LangevinDynamics2D.ipynb

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    "## Single particle Langevin dynamics in two dimensions\n",
    "This notebook solves the two dimensional Langevin equation using a time-splitting integration method from [this paper](https://aip.scitation.org/doi/full/10.1063/1.4802990). The particle density, fluid viscosity, temperature, potential, and the simulation duration can be varied interactively. Particle size is fixed at 2 $\\mu m$ diameter and the time step is fixed at 0.5 ms. "
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   "source": [
    "# Import useful packages\n",
    "import numpy as np\n",
    "import time\n",
    "from collections import deque\n",
    "import ipywidgets as widgets\n",
    "from IPython.display import display\n",
    "from bokeh.io import show, output_notebook, push_notebook\n",
    "from bokeh.plotting import figure \n",
    "import sys\n",
    "output_notebook()"
   ]
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    "# Defines a class for the Langevin simulation\n",
    "class Langevin2D:\n",
    "    \n",
    "    # Initializing variables\n",
    "    def __init__(self, mass_density_of_particle, fluid_viscosity, \n",
    "                 temperature, strength_of_potential, potential_type):\n",
    "        self.partrad = 1e-6\n",
    "        self.dt = 0.5e-3\n",
    "        self.boxsize = 50e-6\n",
    "        self.center = self.boxsize / 2\n",
    "        self.partden = mass_density_of_particle\n",
    "        self.flvisc = fluid_viscosity\n",
    "        self.temp = temperature\n",
    "        self.pot = strength_of_potential * 1e-8\n",
    "        self.potdw = strength_of_potential * 1e8 * 1500\n",
    "        self.pottype = potential_type\n",
    "        self.reset()\n",
    "        self.calc_params()\n",
    "        \n",
    "    # Reseting positions and velocities to zero\n",
    "    def reset(self):\n",
    "        self.x = np.zeros(2)\n",
    "        self.p = np.zeros(2)\n",
    "        self.pmid = np.zeros(2)\n",
    "        self.f = np.zeros(2)\n",
    "        pos_x.clear()\n",
    "        pos_y.clear()\n",
    "        \n",
    "    # Calculating useful parameters used in solving the Langevin equation\n",
    "    def calc_params(self):\n",
    "        self.partmass = self.partden * 1.333 * np.pi * self.partrad**3\n",
    "        self.stokes = 6.0 * np.pi * self.flvisc * self.partrad\n",
    "        self.gamma = self.stokes / self.partmass\n",
    "        self.c1 = np.exp(-self.gamma * self.dt)\n",
    "        self.c2 = np.sqrt(kb * self.temp * (1 - self.c1**2))\n",
    "        \n",
    "    # Run the Langevin update\n",
    "    def run(self, r1, r2, target):\n",
    "        global pos_x, pos_y\n",
    "        pos_x.append(self.x[0] * 1e6)\n",
    "        pos_y.append(self.x[1] * 1e6)\n",
    "        self.calc_force(self.pottype)\n",
    "        self.pmid = self.p - self.dt * self.f / 2\n",
    "        self.x = self.x + self.dt * self.pmid / 2 / self.partmass\n",
    "        self.pmid = self.c1 * self.pmid + self.c2 * np.random.randn(2) * np.sqrt(self.partmass)\n",
    "        self.x = self.x + self.dt * self.pmid / 2 / self.partmass\n",
    "        self.calc_force(self.pottype)\n",
    "        self.p = self.pmid - self.dt * self.f / 2\n",
    "        self.check_boundary(boundary_type='wall')\n",
    "        self.plot_trajectories(r1, r2, target)\n",
    "        \n",
    "    def calc_force(self, force_type):\n",
    "        if force_type == 'quad':\n",
    "            self.f = self.pot * self.x\n",
    "        elif force_type == 'dw':\n",
    "            self.f[0] = self.pot * (self.potdw * self.x[0]**3 - self.x[0])\n",
    "            self.f[1] = self.pot * self.x[1]\n",
    "    \n",
    "    # Apply periodic boundary conditions \n",
    "    def check_boundary(self, boundary_type):\n",
    "        if boundary_type == 'periodic':\n",
    "            # Applying periodic boundaries\n",
    "            for i in range(2):\n",
    "                if self.x[i] <  -self.boxsize * 0.5: \n",
    "                    self.x[i] = self.x[i] + self.boxsize\n",
    "                if self.x[i] >=  self.boxsize * 0.5:\n",
    "                    self.x[i] = self.x[i] - self.boxsize\n",
    "        elif boundary_type == 'wall':\n",
    "            # Applying wall boundaries\n",
    "            for i in range(2):\n",
    "                if self.x[i] <  -self.boxsize * 0.5: \n",
    "                    self.x[i] = -(self.x[i] + self.boxsize)\n",
    "                if self.x[i] >=  self.boxsize * 0.5:\n",
    "                    self.x[i] = -(self.x[i] - self.boxsize)\n",
    "        else:\n",
    "            sys.exit('Error: Unknown Boundary Condition')\n",
    "\n",
    "    \n",
    "    # Update arrays for plotting\n",
    "    def plot_trajectories(self, r1, r2, target):\n",
    "        r1.data_source.data['x'] = pos_x\n",
    "        r1.data_source.data['y'] = pos_y\n",
    "        r2.data_source.data['x'] = [self.x[0] * 1e6]\n",
