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| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "# Mean-field anaylsis of the directed percolation reaction-diffusion process\n", | |
| "\n", | |
| "Directed percolation is the simplest universality class for non-equilibrium phase transitions. Models that belong to this class don't satisfy detailed balance or do not have any steady states and one will observe a phase transition into an absorbing state.\n", | |
| "\n", | |
| "![Directed percolation graph][1]\n", | |
| "\n", | |
| "An embodiment of directed percolation is the bond directed percolation model. It consists out of a lattice structure. We refer to the directed connection between two lattice sites as bonds. \n", | |
| "With probability $p$ such a bond is permeable (open) or with probability $1-p$ such a bond is impermeable (closed). For some critical value $p>p^*$ there will be a connected graph from the top to the bottom. The figure illustrates the lattice and graph structure of the bond directed percolation model. In order to obtain a non-equilibrium system one has to interpret one spatial dimension to represent time. In this case one absorbing state would consist of all bonds being closed.\n", | |
| "\n", | |
| "[1]: https://upload.wikimedia.org/wikipedia/commons/thumb/1/11/Bond_Directed_Percolation.svg/440px-Bond_Directed_Percolation.svg.png\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 71, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/html": [ | |
| "<script type=\"text/javascript\">window.PlotlyConfig = {MathJaxConfig: 'local'};</script><script type=\"text/javascript\">if (window.MathJax) {MathJax.Hub.Config({SVG: {font: \"STIX-Web\"}});}</script><script>requirejs.config({paths: { 'plotly': ['https://cdn.plot.ly/plotly-latest.min']},});if(!window._Plotly) {require(['plotly'],function(plotly) {window._Plotly=plotly;});}</script>" | |
| ], | |
| "text/vnd.plotly.v1+html": [ | |
| "<script type=\"text/javascript\">window.PlotlyConfig = {MathJaxConfig: 'local'};</script><script type=\"text/javascript\">if (window.MathJax) {MathJax.Hub.Config({SVG: {font: \"STIX-Web\"}});}</script><script>requirejs.config({paths: { 'plotly': ['https://cdn.plot.ly/plotly-latest.min']},});if(!window._Plotly) {require(['plotly'],function(plotly) {window._Plotly=plotly;});}</script>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "from plotly import offline as py\n", | |
| "from plotly import graph_objs as go\n", | |
| "\n", | |
| "py.init_notebook_mode(connected=True)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Another directed percolation model is the reaction-diffusion process. The reaction-diffusion process is a discrete time-continous process subject to the following dynamics:\n", | |
| "$$\n", | |
| "\\begin{align}\n", | |
| "A&\\xrightarrow{k_0}A+A & \\text{birth}\\\\\n", | |
| "A&\\xrightarrow{k_1}\\emptyset & \\text{death}\\\\\n", | |
| "A+A&\\xrightarrow{k_2}A & \\text{annhiliation}\\\\\n", | |
| "A+\\emptyset&\\xrightarrow{D}\\emptyset+A & \\text{diffusion}\n", | |
| "\\end{align}\n", | |
| "$$\n", | |
| "For a well-mixed system we can neglect diffusion. The above branches of the process lead to particle number dependent probabilities:\n", | |
| "$$\n", | |
| "\\begin{align}\n", | |
| "nk_0 && \\text{birth}\\\\\n", | |
| "nk_1 && \\text{death}\\\\\n", | |
| "n(n-1)k_2 && \\text{annhiliation}\n", | |
| "\\end{align}\n", | |
| "$$\n", | |
| "\n", | |
| "We are going to use the stochastic simulation algorithm (SSA) from Gillespie in order to sample trajectories subject to these dynamics." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 85, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "from numba import jit\n", | |
| "from math import log\n", | |
| "from random import uniform\n", | |
| "\n", | |
| "def sample(n0, t0, k0, k1, k2, t):\n", | |
| " tt = [t0]\n", | |
