx8RPUBEz3l/l851jhrqWk9EPyWinUS0g4g+FHF9OSL6LRE94dX36SjrU/WmiegxIvp+1PUR0R4ieoqIHvdnXSKur5eI7iaiXd73+Mro6gvpQ9PCTqkt39HEGKrwNQA3zNr3cQAPMvNmAA96r5tBAcCHmfkCAFcBeL93TVHVNwngtcx8CYBLAdxARFdFWJ/PhwDsVK+jru9aZr5USYco67sNwP3MfD6AS+CuM5r6GGAuhdpalZbvaKAchJh5CoDvINRUmPkhACdm7b4JwB1e+Q4Av9+kug4y86NeeQTuIV0bYX3MzH4UZNbbOKr6AICI1gF4I4Avq92R1VeBSOojoiUAXg3gKwDAzFPMPBRVfQDMoomB+QxVWMnMBwHXOQBYUeX4miGiAQCXAfhNlPV5MuZxAEcAPMDMkdYHN1j4UcxcsinK+hjAj4hou+dNHmV9GwEcBfBVTxp+mYg6I6wv8UGVSeho6nIQSgJE1AXgnwH8GTMPR1kXMxeZ+VI4v4etRHRhVHUR0ZsAHGHm7VHVUYarmflyOIn9fiJ6dYR1ZQBcDuCLzHwZgDE0XwYKzG4wOMzWoiSho6nLQahJHCai1QDg/T1S5fjQEFEWrpO5k5m/E3V9Pp6J/zO48aio6rsawFuIaA+c1H0tEX0jwvr8mREw8xG4mZOtEdY3CGDQswoB4G64jie66ysWQ22tShI6msBBiIja4ByE7o2p7nsBvNsrvxtAGBftqpCLWvsKgJ3M/NkY6ltORL1euQPAdQB2RVUfM3+Cmdcx8wDc9/UTZv6jqOojok4i6vbLAF4PF4kc1fUdArCPiPz0qK+DCyqMpL7AYS/B0qnl/WjiClUgom8BuAYuKG0QwC0AbgXwbSL6EwB7AYTzmqrO1QDeBeApb9wEAD4ZYX2rAdzhzeClAHybmb9PRP8WUX2ViOr6VgK4x/XfyAD4JjPfT0QPR1QfAHwQwJ3ej98LAN4L7942vT5/mYgEY57BhtHi9KSW8VVtsz0vyvOjyW/W7RkcJS1v0RjGYocBcMItGutoDKPV4XgWvooS62gMIwG08oxSGGyMxjBaHCK6H26p0DAcY+ZwAzoxYh2NYRiRkwQ/GsMwEo51NIZhRI51NAYAwFsf59955b8mor+b7zYZCwebdTJ8bgHwl0S0Ai6a/C3z3B5jAWGDwUYAEf0cQBeAa7x1cgyjKZh0MgAARHQRXEzUpHUyRrOxjsbwlzS4E26FuDEiun6em2QsMKyjWeQQUR7Ad+DWMN4J4K8A/MW8NspYcNgYjWEYkWMWjWEYkWMdjWEYkWMdjWEYkWMdjWEYkWMdjWEYkWMdjWEYkWMdjWEYkWMdjWEYkfP/AXdrXSV2qTwXAAAAAElFTkSuQmCC",
      "text/plain": [
       "<Figure size 288x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 576x144 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# run simulation for phi0 = 0.2\n",
    "phi0 = 0.2\n",
    "run()\n",
    "#animate(phi_t)\n",
    "plot(phi_t, 10, 100, 1000, 9990)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 288x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 576x144 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# run simulation for phi0 = 0.0\n",
    "phi0 = 0.0\n",
    "run()\n",
    "#animate(phi_t)\n",
    "plot(phi_t, 10, 100, 1000, 9990)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.7"
  },
  "vscode": {
   "interpreter": {
    "hash": "cae271c71fdd3cc871ab10677ba94a2a4bc532d008bc675b95ab71698b6079a6"
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}