{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" }, "tags": [] }, "source": [ "# Random Walk" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "Consider a particle that starts at the origin, $x=0$. At every time step $\\Delta t$, the particle will randomly choose a direction and move for a fixed distance $\\Delta x$. After many time steps, the particle will arrive at a position $x$, which depends on the particular trajectory that it took. If we run the simulation again, there will be a different realization of the *random trajectory*, and the particle may end up at a different position accordingly. Thus, the position $x(t)$ of the particle at a given time $t$ is a *random variable*." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" }, "tags": [] }, "source": [ "## Simulating random walk" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "source": [ "Let us simulate a random walk in 1D. At every time step, the particle can move left or right with equal probability. We are interested in finding the distribution of the random variable $x(t)$ for a given time $t$, and how this distribution changes with time. To do so, we have to simulate many particles, or equivalently repeat the simulation many times. It is convenient to define the simulation as a Python *class*." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "outputs": [], "source": [ "class RandomWalk1D:\n", " \"\"\"\n", " simulate random walk of a particle in 1D.\n", " \"\"\"\n", " \n", " def __init__(self, dt=1., speed=1.):\n", " \"\"\"\n", " initialize the simulation by setting the initial position of the particle.\n", " inputs:\n", " dt: float, time step size.\n", " speed: float, each time step the particle moves a distance dx=dt*speed.\n", " \"\"\"\n", " self.dt = dt\n", " self.dx = dt * speed\n", " self.t = 0. # current time since the beginning of the simulation\n", " self.x = 0. # current position of the particle\n", " \n", " def run(self, T):\n", " \"\"\"\n", " run the simulation until time T (total time since the very beginning).\n", " inputs:\n", " T: float, total amount of time since the beginning of the simulation.\n", " \"\"\"\n", " n = int((T - self.t) / self.dt) # number of time steps needed to simulate\n", " for t in range(n):\n", " direction = np.random.rand() # draw a random number uniformly between 0 and 1\n", " if direction < 0.5: # move left\n", " self.x = self.x - self.dx\n", " else: # move right\n", " self.x = self.x + self.dx\n", " self.t = self.t + self.dt # keep track of time since the beginning of the simulation" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "Let us test this class by creating an *instance* of the class and run it." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "current position = -16.0\n" ] } ], "source": [ "rw1 = RandomWalk1D() # create an instance of class\n", "rw1.run(100) # call class method\n", "print(f'current position = {rw1.x}') # retrieve instance attribute" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here we can see the advantage of using class objects for the simulations --- they can be saved and resumed later. For example, to run the same instance of the simulation up to a later time, we can do:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "rw1.run(200) # resume simulation and run to a later time" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "source": [ "Now we can run many simulations by creating many instances of the class and collecting their results at a given time." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "N = 10000 # number of simulations to run\n", "T = 1000 # amount of time to run\n", "results = [] # to collect results from every simulation\n", "for n in range(N):\n", " rw1 = RandomWalk1D()\n", " rw1.run(T)\n", " results.append(rw1.x)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "To visualize those results, we can plot a histogram, which is a way to estimate the probability distribution." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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UIlYqSZpIWCT5BwyC4uNV9Qet+cuHDy+1n/tb+z5gw9DT17c2SdKYTOJqqAA3AI9X1X8d2rQb2NaWtwF3DLVf1q6KOhd4YehwlSRpDCZxU95bgJ8CPpfkodb2S8DVwG1JrgCeBt7Vtt0FXATMAi8Cl4+12lXieG58krTyjT0squrPgMyz+bw5+hdw5UiLkiQdlXdwS5K6DAtJUpdhIUnqMiwkSV2GhSSpy7CQJHUZFpKkLr8pT9KK5bfsLR5HFpKkLsNCktRlWEiSujxnsUI4EaCkUXJkIUnqMiwkSV2GhSSpy3MWkjSH4z0PuNLu03BkIUnqMiwkSV2GhSSpa9mcs0iyBbgWOAH4aFVdPeGSFp33SkhaqpZFWCQ5AfhN4F8Ae4EHkuyuqscmW9nL+QdfEqy8SQyXRVgA5wCzVfUkQJJbga3AkgsLSTpeSzFolktYrAOeGVrfC7x5uEOS7cD2tvp/knxhTLUdj7XAVyZdxIS5D9wH4D6ARdoH+fBxPf0fzrdhuYRFV1XtBHZOuo5XIslMVU1Puo5Jch+4D8B9AEt/HyyXq6H2ARuG1te3NknSGCyXsHgA2JxkU5KTgEuB3ROuSZJWjWVxGKqqDiV5L3A3g0tnb6yqRydc1mJYVofNRsR94D4A9wEs8X2Qqpp0DZKkJW65HIaSJE2QYSFJ6jIsxiDJO5M8muTvkkwfse39SWaTfCHJBUPtW1rbbJId4696dJL8SpJ9SR5qj4uGts25P1ailfw7PpokTyX5XPvdz7S205LsSfJE+3nqpOtcbEluTLI/ySNDbXN+7gxc1/5tPJzk7MlVPmBYjMcjwNuBe4cbk5zF4MquHwS2AL+V5ISh6U0uBM4C3t36riTXVNUPt8ddMP/+mGSRo7JKfsdH88/b7/7wf552APdU1Wbgnra+0tzE4N/1sPk+94XA5vbYDlw/phrnZViMQVU9XlVz3VG+Fbi1qv6mqr4IzDKY2uTvpzepqm8Ah6c3Wenm2x8r0Wr9Hc9nK7CrLe8CLplcKaNRVfcCB49onu9zbwVuroH7gDVJzhhLofMwLCZrrmlM1h2lfSV5bxte3zh0yGE1fO7DVtNnPVIBn0zyYJumB+D0qnq2LT8HnD6Z0sZuvs+95P59LIv7LJaDJP8LeMMcm365qu4Ydz2TdrT9wWBI/UEGfzQ+CPw68J7xVacJ+6dVtS/JdwN7knx+eGNVVZJVd03/Uv/chsUiqaofPYanHW0ak2U9vclC90eS3wb+qK2upmldVtNn/TZVta/93J/kEwwOyX05yRlV9Ww73LJ/okWOz3yfe8n9+/Aw1GTtBi5NcnKSTQxOZv0lK3x6kyOOvf4EgwsAYP79sRKt6N/xfJK8JslrDy8D5zP4/e8GtrVu24DVMhqf73PvBi5rV0WdC7wwdLhqIhxZjEGSnwA+AkwBdyZ5qKouqKpHk9zG4Hs5DgFXVtU323NW4vQmh/3nJD/M4DDUU8DPABxtf6w0K3gKm57TgU8kgcHfn9+tqj9O8gBwW5IrgKeBd02wxpFIcgvwVmBtkr3AVcDVzP257wIuYnCRx4vA5WMv+AhO9yFJ6vIwlCSpy7CQJHUZFpKkLsNCktRlWEiSugwLSVKXYSFJ6jIspDFI8o/bxImntLuYH03yxknXJS2UN+VJY5LkV4FTgO8A9lbVf5pwSdKCGRbSmLQ5oB4AXgJ+ZKVOZaKVycNQ0vi8HvhO4LUMRhjSsuHIQhqTJLsZfCPeJuCMqnrvhEuSFsxZZ6UxSHIZ8LdV9bvt+7f/PMnbqupTk65NWghHFpKkLs9ZSJK6DAtJUpdhIUnqMiwkSV2GhSSpy7CQJHUZFpKkrv8PWhCSKqKXrVUAAAAASUVORK5CYII=\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "nbins = 20 # number of bins to use\n", "plt.figure()\n", "plt.hist(results, bins=nbins) # make histogram, using the keyword `bins` to specify the number of bins\n", "plt.xlabel('x')\n", "plt.ylabel('count')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "The histogram looks like a bell-shaped curve, such as a Gaussian function. In fact, we can mathematically prove that the distribution is expected to be Gaussian. Let us draw a Gaussian curve on top of the histogram to see the match. We need to calculate the mean and variance of the data, as well as normalize the histogram." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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OutDrKH4xO3soZK+FrM+9jmLqoZAdHFfVZ1V1gKoOSE5O9jqOqWeuavY+u44k83F+H6+j+MX8nLMhIha22l6HqT1/Fo5dQPtyz9u5yyptIyJRQAKQWcM229WwTWNOSOvoDM5s9AVzs8+njEiv4/hFXmkjaH+5M85RWlTzCsaU48/CsQboJiKdRCQGGAdUvIblfGCC+3gMsMQdu6iUqu4B8kTkp+7RVNcBb9Z9dBPOxjT9gAhRXs2q32eK16jLRCjOgZ3zvE5i6hm/FQ53zGIS8B6wAZijqutE5EERucRtNh1IEpEtwF3AsUN2RSQNmAZcLyLp5Y7IuhV4DtgCbAXe8ddnMOFHKOPKph/w8cHepBe38jqOf7U8F+JTbJDc1JpfZ8dV1YXAwgrLHij3uAi4sop1U6pYngr0qruUxnxvUPw3dIjdx9R9472O4n8SAZ1vgK//CIe2Q3xHrxOZeiJkB8eNOR6XNf2Qg6UNeC/3dK+jBEana537tJe9zWHqFSscxrhi5TAXJ3zMu7lDKNI4r+MERqNOkHwGbHsRqh5eNOYHrHAY47qgyWqaRBYwL8TOFK9RynjI2wDZX3idxNQTVjiMcV3a9EP2HEliZf6pXkcJrI5jISIGts30OompJ6xwGAM0i8zlnMaf8WbO2SF77kaVYppCmxGw/WUoK/E6jakHrHAYA4xIXE60lDIv5zyvo3ij07VQtA/2LvY6iakHrHAYA1ye+CEbClPYVJTidRRvtBkO0YmQ9qLXSUw9YIXDmLzN9IvfxLycc71O4p3IWGesY+c8KM73Oo0JclY4jEmbSZkKb2af7XUSb6VcC6UFkG5TkJjqWeEw4U0V0mbySX5v9pU09zqNt5IHO1OQ2NFVpgZWOEx4y1wF+d/xRjh3Ux0lEc45Hfs+gMK9XqcxQcwKhwlv22dBRGz4TDFSk5SrQctgx1yvk5ggZoXDhK+yUtgxB9oM52BZvNdpgkNCD0g8Fba/4nUSE8SscJjwlbEcCvdAx3FeJwkuHa+GA584M+YaUwkrHCZ8bZ8FUfHQdoTXSYJLx6uc++2zvc1hgpYVDhOeyoph51xoe4lTPMz3GnWGpIFOYTWmElY4THjauxgOZ1o3VVU6jnNmy83b5HUSE4SscJjwtH0WRCdA64u8ThKcOowFxPY6TKWscJjwU3rYOTu6/WXOVBvmxxq2hRZnOYXDLvBkKrDCYcLPnnehOA86WDdVtTqOg7yNkPOV10lMkPG5cIhIWxEZLCJnHb35M5gxfrN9FsQ2h1ZhOoW6r9qPAYm0czrMj0T50khEHgWuAtYDpe5iBZb5KZcx/lFyCNLnQ6frICLa6zTBLa45tLrQKbR9HgYRrxOZIOFT4QAuBbqr6mE/ZjHG/3a97cwAa0dTkTJ5QY1trmh6ClPbv8tlD03ji4KTjy1Pe8TOfQlnvnZVfQfU+s8zERkmIptEZIuITK7k9VgRme2+vkpEUsq9dq+7fJOIXFRu+a9FZJ2IfCMir4hIXG1zmTC2fRY0aAPJZ3idpF54P/enHC6L5pLEj7yOYoKIr4WjAFgrIs+IyONHb9WtICKRwJPAxUAP4GoR6VGh2UQgW1W7Ao8Bj7rr9gDGAT2BYcBTIhIpIm2BO4ABqtoLiHTbGVOzI7mwe6FzqGlEmF1X/DgdLIvnw4MDGJGwgohjvdQm3PlaOOYDDwGfAJ+Vu1VnILBFVb9T1SPALGB0hTajgRnu47nA+SIi7vJZqnpYVbcBW9ztgdO91kBEooCGwG4fP4MJd+lvQNkR66aqpbdyzqJFdDaD4td5HcUECZ/GOFR1hojEACe5izapanENq7UFdpZ7ng4MqqqNqpaISC6Q5C5fWWHdtqr6qYj8HdgBFALvq+r7lb25iNwE3ATQoUOHGqKasLB9lnOhoqSBNTY131uc9xMOlcYxKvEjPj3U2+s4Jgj4tMchIucAm3G6np4CvvXicFwRaYqzN9IJaAPEi8j4ytqq6rOqOkBVByQnJwcypglGRQdg7yJnb8OODqqVIo1jUd4gLk74hGip6e9FEw587aqaCgxV1bNV9SzgIpwxiersAtqXe97OXVZpG7frKQHIrGbdC4Btqprh7vG8Dgz28TOYcLbzNdBS66Y6Tm/lnEXTqIOc0Wit11FMEPC1cESr6rHZzlT1W2o+ymoN0E1EOrndXONwxkrKmw9McB+PAZaoqrrLx7lHXXUCugGrcbqofioiDd2xkPOBDT5+BhPOts+CJidDonW1HI9l+f3JLYlnVKKdumV8P48jVUSeA45exf4aILW6Fdwxi0nAezhHPz2vqutE5EEgVVXnA9OBF0VkC5CFe4SU224OzgmHJcBtqloKrBKRucDn7vIvgGd9/7gmLBXshv0fwal/tG6q41Ss0byTN4QRCcuJFTudK9z5WjhuAW7DORQWYDnOWEe1VHUhsLDCsgfKPS4Crqxi3SnAlEqW/xH4o4+5jYEdrwIKHa7yOkm99nbOmYxr9j7nNP4MuNzrOMZDvh5VdRiY5t6MqV+2z4KmfSHh5Bqbmqp9mt+bAyUJ1l1lqi8cIjJHVceKyNc4c1P9gKpah7EJClVNn9Euei8rTlnJI3uu598f1TzFhqlaKZEszDmDK5t9AMUHIbqx15GMR2ra4/iVez/S30GM8YeRiSsAp5vFnLi3cs7kuuYLYNdbkPIzr+MYj1R7VJWq7nEf3qqq28vfgFv9H8+YEzMqcRmfH+pOenFLr6OEhNSCHuw5kmRXBgxzvh6Oe2Elyy6uyyDG1LXOsen0bPAdb+XYpWPqihLB27lnOhfDOpLtdRzjkWoLh4jc4o5vnCwiX5W7bQPssmAmqI1KWEaZCgtybSbcuvRWzllQVgw753kdxXikpjGOl4F3gIeB8tOiH1TVLL+lMuaEKZckLmPVoV7sL0nyOkxI+aqwGzTq7HRXdfm513GMB2oa48hV1TTgdzhHVR29NRIRmznQBK0ecdvoEpdu3VR+Ic7ULfsWQ9F+r8MYD/g6xrEAeNu9X4xzYad3/BXKmBM1KnEZxRrJO7k2lZlfdBwHWgY75nqdxHjAp8Khqqeqam/3vhvOtTE+9W80Y46XMipxGSsO9iW7NMHrMKEpoRck9LCjq8KUr3scP6Cqn/Pja2sYExT6N9xIu5j91k3lTyLQYRxkLIeCdK/TmADzacoREbmr3NMIoD925T0TpEYlLuNwWTTv553udZTQ1vEq+PoB2D4HTrmr5vYmZPi6x9G43C0WZ6yj4mVgjfFcBKWMSFjBkoM/Ib+soddxQluTk6Bpf+uuCkO+TnL4ZwARaeI81YN+TWXMcRoU/w0torOtmypQOo6DtffAwa3QuIvXaUyA+Hrp2AHuiYBfAV+LyJciMsC/0YypvVGJy8gvbcCSPPv1DIiOY537HbO9zWECyteuqudx5qtKUdUUnGtzPO+3VMYch2gpZnjCxyzKG0SRxnkdJzzEd4Tmg627Ksz4WjhKVXX50SequgLnCnzGBI0zGq0lMSrfuqkCreM4yPkactZ5ncQESE1zVfUXkf7ARyLyjIicIyJni8hTwNKAJDTGR6MSl5FT0ojl+f28jhJeOlwJEmHdVWGkpsHxqRWel79k648u7GSMV+KkiKFNVvJWzpkUa7TXccJLg1bQ4hzYPhtO/bNd1z0MVFs4VPXcQAUx5kSc2ySVRpGFvJVr3VSe6DgOVt8E2Wuhme3xhbqaLh07XlVnVjgB8BhVtWuQm6AwKmEZGcWJrMw/1eso4an95bDmVmeQ3ApHyKtpcDzevW9cxc0Y7xXncX6TNbydeyZlRHqdJjzFJkHroU7hUOvFDnU1dVU9IyKRQJ6qPhagTMbUTvqbxEYU29FUXus4DnZfBwdWQrJN9xLKajwcV1VLgauPZ+MiMkxENonIFhGZXMnrsSIy2319lYiklHvtXnf5JhG5qNzyRBGZKyIbRWSDiNhvaLhLe4X0I8l8XnCy10nCW7vREBEL21/xOonxM1/P4/hYRJ4QkTOPHqLrHqZbJXdP5Umca5P3AK4WkR4Vmk0EslW1K/AY8Ki7bg9gHNATGAY85W4P4J/Au6p6MtAH2ODjZzChqHAf7H2f+TlnA3Y0j6eim0DbUU53VVmx12mMH/k0VxXQ171/sNwyBc6rZp2BwBZV/Q5ARGbhTIy4vlyb0cCf3MdzgSdERNzls1T1MLBNRLYAA0VkPXAWcD2Aqh4Bjvj4GUwo2jEbtJTXs6v7VTQB0+la2DkX9rwPbUd4ncb4ia+FY+LRAnCUiHSuYZ22wM5yz9P58TU8jrVR1RIRyQWS3OUrK6zbFigEMoD/ikgf4DPgV6p6qOKbi8hNwE0AHTrYVW5D1rYXoWk/thy2n3FQaD3MGShPm2mFI4T52lVV2fUhX63LID6KwrkWyNOq2g84BPxo7ARAVZ9V1QGqOiA5OTmQGU2g5G6ErFTnr1wTHCJjoMNVkP4GFOd5ncb4SU1TjpwsIlcACSJyebnb9UBNs8jtAtqXe97OXVZpGxGJAhKAzGrWTQfSVXWVu3wuTiEx4ShtpjPVRcdxXicx5aWMh9Ii2Pm610mMn9S0x9EdGAkkAqPK3foDv6hh3TVANxHpJCIxOIPd8yu0mQ9McB+PAZaoqrrLx7lHXXUCugGrVXUvsFNEurvrnM8Px0xMuNAySHsJWl0IDVp7ncaU1/yn0KiL041oQlJN53G8CbwpIqer6qe12bA7ZjEJeA+IBJ5X1XUi8iCQqqrzgenAi+7gdxZOccFtNwenKJQAt7mHBQPcDrzkFqPvgBtqk8uEiIyP4VAa9H7I6ySmIhFnr+ObB53rkTds53UiU8d8HRy/TETW4QxOvwv0Bn6tqjOrW0lVFwILKyx7oNzjIuDKKtadAkypZPlawK7SE+7SZkJUPLS/zOskpjKdxsM3f4a0l6HHPV6nMXXM18Hxoaqah9NtlQZ0Be72VyhjqlV6GLbPgXaXOcXDBJ/GXSHpp06BNyHH1z2Oo/NUjwBeVdVcsamTjVd2L4DiHDuaykMpkxfU2GZ8Uj/+0vZpLv7Tv9hQ9P3R+2mP2GG69Z2vexxvichG4DRgsYgkA0X+i2VMNbbNhLhW0NJO+gtmC3LOoFgjubTpUq+jmDrmU+FQ1cnAYGCAqhbjnD8x2p/BjKnU4SzY/Tak/AwifN1hNl7ILk1gad4ARicuJYLSmlcw9UZN53Gc595fDpwDjHYfD8MpJMYE1o5XnXmQUsZ7ncT4YF7OubSKzuL0Rl97HcXUoZr+ZDsLWIJz7obizCJX/t7O8DGBlfYiJPSEpn29TmJ8sDhvIHmlDbk8cQkf5/f1Oo6pIzV1VR10r/73TbnbOuBr97ExgZO32Tl/o9O1dl3reuKwxrAg5wyGJXxCfESB13FMHampcDTCudLfacAtQGugDXAzNtWHCbTv/gsSCZ2u8zqJqYVXsy8kPrKIEQkrvI5i6khNZ47/GUBElgH9VfWg+/xPQM3H4xlTV8pKYNsMaH2xTTFSz3xecDJbitoxttki5mQP9TqOqQO+Ho7bkh9e9+KIu8yYwNjzPhTuhi4/9zqJqTVhTvaFDIjfQJfYnTU3N0HP18LxP2C1iPzJ3dtYBbzgr1DG/Mh3z0NcC2g70usk5ji8nn0eJRrBlU0/8DqKqQM+HQivqlNE5B3gTHfRDar6hf9iGVNOUQbsmg8n3QER0TW3N0HnQElTluQN5Iqmi53Dqe3nWK/5fAaVqn4OfO7HLMZULm2m859NZ5sIuT6bk30hQxNWwu53od0or+OYE+BrV5Ux3lCFrdMhaRAk9vQ6jTkBS/NOI6M40el2NPWaFQ4T3LJSIXedDYqHgBKieC37PNj1NhTu8zqOOQFWOExw2/o8RDZwrmNt6r1Xsy8ELbHp1us5KxwmeBXnw/aXof0YiEnwOo2pA1sPt4fmg2Hrf5xuSFMvWeEwwWv7y1CcB91u9jqJqUtdfwl5m2Dfh14nMcfJCocJTqqw+d+Q2Buan+51GlOXOo6FmGaw5d9eJzHHyQqHCU6ZqyH7C+h2i01oGGoi45xDq3fOg8I9Xqcxx8EKhwlOm5+GqEaQco3XSYw/dP2lM0i+dbrXScxxsMJhgs/hLNgx27lYU3Rjr9MYf2jSDVpdAFuehTK7OmB9Y4XDBJ9tM6C0yOmmMqGr2y1QsBN2L/Q6iaklv160WUSGAf8EIoHnVPWRCq/H4kygeBqQCVylqmnua/cCE4FS4A5Vfa/cepFAKrBLVW3WuxCRMnkBoCzuPpXsklMY8+hOwGZTDVltL4EGbZxuSZuCpF7x2x6H+5/7k8DFQA/gahHpUaHZRCBbVbsCjwGPuuv2AMYBPXGub/6Uu72jfgVs8Fd2453Bjb6kS+wuZmYO9zqK8beIKOjyC9jzLuRv8zqNqQV/dlUNBLao6neqegSYBYyu0GY0MMN9PBc4X0TEXT5LVQ+r6jZgi7s9RKQdMAJ4zo/ZjUeuTVpIVkkT3skd4nUUEwhdfwES4Rx6beoNfxaOtvywnyHdXVZpG1UtAXKBpBrW/QdwD1BW3ZuLyE0ikioiqRkZGcf5EUwgtYvey9AmK5mdNZTDGuN1HBMIDdtCu0udM8lLDnmdxvioXg2Oi8hIYL+qflZTW1V9VlUHqOqA5OTkAKQzJ+r65m9RhjAj04atwsrJd8GRbPhuRs1tTVDw5+D4LqB9ueft3GWVtUkXkSggAWeQvKp1LwEuEZHhQBzQRERmqup4/3wEEzDFeVzV7H0W5JzB3uLmXqcxfuQcBFGe8kbXk2jy8RTOf749Ws3fs2mPjPBvOOMTf+5xrAG6iUgnEYnBGeyeX6HNfGCC+3gMsERV1V0+TkRiRaQT0A1Yrar3qmo7VU1xt7fEikaI2DqdxpGFTD9wqddJTMAJ0zNG0zl2N+c1XuN1GOMDvxUOd8xiEvAezhFQc1R1nYg8KCKXuM2mA0kisgW4C5jsrrsOmAOsB94FblNVO0soVJWVwKbHWZXfk68Lu3mdxnjgndwh7DqSzI3Jb3gdxfjAr+dxqOpCYGGFZQ+Ue1wEXFnFulOAKdVseymwtC5yGo+lvwGH0nj+wH1eJzEeKSGKGZkjua/1f+kZt5V1RV28jmSqUa8Gx02I2vgYNOrMorxBXicxHpqVeRGHSuP4ue11BD0rHMZbB1bBgU+g+68oI7Lm9iZk5ZU1Yk72hYxKWE5yVJbXcUw1rHAYb22cCtEJzjTbJuz998AlREkpNzSveByNCSZWOIx38r6FHXOh2602C64BYMeR1izMHcK1SQtoEpnvdRxTBSscxjvrH4XIWDj5Tq+TmCDyxP6raBxZyPVJb3kdxVTBCofxxqEdsO1/ziR3cS28TmOCyMaiTizKHcTPm79Jo4gCr+OYSljhMN7Y8Hfn/pTfepvDBKUn9o8lMSqf8Ul2rY5gZIXDBF7RfmdSu07XQnwHr9OYIPRlYXeWHezHjcnziJMir+OYCqxwmMBb/zcoOwI9JnudxASxx/eNo3lULlc3e6/mxiagrHCYwCrcA5ufdK4n3uQkr9OYIJZa0JOV+b34ZYvXiJFir+OYcqxwmMBa9zCUFUOvB2pua8Lev/aPo1V0FlfZXkdQ8etcVSb8/HjK7O+1js5gafenmZdzPpOnbAQ2Bi6YqZc+zu/D6kM9uL3FbOZmXeB1HOOyPQ4TMJNazEaAf+0b53UUU28If9szgRbR2Vzf3M7rCBZWOExAdIrZxdhmi5idNZRdxXbehvFdakFPFuf9hJuT5zpXCjSes8JhAuLuVjM4UhbN4/uv9jqKqYf+vvdaEqIOwfr/8zqKwQqHCYD+DTcwPPETnsm4goySpl7HMfXQhqLOvJF9Nmz6BxSkex0n7FnhMH6m3Nf6efYXN+U5uyysOQF/33staCmstQt+ec0Kh/Gri5p8yoD4DUzbdw0FZQ28jmPqsfTiVnDyXZD2ImTatcm9ZIXD+E2sHOYPbaazqagDr2Zd6HUcEwp63gtxLeGzO0HV6zRhywqH8Zubk1+jfcw+/rjrZkrt6n6mLkQ3gd5/ca4auX2212nClhUO4xftovdxS4u5vJVzJisP9fY6jgklnW+Apn3hi99A8UGv04QlKxzGL/7Q5jnKEKbsmeh1FBNqIiLhJ/925j37yqau8YIVDlPnLmiyimEJn/LEvqvYW9zc6zgmFDUfBF1/Cd8+DllfeJ0m7Pi1cIjIMBHZJCJbRORHc2iLSKyIzHZfXyUiKeVeu9ddvklELnKXtReRD0VkvYisE5Ff+TO/qb1GEQU81OYpNhSm8J8Dl3kdx4Syvn+F2Oaw5mYoK/U6TVjxW+EQkUjgSeBioAdwtYj0qNBsIpCtql2Bx4BH3XV7AOOAnsAw4Cl3eyXAb1S1B/BT4LZKtmk89LtWL9AyOovJ6bdTrNFexzGhLKYp9JsKmavh2ye8ThNW/LnHMRDYoqrfqeoRYBYwukKb0cAM9/Fc4HwREXf5LFU9rKrbgC3AQFXdo6qfA6jqQWAD0NaPn8HUxv4VXNt8If89cAlfFnb3Oo0JBynXQJsR8OW9kLfZ6zRhw5/TqrcFdpZ7ng4MqqqNqpaISC6Q5C5fWWHdHxQIt1urH7CqTlOb41OcDytvYOeRlkzdN97rNCZEVTZtf8uoK3n/pKV8O/tSrtr6MGVVHPqd9sgIf8cLG/VycFxEGgGvAXeqal4VbW4SkVQRSc3IyAhswHD0+V2Qv5Xf7Py1nSFuAmpfSXP+vPsmfhK/nhuaz/c6TljwZ+HYBbQv97ydu6zSNiISBSQAmdWtKyLROEXjJVV9vao3V9VnVXWAqg5ITk4+wY9iqpU+H7b+B065m9WHenmdxoSh13POY1HuIO5pNYMecd95HSfk+bNwrAG6iUgnEYnBGeyu+OfAfGCC+3gMsERV1V0+zj3qqhPQDVjtjn9MBzao6jQ/Zje+KtwLq26ExD7Q+0Gv05iwJfwu/Q5ySpvwrw5/o2FEodeBQprfCoeqlgCTgPdwBrHnqOo6EXlQRC5xm00HkkRkC3AXMNlddx0wB1gPvAvcpqqlwBDgWuA8EVnr3ob76zOYGpSVwMdXQ0k+DH4JImO9TmTCWFZpAnfu+C2dYnfx5zbPeB0npPn1muOquhBYWGHZA+UeFwFXVrHuFGBKhWUrAKn7pOa4fHU/7F8KP30BEnt6ncYYPj3Umyf2j+WOlrP5NP9UXs853+tIIaleDo6bIJD+Fqx/BLr8AjpPqLm9MQHyz30/49P8U3m43ROc2sAO0fUHKxym9nLXw6fXQtP+MOBxr9MY8wOlRHLb9slklCTyTMcpNI+y65TXNSscpnaKMmDpSIiMg7PmOffGBJms0gR+mfYHmkYd5KmODxMjxV5HCil+HeMw9VNlJ1kBxEgxMzv/nt4NdnHV1of5cvXXwNeBDWeMj9YVdeGenXfwr47/x9T200AvAbG/leuCfYvGJxGUMq39VAbGr+c3O39tU4qYeuGt3LN5eM/1jEpcDl/c43WckGF7HMYHyl/bPcnIxBU8tHsiC3LP9DqQMT57JuMKWkVncsPGqdCgNZzyG68j1Xu2x2FqoPy+9XTGNXuff+4bx3SbKt3UO8JDu2+EDmPhi9/CJjug40TZHoephnJ/6+eYmPwm/z0wisf2XeN1IGOOSxmRMHgmlBXDZ78CiYSTbvM6Vr1lexymUkIZU9o+eaxo/Hn3Tdi5l6Zei4iGIbOg3WhInQQb/u51onrLCof5kRgp5rH2U7km6V2e2j/GioYJHZExMGSO2211t3PTMq9T1TvWVWV+6HAmL3b6A4MareNve67jqYwrsaJhQkpkDAx5BeJaOHsdhXtg0HN2TlItWOEw38tdD8supW/DNG7ffjdv5Z7tdSJj/EMi4LTHnaOsvvw95H3rnNDa0C4o6gtxZjEPbQMGDNDU1FSvYwS3tJdh1S8gKp4r1t3NZwV2KXcTHoY2+ZRp7adRWBbLbTsmV3lNmXC8gqCIfKaqAyoutzGOcFdyCFbfAp9cA836w8VrrWiYsPJ+3ulctuXvHCxryCud7+Ouli8SSanXsYKaFY5wlvExLOwDW56BU+6G85dAwzZepzIm4DYf7siozf/g9ezzuKPlbF7tcg9dY3d4HStoWeEIR0dyIPV2WHSmc0TJBUuh39+cwxWNCVOHyhpyd/qd3L79bjrH7mJhtzu4s+VLNkFiJWxwPERVNlFhBKVc0XQxv2s9g6aRB3kxcwR/+2YCBZ8eBCqf2NCYcPNW7tl8kt+HB9r8hztbvsLoxKU8vOfnoMNB7AhDsMIRJpShTVbym1Yv0j1uB6mHTuG6Xbewvqiz18GMCUqZpYn8aufdvJZ9Hn9oM51nU6bA4uXQ+y/QwuZqs8IRwqKlmJEJy5mY/Ca9Gmxla1E7bt0+mXdyB6PWS2lMjZbln8bF3/ZlXLP3mBL3GnxwFrQ4G3reB60uDNs9ECscoagog1uT5zCh+du0jM5ic1F77t75K17PPo9SIr1OZ0y9UkokL2UNZ8ol/wdb/gMb/gYfXgSNT4Jut0Dn6yEm0euYAWXncYSKkgJInw9pM2HPu6ClLDvYj+kZl/JRfn/s7G9j6kasHOHihI+5NmkBp8VvpLAsloU5Q3g790xW5PelWH98kEl9PQekqvM4bI+jPivc6xSJ3Qth9ztQkg8N28Epv+XCeR3YfLij1wmNCTmHNYY3cs7ljZxz6Rm3lWuSFjIycQVXNFtCbkk8i/J+yqK8QXya35u8skZex/UL2+OoTwr3OudeHPgE9i2F7M+d5Q1aQ5sRkPIzp/9VIqq8/Ksxpu5FSzFDGq1lZMIKhiaspEnkIUo1gq8KurEivy+3X3UDJP0EYpt5HbVWbI+jPikrhkPbIedryPnKuWV9AYe2Oa9HxELSQOgzBdoMh8Q+YTtIZ0wwKNZolh78CUsP/oToXcX0bbCJMxqv5YxGa7mlxauwdLbTsFFX599u4qmQcAo0OQUadYaI+vVfsV/3OERkGPBPIBJ4TlUfqfB6LPA/4DQgE7hKVdPc1+4FJgKlwB2q+p4v26xMUO9x5HwDuxdA/ndwcKtzX7AD1JnyoEyFbUfasLEwhc8LTubzglP4prBLpf2oxpjg0yiigG/uSILM1c4taw0UpH/fICIaGndzikp8R4hPgQ5jIL6DZ5mPqmqPw2+FQ0QigW+BC4F0YA1wtaquL9fmVqC3qt4sIuOAy1T1KhHpAbwCDATaAB8AJ7mrVbvNynhVOHzpLrqi6WKmtn+MAyUJ7Dzcih1HWrH9SCt2HmnFpqKOfFvUgSK16Z6NCSWNIw7RJTadLnHpdI3dSdfYnbSP2UvbmP00jixkzJZHSS3oCXg7sO5FV9VAYIuqfucGmAWMBsr/Jz8a+JP7eC7whIiIu3yWqh4GtonIFnd7+LDNemVhzhDeyz2d/LKGXkcxxgTIwbJ41hZ2Z21h9wqvKE0iD1FYFntsyYmMV/qr6PizcLQFdpZ7ng4MqqqNqpaISC6Q5C5fWWHdoxPl17RNAETkJuAm92m+iGw6js8QSM2BA16H8Jh9B/YdgH0HUEffgTx6wjkqPTSzfo3I1IKqPgs863UOX4lIamW7hOHEvgP7DsC+Awj+78Cf807sAtqXe97OXVZpGxGJAhJwBsmrWteXbRpjjPEjfxaONUA3EekkIjHAOGB+hTbzgQnu4zHAEnVG6+cD40QkVkQ6Ad2A1T5u0xhjjB/5ravKHbOYBLyHc+js86q6TkQeBFJVdT4wHXjRHfzOwikEuO3m4Ax6lwC3qTrHp1a2TX99hgCrN91qfmTfgX0HYN8BBPl3EBZnjhtjjKk7Nre2McaYWrHCYYwxplascASYiFwpIutEpExEBlR47V4R2SIim0TkonLLh7nLtojI5MCn9i8R+ZOI7BKRte5teLnXKv1OQlGo/5yrIiJpIvK1+7NPdZc1E5FFIrLZvW/qdc66JCLPi8h+Efmm3LJKP7M4Hnd/L74Skf7eJXdY4Qi8b4DLgWXlF7rTrIwDegLDgKdEJNKduuVJ4GKgB3C12zbUPKaqfd3bQqj6O/EypL+E0c+5Kue6P/ujf0xNBharajdgsfs8lLyA8ztdXlWf+WKcI0u74ZzU/HSAMlbJCkeAqeoGVa3sLPZj06yo6jbg6DQrx6ZuUdUjwNFpVsJBVd9JKArnn3NlRgMz3MczgEu9i1L3VHUZzpGk5VX1mUcD/1PHSiBRRFoHJGgVrHAEj8qmaGlbzfJQM8ndDX++XLdEuHx2CK/PWpEC74vIZ+5UQQAtVXWP+3gv0NKbaAFV1WcOut+NkJ1yxEsi8gHQqpKXfq+qbwY6TzCo7jvB2fV+COc/kIeAqcDPA5fOeOwMVd0lIi2ARSKysfyLqqoiElbnDQT7Z7bC4QeqesFxrFbddCr1fpoVX78TEfkP8Lb7NJymmAmnz/oDqrrLvd8vIvNwuu32iUhrVd3jdsvs9zRkYFT1mYPud8O6qoJH2E6zUqG/9jKcAwig6u8kFIX8z7kyIhIvIo2PPgaG4vz8y09HNAEIhz31qj7zfOA69+iqnwK55bq0PGF7HAEmIpcB/wKSgQUislZVLwrTaVaO+puI9MXpqkoDfgnVTz0TaqqaosfjWIHQEpjnXIaHKOBlVX1XRNYAc0RkIrAdGOthxjonIq8A5wDNRSQd+CPwCJV/5oXAcJyDQwqAGwIeuAKbcsQYY0ytWFeVMcaYWrHCYYwxplascBhjjKkVKxzGGGNqxQqHMcaYWrHCYYwxplascBhjjKkVKxzGBJiI/MSd0DHOPXN6nYj08jqXMb6yEwCN8YCI/AWIAxoA6ar6sMeRjPGZFQ5jPODOR7UGKAIGh+pUKiY0WVeVMd5IAhoBjXH2PIypN2yPwxgPiMh8nKv8dQJaq+okjyMZ4zObHdeYABOR64BiVX3Zvdb4JyJynqou8TqbMb6wPQ5jjDG1YmMcxhhjasUKhzHGmFqxwmGMMaZWrHAYY4ypFSscxhhjasUKhzHGmFqxwmGMMaZW/h+33yJxPGB9jAAAAABJRU5ErkJggg==\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "xmean = np.mean(results) # calculate the mean of data\n", "xvar = np.var(results) # calculate the variance of data\n", "x_array = np.linspace(xmean-100, xmean+100, 201) # choose sample points to draw the curve\n", "g_array = 1./np.sqrt(2*np.pi*xvar) * np.exp(-0.5*(x_array-xmean)**2/xvar) # calculate Gaussian curve\n", "\n", "plt.figure()\n", "plt.hist(results, bins=nbins, density=True) # use the `density` keyword to normalize the distribution\n", "plt.plot(x_array, g_array, color='orange', label='Gaussian fit') # specify `color` and `label`\n", "plt.xlabel('x')\n", "plt.ylabel('distribution')\n", "plt.legend() # add legend for labeled curves\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "Now we would like to see how the distribution changes over time. It turns out that the distribution remains Gaussian for all time $t$, as we just saw for $t=1000$. Therefore, all we need to know is how the mean and variance change over time. To find that, we have to run the simulation for different periods of time. Here we see the advantage of using class objects for the simulations --- they can be saved and resumed later." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "outputs": [], "source": [ "rw_list = [RandomWalk1D() for n in range(N)] # create and save N instances of the class\n", "T_list = [0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000] # time points at which we check the distribution\n", "xmean_list = [0] # list to collect the mean of the distribution at each time point above; first value is 0 at T=0\n", "xvar_list = [0] # list to collect the variance of the distribution at each time point above; first value is 0 at T=0\n", "\n", "for T in T_list[1:]: # skip the first time point T=0\n", " results = [] # reset the list to collect new data for each time point\n", " for rw1 in rw_list:\n", " rw1.run(T) # run each simulation until time T\n", " results.append(rw1.x)\n", " xmean = np.mean(results) # calculate the mean at each time point\n", " xvar = np.var(results) # calculate the variance at each time point\n", " xmean_list.append(xmean) # store mean value at this time point\n", " xvar_list.append(xvar) # store variance value at this time point" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "Let us plot the mean and variance as functions of time." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots(1,2, figsize=(12,4)) # make 2 subplots\n", "ax[0].plot(T_list, xmean_list) # ax[0] is the first subplot\n", "ax[0].set_ylim(-10, 10) # note the convention: ax.set_ylim() vs plt.ylim(); similarly below\n", "ax[0].set_xlabel('time')\n", "ax[0].set_ylabel('mean')\n", "ax[1].plot(T_list, xvar_list) # ax[1] is the second subplot\n", "ax[1].set_xlabel('time')\n", "ax[1].set_ylabel('variance')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "source": [ "We see that the mean of the distribution basically remains 0, whereas the variance increases linearly with time. The latter is a characteristic feature of diffusion. The slope of the variance versus time is twice the \"diffusion coefficient\". In our case, we can read off from the plot that the diffusion coefficient is $D = 0.5$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "To better visualize the temporal changes of the distribution, we may plot the Gaussian distributions at every time point." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.figure()\n", "for i in range(1,len(T_list)): # skip the first time point T=0\n", " xmean = xmean_list[i] # get the mean of distribution at given time point\n", " xvar = xvar_list[i] # get the variance of distribution at given time point\n", " x_array = np.linspace(xmean-100, xmean+100, 201) # choose sample points to draw the curve\n", " g_array = 1./np.sqrt(2*np.pi*xvar) * np.exp(-0.5*(x_array-xmean)**2/xvar) # calculate Gaussian curve\n", " plt.plot(x_array, g_array, label=f'T={T_list[i]}') # label curves by time point\n", "plt.xlabel('x')\n", "plt.ylabel('distribution')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "source": [ "We see that the distribution is always centered at zero, but the width increases over time. This means that, early on, the particle is typically found in a small region around the origin; but as time goes by, the particle will \"diffuse\" over a larger region." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "If we accept that the distribution is Gaussian, with zero mean and a variance that increases linearly with time like $\\sigma^2 = 2Dt$ (which can be proved mathematically), then we can write down the distribution function as:\n", "\\begin{equation}\n", "P(x,t) = \\frac{1}{\\sqrt{4\\pi Dt}} \\, \\mathrm{e}^{-\\frac{x^2}{4Dt}}\n", "\\end{equation}\n", "One may check that it is a solution to the **diffusion equation**:\n", "\\begin{equation}\n", "\\frac{\\partial}{\\partial t} P(x,t) = D \\frac{\\partial^2}{\\partial x^2} P(x,t)\n", "\\end{equation}\n", "This solution describes diffusion of particles that all start from the origin." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" }, "tags": [] }, "source": [ "## Adding a bias (drift)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "source": [ "So far we have assumed that the random walk is unbiased, i.e., there is equal probability of going left or right at each step. Let us now introduce a bias by specifying an unequal probability of going left or right. Let the probability of going left be $p$, then that of going right is $1-p$. We can modify the Python class as follows." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "outputs": [], "source": [ "class RandomWalk1D:\n", " \"\"\"\n", " simulate biased random walk of a particle in 1D.\n", " \"\"\"\n", " \n", " def __init__(self, dt=1., speed=1., prob=0.5): # notice new keyword `prob`\n", " \"\"\"\n", " initialize the simulation by setting the initial position of the particle.\n", " inputs:\n", " dt: float, time step size.\n", " speed: float, each time step the particle moves a distance dx=dt*speed.\n", " prob: float, probability of going left, should be between 0 and 1.\n", " \"\"\"\n", " self.dt = dt\n", " self.dx = dt * speed\n", " self.p = prob\n", " self.t = 0. # current time since the beginning of the simulation\n", " self.x = 0. # current position of the particle\n", " \n", " def run(self, T):\n", " \"\"\"\n", " run the simulation until time T (total time since the very beginning). By defining the argument `T` this way,\n", " we can pick up the simulation where we left last time and continue to run it further. \n", " inputs:\n", " T: int, total amount of time since the beginning of simulation.\n", " \"\"\"\n", " n = int((T - self.t) / self.dt) # number of time steps needed to simulate\n", " for t in range(n):\n", " direction = np.random.rand() # draw a random number uniformly from between 0 and 1\n", " if direction < self.p: # move left with probability p\n", " self.x = self.x - self.dx\n", " else: # move right with probability 1-p\n", " self.x = self.x + self.dx\n", " self.t = self.t + self.dt # keep track of time since the beginning of the simulation" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "Let us see how the mean and variance change with time for a biased random walk." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "p = 0.3 # probability of going left\n", "\n", "rw_list = [RandomWalk1D(prob=p) for n in range(N)] # note that we provided a value to the keyword `prob`\n", "T_list = [0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000] # time points at which we check the distribution\n", "xmean_list = [0] # list to collect the mean of the distribution at each time point above; first value is 0 at T=0\n", "xvar_list = [0] # list to collect the variance of the distribution at each time point above; first value is 0 at T=0\n", "\n", "for T in T_list[1:]: # skip the first time point T=0\n", " results = [] # collect results from every simulation\n", " for rw1 in rw_list:\n", " rw1.run(T) # run each simulation until time T\n", " results.append(rw1.x)\n", " xmean = np.mean(results) # calculate the mean at each time point\n", " xvar = np.var(results) # calculate the variance at each time point\n", " xmean_list.append(xmean)\n", " xvar_list.append(xvar)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "outputs": [ { "data": { "image/png": 