| " nn = [n0]\n", | |
| " \n", | |
| " while tt[-1] < t and nn[-1] > 0:\n", | |
| " ti = tt[-1]\n", | |
| " ni = nn[-1]\n", | |
| " \n", | |
| " r1 = uniform(0, 1)\n", | |
| " r2 = uniform(0, 1)\n", | |
| " \n", | |
| " alpha1 = ni * k0\n", | |
| " alpha2 = ni * k1\n", | |
| " alpha3 = ni * (ni - 1) * k2\n", | |
| " alpha0 = alpha1 + alpha2 + alpha3\n", | |
| " \n", | |
| " dt = log(1 / r1) / alpha0\n", | |
| " dn = 0\n", | |
| " \n", | |
| " if r2 < alpha1 / alpha0:\n", | |
| " dn += 1\n", | |
| " else:\n", | |
| " dn -= 1\n", | |
| " \n", | |
| " tt.append(ti + dt)\n", | |
| " nn.append(ni + dn)\n", | |
| " \n", | |
| " return tt, nn" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "We implement the Gillespie algorithm through iteration of\n", | |
| "\n", | |
| "1. sample $r_1,r_2$ from $\\mathcal{U}(0,1)$\n", | |
| "2. calculate probabilities $\\alpha_1,\\dots,\\alpha_n$\n", | |
| "3. calculate $\\alpha_0=\\sum_{i=1}^n\\alpha_i$\n", | |
| "4. calculate time step $\\Delta t=\\log(1/r_1)/\\alpha_0$\n", | |
| "5. choose population step $\\Delta n$ related to $r_2\\alpha_0\\in[\\alpha_1+\\dots+\\alpha_m,\\alpha_0]$\n", | |
| "\n", | |
| "until $n(t)=0$ or one reaches the desired time $t$." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 88, | |
| "metadata": {}, | |
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| "linkText": "Export to plot.ly", | |
| "plotlyServerURL": "https://plot.ly", | |
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{\"text\": \"n(t)\"}}}, {\"showLink\": false, \"linkText\": \"Export to plot.ly\", \"plotlyServerURL\": \"https://plot.ly\"})});</script><script type=\"text/javascript\">window.addEventListener(\"resize\", function(){window._Plotly.Plots.resize(document.getElementById(\"37dec3ab-0229-400c-9c22-b8851d3993d1\"));});</script>" | |
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| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "n0 = 100\n", | |
| "t = 10\n", | |
| "\n", | |
| "t1, n1 = sample(n0, 0, .1, 0, 0, t)\n", | |
| "t2, n2 = sample(n0, 0, 0, .1, 0, t)\n", | |
| "t3, n3 = sample(n0, 0, 0, 0, .1, t)\n", | |
| "\n", | |
| "layout = go.Layout(\n", | |
| " title='Branches',\n", | |
| " xaxis=dict(title='t'),\n", | |
| " yaxis=dict(title='n(t)'),\n", | |
| ")\n", | |
| "\n", | |
| "figure = go.Figure([\n", | |
| " go.Scatter(x=t1, y=n1, name='birth'),\n", | |
| " go.Scatter(x=t2, y=n2, name='death'),\n", | |
| " go.Scatter(x=t3, y=n3, name='anhiliation'),\n", | |
| "], layout)\n", | |
| "\n", | |
| "py.iplot(figure)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "The above figure illustrates the contribution of the different branches of the reaction-diffusion process in the mean-field approximation. The decay branch leads to a linear decrease of the number of particles. The creation branch leads to a linear increase of the number of particles. The anhiliation branch, however, leads to a nonlinear decrease of the number of particles." | |
| ] | |
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| { | |
| "cell_type": "code", | |
| "execution_count": 91, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "def simulate(n0, t0, k1, k2, k3, t, N):\n", | |
| " tt = np.linspace(t0, t)\n", | |
| " nn = np.array([np.interp(tt, *sample(n0, t0, k1, k2, k3, t)) for _ in range(N)])\n", | |
| "\n", | |
| " return tt, nn" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
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| "In `simulate` we sample $N$ trajectories using `sample` and interpolate the sampled trajectory to equally spaced time steps. The interpolation allows us later to easily mean and variance for the different trajectories." | |
| ] | |
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| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "t, n = simulate(100, 0, .5, .1, .01, 20, 10)\n", | |
| "\n", | |