    "        r2.data_source.data['y'] = [self.x[1] * 1e6]\n",
    "        push_notebook(handle=target)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [],
   "source": [
    "kb = 1.380e-23  # Boltzmann constant\n",
    "# Queue for holding recent particle positions\n",
    "pos_x = deque('', 100)\n",
    "pos_y = deque('', 100)\n",
    "\n",
    "# Start the Langevin simulation\n",
    "def start_Langevin_sim(mass_density_of_particle=1000.0, \n",
    "                       fluid_viscosity=9e-4, temperature=300, strength_of_potential=1.0,\n",
    "                       simulation_duration=0.5, potential_type='quad'):\n",
    "    pos_x.clear()\n",
    "    pos_y.clear()\n",
    "    # Create langevin class object\n",
    "    l2d = Langevin2D(mass_density_of_particle, fluid_viscosity, temperature, \n",
    "                     strength_of_potential, potential_type)\n",
    "    # Initialize figure for plotting trajectories\n",
    "    p1 = figure(plot_width=500, plot_height=400, tools='',\n",
    "                x_range=(-l2d.boxsize*1e6/2, l2d.boxsize*1e6/2), \n",
    "                y_range=(-l2d.boxsize*1e6/2, l2d.boxsize*1e6/2))\n",
    "    p1.xaxis.axis_label = 'x (um)'\n",
    "    p1.yaxis.axis_label = 'y (um)'\n",
    "    p1.toolbar.logo = None\n",
    "    runtime = simulation_duration * 0.1 \n",
    "    r1 = p1.circle(pos_x, pos_y, size=1, color='chocolate', alpha=1)\n",
    "    r2 = p1.circle([l2d.x[0] * 1e6], [l2d.x[1] * 1e6], \n",
    "                   radius=int(l2d.partrad*1e6), alpha=0.5, color='maroon')\n",
    "    target = show(p1, notebook_handle=True)\n",
    "    print('Diffusion Coefficient (um^2/s) = ', 1e12 * kb*l2d.temp / (l2d.stokes))\n",
    "    # Langevin time-steping loop\n",
    "    for t in range(int(runtime/l2d.dt)):\n",
    "        l2d.run(r1, r2, target)\n",
    "        time.sleep(0.05)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Interactive controls\n",
    "style = {'description_width': 'initial'}    \n",
    "mass_density_of_particle = widgets.IntSlider(description='Particle mass density', \n",
    "                                              style=style, min=500, max=5000, \n",
    "                                              step=100, value=1000, continuous_update=False)\n",
    "fluid_viscosity = widgets.FloatLogSlider(description='Dynamic viscosity', \n",
    "                                         style=style, base=10, min=-6, max=-1, \n",
    "                                         step=-4, value=9e-4, continuous_update=False)\n",
    "temperature = widgets.IntSlider(description='Temperature (K)', \n",
    "                                style=style, min=50, max=600, \n",
    "                                step=50, value=300, continuous_update=False)\n",
    "strength_of_potential = widgets.IntSlider(description='Potential',\n",
    "                                            style=style, min=0, max=4, \n",
    "                                            step=1, value=2, continuous_update=False)\n",
    "potential_type = widgets.ToggleButtons(options=[('Quadratic', 'quad'), ('Double Well', 'dw')],\n",
    "                                            description='Type of Potential', disabled=False, style=style)\n",
    "simulation_duration = widgets.ToggleButtons(options=[('Short', 0.5), ('Regular', 2), ('Long', 5)],\n",
    "                                            description='Simulation duration', disabled=False, style=style)\n",
    "\n",
    "# Creating the interactive controls ui\n",
    "widget_ui = widgets.VBox([widgets.HBox([mass_density_of_particle, fluid_viscosity]), \n",
    "                          widgets.HBox([temperature, strength_of_potential]), \n",
    "                          potential_type, simulation_duration])\n",
    "widget_out = widgets.interactive_output(start_Langevin_sim, \n",
    "                                        {'mass_density_of_particle': mass_density_of_particle,\n",
    "                                         'fluid_viscosity': fluid_viscosity,\n",
    "                                         'temperature': temperature,\n",
    "                                         'strength_of_potential': strength_of_potential,\n",
    "                                         'potential_type': potential_type,\n",
    "                                         'simulation_duration': simulation_duration})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "c27d567b050e43f896085fd8f63d1161",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "VBox(children=(HBox(children=(IntSlider(value=1000, continuous_update=False, description='Particle mass densit…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "0726cb4306fa45939225d3130316f583",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Output()"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Display the controls and output\n",
    "display(widget_ui, widget_out)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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