| "layout = go.Layout(\n", | |
| " title='Trajectories',\n", | |
| " xaxis=dict(title='t'),\n", | |
| " yaxis=dict(title='n(t)'),\n", | |
| " showlegend=False,\n", | |
| ")\n", | |
| "\n", | |
| "figure = go.Figure([\n", | |
| " go.Scatter(x=t, y=n[i])\n", | |
| " for i in range(10)\n", | |
| "], layout)\n", | |
| "\n", | |
| "py.iplot(figure)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "The above figures shows the interpolated trajectories obtained through `simulate`." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 115, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "def mean(n0, t0, k0, k1, k2, t):\n", | |
| " t = np.linspace(t0, t)\n", | |
| " \n", | |
| " delta = k0 - k1\n", | |
| " nstar = delta / k2\n", | |
| " \n", | |
| " n = n0 * nstar / (n0 + (nstar - n0) * np.exp(- delta * (t - t0)))\n", | |
| " \n", | |
| " return t, n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Applying the mean-field approximation,\n", | |
| "$$\n", | |
| "\\mathbb{E}\\left[(N(t)-\\mathbb{E}[N(t)])^2\\right]\n", | |
| "=\n", | |
| "\\mathbb{E}\\left[N(t)^2\\right]\n", | |
| "-\n", | |
| "\\mathbb{E}\\left[N(t)\\right]^2\n", | |
| "\\approx\n", | |
| "\\mathbb{E}\\left[N(t)\\right]^2,\n", | |
| "$$\n", | |
| "we were able to derive an expression for the mean particle number\n", | |
| "$$\n", | |
| "\\overline{n}(t)\n", | |
| "=\n", | |
| "\\frac{n_0n^*}{n_0+(n^*-n_0)e^{-\\Delta(t-t_0)}},\n", | |
| "$$\n", | |
| "wherein $\\Delta=k_0-k_1$ and $n^*=\\Delta/k_2$. We implemented the mean particle number in `mean`." | |
| ] | |
| }, | |
| { | |
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| "linkText": "Export to plot.ly", | |
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| "source": [ | |
| "n0 = 100\n", | |
| "t0 = 0\n", | |
| "\n", | |
| "k0 = .2\n", | |
| "k1 = .1\n", | |
| "k2 = .01\n", | |
| "\n", | |
| "t = 200\n", | |
| "\n", | |
| "ts, ns = simulate(n0, t0, k0, k1, k2, t, 100)\n", | |
| "tt, nn = mean(n0, t0, k0, k1, k2, t)\n", | |
| "\n", | |
| "layout = go.Layout(\n", | |
| " title=f'Mean (k0={k0}, k1={k1}, k2={k2})',\n", | |
| " xaxis=dict(title='t'),\n", | |
| " yaxis=dict(title='n(t)'),\n", | |
| " showlegend=False,\n", | |
| ")\n", | |
| "\n", | |
| "figure = go.Figure([\n", | |
| " go.Scatter(x=ts, y=ns.mean(0), error_y=dict(type='data', array=ns.std(0), visible=True)),\n", | |
| " go.Scatter(x=tt, y=nn, line=dict(dash='dot')),\n", | |
| "], layout)\n", | |
| "\n", | |
| "py.iplot(figure)" | |
| ] | |
| }, | |
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| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "For small $\\Delta>0$ we observe that the theoretical mean (orange dotted) approaches a constant while the simulated mean approaches $0$ slowly." | |
| ] | |
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| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "n0 = 100\n", | |
| "t0 = 0\n", | |
| "\n", | |
| "k0 = .5\n", | |
| "k1 = .1\n", | |
| "k2 = .01\n", | |
| "\n", | |
| "t = 200\n", | |
| "\n", | |
| "ts, ns = simulate(n0, t0, k0, k1, k2, t, 100)\n", | |
| "tt, nn = mean(n0, t0, k0, k1, k2, t)\n", | |
| "\n", | |
| "layout = go.Layout(\n", | |
| " title=f'Mean (k0={k0}, k1={k1}, k2={k2})',\n", | |
| " xaxis=dict(title='t'),\n", | |
| " yaxis=dict(title='n(t)'),\n", | |
| " showlegend=False,\n", | |
| ")\n", | |
| "\n", | |
| "figure = go.Figure([\n", | |
| " go.Scatter(x=ts, y=ns.mean(0), error_y=dict(type='data', array=ns.std(0), visible=True)),\n", | |
| " go.Scatter(x=tt, y=nn, line=dict(dash='dot')),\n", | |
| "], layout)\n", | |
| "\n", | |
| "py.iplot(figure)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "For $n^*=10$ we observe that $\\overline{n}(t)\\xrightarrow{t\\to\\infty}40$." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 112, | |
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| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "k0 = .1 + .1 * np.arange(1, 11)\n", | |
| "k1 = .1\n", | |
| "\n", | |
| "data = []\n", | |
| "\n", | |
| "for i in range(10):\n", | |
| " t, n = simulate(500, 0, k0[i], k1, .01, 500, 100)\n", | |
| " \n", | |
| " data.append((t, n.mean(0), n.std(0)))\n", | |
| "\n", | |
| "layout = go.Layout(\n", | |
| " xaxis=dict(title='t'),\n", | |
| " yaxis=dict(title='n(t)'),\n", | |
| ")\n", | |
| "\n", | |
| "figure = go.Figure([\n", | |
| " go.Scatter(x=t, y=mean, name=f'Δ={np.round(k0[i]-k1, 2)}')\n", | |
| " for i, (t, mean, sigma) in enumerate(data)\n", | |
| "], layout)\n", | |
| "\n", | |
| "py.iplot(figure)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "In order to show scaling behaviour we plot $\\overline{n}(t)/\\Delta^\\beta$ against $t \\Delta^{\\nu z}$ and try to find $\\beta,\\nu z$ such that the data points collapse on a straight line in a double logarithmic plot." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 113, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "application/vnd.plotly.v1+json": { | |
| "config": { | |
| "linkText": "Export to plot.ly", | |
| "plotlyServerURL": "https://plot.ly", | |
| "showLink": false | |
| }, | |
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(\\u03b2=1, \\u03bdz=1)\"}, \"xaxis\": {\"title\": {\"text\": \"t \\u0394^\\u03bdz\"}, \"type\": \"log\"}, \"yaxis\": {\"title\": {\"text\": \"\\u00f1(t)/\\u0394^\\u03b2\"}, \"type\": \"log\"}}, {\"showLink\": false, \"linkText\": \"Export to plot.ly\", \"plotlyServerURL\": \"https://plot.ly\"})});</script><script type=\"text/javascript\">window.addEventListener(\"resize\", function(){window._Plotly.Plots.resize(document.getElementById(\"2f1c72e3-b58b-49e2-ab84-f4c7e03b2064\"));});</script>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "beta = 1\n", | |
| "nuz = 1\n", | |
| "\n", | |
| "delta = k0 - k1\n", | |
| "\n", | |
| "layout = go.Layout(\n", | |
| " title=f'Scaling (β={beta}, νz={nuz})',\n", | |
| " xaxis=dict(title='t Δ^νz', type='log'),\n", | |
| " yaxis=dict(title='ñ(t)/Δ^β', type='log'),\n", | |
| ")\n", | |
| "\n", | |
| "figure = go.Figure([\n", | |
| " go.Scatter(x=t*delta[i]**nuz, y=mean/delta[i]**beta, mode='markers', name=f'Δ={np.round(k0[i]-k1, 2)}')\n", | |
| " for i, (t, mean, sigma) in enumerate(data)\n", | |
| "], layout)\n", | |
| "\n", | |
| "py.iplot(figure)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "#### Questions\n", | |
| "\n", | |
| "> Does the mean-value evolve according to the analytic solution from the sheet? What does this tell us?\n", | |
| "\n", | |
| "Yes. It tells us that we did everything right or that we made the same mistakes in the simulation and the theory.\n", | |
| "\n", | |
| "> Do the scaling relations meet your expectations?\n", | |
| "\n", | |
| "For $\\Delta>=0.2$ they do. For $\\Delta=0.1$ we found a different scaling (blue points).\n", | |
| "\n", | |
| "> What do you observe for small $\\Delta>0$ ($n^*\\approx10$)? How can you explain that?\n", | |
| "\n", | |
| "We observed that the theoretical mean and the simulated mean start to diverge for very large times. We believe that stochastic effects are to dominant in this configuration then that they could be neglected in a mean approximation.\n", | |
| "\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [] | |
| } | |
| ], | |
| "metadata": { | |
| "kernelspec": { | |
| "display_name": "notebooks", | |
| "language": "python", | |
| "name": "notebooks" | |
| }, | |
| "language_info": { | |
| "codemirror_mode": { | |
| "name": "ipython", | |
| "version": 3 | |
| }, | |
| "file_extension": ".py", | |
| "mimetype": "text/x-python", | |
| "name": "python", | |
| "nbconvert_exporter": "python", | |
| "pygments_lexer": "ipython3", | |
| "version": "3.7.1" | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 2 | |
| } |
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