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IUEQklhUWOn/+10/89qMl1K5ehTHX9eK8zk3CDivuRfPOdC5wlrvvMLOqwD/N7APgHmCku483s/8DhgCvB1+z3L2dmQ0GngcGRTE+ESmBfYsM376lLx2bqMhQRCSWZWTt4r6Jc/l2+RbO7dSYZ6/oqk8SS0nUkml3d2BHsFk1eDhwFnBN0P4m8BiRZLp/8BxgEvCqmVnwPiISA5as287Q8SksXqciQxGReODuTJ6TyePTFlDozm+v6saAXs31SWIpiuqYaTOrTGQoRzvgNWAZsNXd84NdMoBmwfNmwGoAd883s2wiQ0E2RTNGETk0d+etb1by9IxF1KlRRUWGIiJxYPOOXB5+Zx4fLVhP79YNeHFAd1o0qBl2WOVOVJNpdy8AEs2sHvAOcPyRvqeZ3QbcBpCQkHCkbycih6AiQxGR+PPZovU8OHke23bn8fBFxzPklDZU1gIsUVEms3m4+1Yzmwn0A+qZWZXg7nRzIDPYLRNoAWSYWRWgLpFCxH3fawwwBiApKUlDQESiSEWGIiLxZUduPk9NX8j471dzQtM6/P2W3hzfpE7YYZVr0ZzNoxGQFyTSRwHnEikqnAlcRWRGjxuAqcEh04Ltb4LXP9d4aZFwFC0y7NhYRYYiIvHg+xVbuGdCKhlZu7nj9LYMP7e9FmApA9G8M90UeDMYN10JmODu081sITDezJ4CUoA3gv3fAP5mZunAFmBwFGMTkQMoWmR440mtGHGhigxFRGJZbn4BIz9Zyh++XEaL+jWZcHs/TmzVIOywKoxozuaRBvTYT/tyoPd+2nOAAdGKR0QOTkWGEtS3/AnoQmT2pZuBJUAy0ApYAQx09yyLjPcZDVwE7AJudPc5ZR+1SMW2eN02ho1PZfG67VzduwW/urgTR1fXmnxlSf/aIsKmHbk8MCmNzxdv4MyOjfjtVd1pVFtFhhXQaOBDd7/KzKoBNYGHgc/c/TkzGwGMAB4ELgTaB48+RKY47RNO2CIVT0Gh86evlvPixz9S56iqvHFDEmef0DjssCokJdMiFdwXSzZw38Q0tuXkqciwAjOzusBpwI0A7r4H2GNm/YEzgt3eBL4gkkz3B94Kalu+NbN6ZtbU3deWcegiFc7qLbu4d8JcvluxhQs6N+Hpy7vQULMshUbJtEgF9b9Fhn1UZFixtQY2An8xs+5E1ggYCjQukiCvA/be+vr32gCBvesGKJkWiRJ3Z+IPGTz+3gIqmfHigO5c0bOZboCETMm0SAWkIkPZjypAT+CX7j7LzEYTGdLxb+7uZnZYsyxpbQCR0rFpRy4jJs/j00Xr6dumAS8M6E7z+lqAJRYomRapQPYWGT4zYxG1VWQo/y0DyHD3WcH2JCLJ9Pq9wzfMrCmwIXh979oAexVdN+DftDaAyJH7ZOF6RkxOY3tuPo9cfAI3n9yaSlqAJWYomRapIDZuz+WBSf9ZyVBFhlKUu68zs9Vm1tHdlwBnAwuDxw3Ac/zv2gB3mdl4IoWH2RovLVK6cvIKeGbGIt76ZiWdmtZh3OBEOjTWcLxYo2RapAKYuXgD90+ay/acfJ7o35nr+qrIUPbrl8DbwUwey4GbCNYJMLMhwEpgYLDvDCLT4qUTmRrvprIPV6T8St+wnbvGRobj3XJKax644HiqVakUdliyH0qmRcqxnLwCnp2xiDe/WcnxTWoz9ta+uqshB+TuqUDSfl46ez/7OnBntGMSqWjcneTvV/PYewuoVU3D8eKBkmmRcmrR2m0MHZ/Cj+t3MOSU1tx/fkcVGYqIxLDs3Xk8PGUe789byyntjuGlgd05tk6NsMOSQ1AyLVLOFBY6f/l6Bc9/sJi6Navy1s29Oa1Do7DDEhGRg5i9Mou7x6WwflsOD15wPLef1kZFhnFCybRIObJhWw73TUrjyx83cs4JjXn+yq6ayF9EJIYVFDr/949lvPTJjxxXrwYT7+hHj4T6YYclh0HJtEg58cnC9Tw4OY1de/J5+vIuXNM7QUWGIiIxbP22HIYnp/L1ss38rPtxPH15F+rUqBp2WHKYlEyLxLndewp46v2FvD1rFZ2Pq8PowT1od+zRYYclIiIH8fni9dw3MY3dewr47ZXdGJDUXDdA4pSSaZE4Nj8zm6HjU1i2cSe3n9aGe87rQPUqKjIUEYlVufkFPPfBYv7yrxWc0LQOr1ytGyDxTsm0SBwqLHT+9M/l/O6jJTSoVY23b+nDye2OCTssERE5iOUbd/DLcSksWLONG09qxYgLj9csS+WAkmmROLMuO4d7J6byr/TNXNC5Cc9e0ZX6taqFHZaIiByAuzNpdgaPTltA9SqV+NP1SZzTqXHYYUkpiVoybWYtgLeAxoADY9x9tJk9BtwKbAx2fdjdZwTHPAQMAQqAu939o2jFJxKPPpy/jhFT0sjNK+T5K7syMKmFxtiJiMSw7Tl5PPLufKamrqFvmwaMGtSDJnU1d3R5Es070/nAve4+x8xqA7PN7JPgtZHu/kLRnc2sEzAY6AwcB3xqZh3cvSCKMYrEhZ25+Tw5fSHjv19Nt+Z1GTUokTaNNMZORCSWpa7eyt3jUsjcupt7z+3AL85sR2XNHV3uRC2Zdve1wNrg+XYzWwQ0O8gh/YHx7p4L/GRm6UBv4JtoxSgSD9IytjJ0fCorNu/kF2e0Zfi5HahauVLYYYmIyAEUFjpjvlrOCx8toXGdGiTf1pekVg3CDkuipEzGTJtZK6AHMAs4GbjLzK4HfiBy9zqLSKL9bZHDMthP8m1mtwG3ASQkJEQ3cJEQFRQ6f/hyGS99/CPH1q7OuFv70rdNw7DDEhGRg9iwPYd7J8zlq6WbuLBLE567oht1a2ru6PIs6sm0mR0NTAaGufs2M3sdeJLIOOongReBm4v7fu4+BhgDkJSU5KUfsUj41mzdzfDkVGb9tIWLuzXlmcu6qjMWEYlx//hxI/dOSGV7Tj7PXN6Vq3urrqUiiGoybWZViSTSb7v7FAB3X1/k9T8C04PNTKBFkcObB20iFcr0tDU8PGUeBYXOCwO6c2XPZuqMRURi2J78Ql74eAljvlxOx8a1GXtrXzo0rh12WFJGojmbhwFvAIvc/aUi7U2D8dQAlwPzg+fTgLFm9hKRAsT2wHfRik8k1uzIzeexaQuYNDuDxBb1GD04kZYNa4UdloiIHMSKTTu5e3wKaRnZ/L++CTxycSfNHV3BRPPO9MnAdcA8M0sN2h4GrjazRCLDPFYAtwO4+wIzmwAsJDITyJ2ayUMqipRVWQxLTmX1ll3cfVY7fnl2exUZiojEuHdTMvnVO/OoXMn4v//Xkwu6NA07JAlBNGfz+Cewv8+mZxzkmKeBp6MVk0isKSh0fj8znVGfLaVJnRok396PE1XxLSIS03bm5vPrqfOZMieTE1vVZ9TgHjSrd1TYYUlItAKiSEgysnYxPDmV71dk0T/xOJ68rAt1aqjIUI5cMMzuWqCNuz9hZglAE3fX0DmRI7Ro7TbuHDuHFZt2MvTs9vzyrHZU0SeJFZqSaZEQTJu7hl+9Mw93GDUokct6HGwKdpHD9nugEDgLeALYTqQY/MQwgxKJZ+7OuO9W8/h7C6h7VFXevqUv/dpqulJRMi1Sprbn5PHo1AVMScmkV8v6jBqUSIsGNcMOS8qfPu7e08xSANw9y8yqhR2USLzakZvPw1PmMW3uGk5tfwwjByVyzNHVww5LYoSSaZEyMntlFsOSU8jM2s2wc9pz15n6aFCiJs/MKhMp9MbMGhG5Uy0ih2nBmmzuGpvCys07uf/8jvz89LZU0pLgUoSSaZEoyy8o5LWZy3j586UcV68GE+/oR6+WKjKUqHoZeAc41syeBq4CHgk3JJH44u68PWsVT0xfSP2aVRl3a1/6aBVa2Q8l0yJRtHpLpMjwh5VZXNGjGY/370xtFRlKlLn722Y2GzibyKxKl7n7opDDEokb23PyGDFlHu+nreW0Do0YObA7DTWsQw5AybRIlLybksmv342sSTR6cCL9E1VkKGXDzPoCC9z9tWC7jpn1cfdZIYcmEvPmZ2Zz59g5ZGTt5oELOnLHaRrWIQenZFqklG3LyeM3787n3dQ1nNiqPi8NVJGhlLnXgZ5Ftnfsp01EinB3/vbtSp6avogGtaox/ra+mvdfiqXYybSZnQS0KnqMu78VhZhE4tYPK7YwLDmVtdk53HNuB35xRlsVGUoYzN1974a7F5qZbp6IHMC2nDxGTE5jxrx1nNmxES8OTKRBLU2AI8VTrM7VzP4GtAVSgb1LfDugZFqESJHhy5+n8+rnS2levyYT7+hHz4T6YYclFddyM7ubyN1ogF8Ay0OMRyRmpWVs5a6xKWRu3c1DFx7Prae20bAOOSzFvVORBHQqeqdDRCJWbd7F0OQUUlZt5cqezXns0k4qMpSw3UFkRo9HiNz4+Ay4LdSIRGKMu/Pm1yt4esYiGh1dnQm399VMS1IixU2m5wNNgLVRjEUkrrg776Rk8pupCzCDV67uwc+6Hxd2WCK4+wZgcNhxiMSq7N15PDgpjQ8XrOPs44/lhQHdqa9hHVJCxU2mjwEWmtl3QO7eRne/NCpRicS47N15PPLufN6bu4berRowcnAizeodFXZYIsC/F2m5lf+tc7k5rJhEYkXq6q3cNXYO67Jz+NVFJ3DLqa0x07AOKbniJtOPRTMIkXjy3U9bGJ6cyrptOdx/fkfuOL0tlTW+TmLLVOAr4FP+U+ciUqG5O3/51wqe/WARx9auwQTVtkgpKVYy7e7/iHYgIrEur6CQlz9bymsz02nRoCaTf34SiS3qhR2WyP7UdPcHww5CJFZk78rj/klz+Xjhes45oTEvDOhGvZoa1iGlo7izefQFXgFOAKoBlYGd7l4nirGJxIyVm3cydHwqqau3MqBXcx69tDNHV9dMYxKzppvZRe4+I+xARMKWsiqLu8amsGF7Dr++pBM3n9xKwzqkVBU3G3iVSDHLRCIze1wPdDjYAWbWgsjUeY2JVJOPcffRZtYASCYylm8FMNDdsyzykz0auAjYBdzo7nMO94RESpO7M2l2Bo9NW0DlSsZr1/Tk4m5Nww5L5FCGAg+bWS6QR2RJcdcNEKlI3J03/vkTz32wmCZ1azDxDn2aKNFR7Ftr7p5uZpXdvQD4i5mlAA8d5JB84F53n2NmtYHZZvYJcCPwmbs/Z2YjgBHAg8CFQPvg0YfI/Kh9SnJSIqUhe1ceD787j/fT1tKndQNGDkrkOBUZShxw99phxyASpq279nDfxLl8umgD53duzG+v6k7dozRlqURHcZPpXWZWDUg1s98SmSLvoMu6ufvaYD/cfbuZLQKaAf2BM4Ld3gS+IJJM9wfeCuay/tbM6plZ0+B9RMrUrOWbGZ6cyobtuTxwQUduP01FhhJfzKw+kZsTNfa2ufuX4UUkUjZmr8zil2PnsHFHLo/+rBM3nqRhHRJdxU2mryOSPN8FDAdaAFcW95uYWSugBzALaFwkQV5HZBgIRBLt1UUOywja/iuZNrPbCBYfSEhIKG4IIsWSV1DI6E+X8toX6bQMigy762NBiTNmdguRoR7Niaxc2xf4BjgrxLBEoqqw0PnjV8v53UdLaFqvBpN/fhLdmtcLOyypAIo7m8dKMzsKaOrujx/ONzCzo4HJwDB331b0f4fu7mZ2WKsquvsYYAxAUlKSVmSUUlO0yHBgUnMe/VlnaqnIUOLTUOBE4Ft3P9PMjgeeCTkmkajZtCOXByal8fniDVzYpQnPXdlNwzqkzBR3No+fAS8QmcmjtZklAk8catEWM6tKJJF+292nBM3r9w7fMLOmwIagPZPIHe+9mgdtIlG1dyXDX787X0WGUl7kuHuOmWFm1d19sZl1LM6BZlYZ+AHIdPdLzKw1MB5oCMwGrnP3PWZWnUiReS9gMzDI3VdE5WxEDuLThesZMSWNbTn5PH5pZ67v11LDOqRMHXTccxGPAb2BrQDungq0PtgBwewcbwCL3P2lIi9NA24Int9AZHGBve3XW0RfIFvjpSXatuXkMXR8KvdMmEvnZnX5YNhpSqSlPMgws3rAu8AnZjYVWFnMY4cCi4psPw+MdPd2QBYwJGgfAmQF7SOD/UTKzM7cfB6aMo9b3vqBRrVr8N5dp3CDxkdLCIr7GXaeu2fv8wN6qCEWJxMZaz3PzFKDtoeB54AJZjaESOc+MHhtBpFp8dKJTI13UzFjEymRH1ZsYVhyKmuzc7jvvA78/Ix2KjKUcsHdLw+ePmZmM4G6wIeHOs7MmgMXA08D9wQ3Rc4Crgl2eZPIzZXXiRSNPxa0TwJeNTMLishFomrOqizuSU5l5ZZd3H56G+45twPVq1QOOyypoIqbTC8ws2uAymbWHrgb+PpgB7j7P4nMbbo/Z+9nfwfuLGY8IiWWX1DIK5+n88rnS2levyYTtaSslBNmVieoTWlQpHle8PVoYMsh3mIU8ACwd2q9hsBWd88PtvcWhkORonF3zzez7GD/TfvEpKJxKTV5Qf/92sx0mtSpwfhb+9KnTcOww5IKrrjJ9C+BXwG5wFjgI+DJaAUlEi2rt+xiWHIqs1dmcUWPZjzevzO1a6hIRcqNscAlRMY2O8FiLUW+tjnQgWZ2CbDB3Web2RmlFZCKxqW0LNu4g3uSU5mbkc0VPZvx2KWdqaP+W2JAcZPpTsGjSvDoD1wKdItSXCKlbmpqJo+8Mx+A0YMT6Z/Y7BBHiMSXoGDQgNPdfdVhHn4ycKmZXURkbuo6RFalrWdmVYK700ULw/cWjWeYWRUiQ0k2l8Z5iBTl7vx91iqefn8hNapW5vfX9uSirqptkdhR3GT6beA+YD5QGL1wRErfjtx8fjN1PlPmZNKrZX1GDUqkRYOaYYclEhXBlKPvA10P87iHCFa1De5M3+fu15rZROAqIjN67Fs0fgOR+auvAj7XeGkpbRu25/DApDS+WLKR0zo04ndXdaNxnRqHPlCkDBU3md7o7u9FNRKRKEhZlcXQ8alkZO1i2DntuevMdlSpXNxJbETi1hwzO9Hdvy+F93oQGG9mTwEpRGZpIvj6NzNLJzIWe3ApfC+Rf/tw/joempLGrj0FmvJOYlpxk+lHzexPwGdExk0DUGTuaJGYUlDovP5FOiM/XUqTOjWYcHs/klo1OPSBIuVDH+BaM1sJ7CQYM+3uxRqa5+5fAF8Ez5cTmRp1331ygAGlFK/Iv23PyeOJ9xYycXYGXZvVZeSgRNode3TYYYkcUHGT6ZuA44Gq/GeYhwNKpiXmrNm6m2HJqXz30xZ+1v04nrqsi1bCkorm/LADECmJ71dsYXhyKmu27uauM9tx99ntqVZFnyZKbCtuMn2iuxdr9SyRML2ftpaHpqRRUOi8OKA7V/Rspo8FpcJx95UAZnYskWJCkZi2J7+QUZ/+yP/9YxnN69fUp4kSV4qbTH9tZp3cfWFUoxEpoZ25+Tz+3gIm/JBBYot6jB6cSMuGtcIOSyQUZnYp8CJwHLABaElkVcPOYcYlsj9L129nWHIqC9ZsY1BSC379s04cXb246YlI+Ir709oXSDWzn4iMmT6s8Xci0ZSWsZWh41NZsXknd53ZjqHntKeqigylYnuSSL/9qbv3MLMzgf8Xckwi/6Ww0HnzmxU898FialWvwh+u68X5nZuEHZbIYStuMn1BVKMQKYHCQmfMV8t54aMlNKpdnXG39qWvVsISAchz981mVsnMKrn7TDMbFXZQInuty87h/klz+WrpJs46/lieu7Irx9bWiCSJT8VKpveOvxOJFeuyc7hnQipfL9vMRV2b8Ozl3ahbU0WGIoGtZnY08BXwtpltIDKrh0jopqet4VfvzGdPfiFPX96Fa3onqLZF4poGJUnc+WjBOh6cnEZuXiG/vbIbA5KaqyMW+W8ziaxIOJTI8I66wBOhRiQVXvbuPB6btoB3UjLp3qIeIwd2p00jTXkn8U/JtMSNXXvyeXL6IsZ9t4quzeoyenCiOmKR/asCfExkMZVkINndtdS3hOabZZu5d0Iq67fnagEtKXeUTEtcmJ+ZzdDxKSzftJPbT2/Dved21NyjIgfg7o8Dj5tZN2AQ8A8zy3D3c0IOTSqY3PwCXvz4R/741XJaNazFpDv60SOhfthhiZQqJdMS0woLnT/9czm/+2gJDWpV4+9D+nByu2PCDkskXmwA1gGbgWNDjkUqmMXrtjFsfCqL123n2j4J/OriE6hZTWmHlD/6qZaYtX5bDvdOmMs/0zdxfufGPHdFN+rXqhZ2WCIxz8x+AQwEGgETgVu1ToCUlYJC541/LueFj36kzlFV+PONSZx1fOOwwxKJmqgl02b2Z+ASYIO7dwnaHgNuBTYGuz3s7jOC1x4ChgAFwN3u/lG0YpPYV7TI8LkrujLoxBYqMhQpvhbAMHdPDTsQqVgysnZx74S5zPppC+d1asyzV3Sl4dHVww5LJKqieWf6r8CrwFv7tI909xeKNphZJ2AwkdW5jgM+NbMO7l4QxfgkBu1bZDhqcCJtVWQocljc/aGwY5CKxd2ZMieTx6YtwIHfXdWNq3pppiWpGKKWTLv7l2bWqpi79wfGu3su8JOZpQO9gW+iFZ/EnvmZ2dw9PoWfNu3kjtPbcs+5HVRkKCIS47bs3MOv3pnHB/PX0btVA14c2J0WDWqGHZZImQljzPRdZnY98ANwr7tnAc2Ab4vskxG0/Q8zuw24DSAhISHKoUpZ2LuS4YsfL6Fhreq8fUsfTmqrIkMRkVg3c8kGHpiUxtZdexhx4fHcemobKlfS3WipWMo6mX4deBLw4OuLwM2H8wbuPgYYA5CUlOSlHaCUrbXZu7l3wly+XraZC7s04dkrulKvpooMRURi2a49+Tz9/iLenrWKjo1r8+ZNvel0XJ2wwxIJRZkm0+6+fu9zM/sjMD3YzCRSMLNX86BNyrEP56/lwcnzyCvQSoYiIvEiZVUWw5NTWbllF7ed1oZ7zu1AjaqVww5LJDRlmkybWVN3XxtsXg7MD55PA8aa2UtEChDbA9+VZWxSdnbm5vPEewtJ/mE13ZrXZfTgHrQ+plbYYYmIyEHkFRTyymdLee2LZTSpU4Oxt/SlX9uGYYclErpoTo03DjgDOMbMMoBHgTPMLJHIMI8VwO0A7r7AzCYAC4F84E7N5FE+pWVsZej4VFZs3skvzmjL8HM7UFVLyoqIxLT0DTsYnpzKvMxsrujZjMcu7UydGlXDDkskJkRzNo+r99P8xkH2fxp4OlrxSLgKCp0/fLmMlz7+kUa1qzPu1r70baM7GiIisayw0Pnbtyt5ZsYialarzOvX9uTCrk3DDkskpmgFRIm6NVt3c8+EVL5dvoWLuzblmcu7Urem7miIiMSyddk53D9pLl8t3cQZHRvx2yu7cWydGmGHJRJzlExLVM2Yt5aHpkSKDDWJv4hIfHhv7hoeeXc+e/ILeeqyLlzbJ0F9t8gBKJmWqNiZm89j0xYwcXYG3YMiw1YqMhQRiWnZu/L4zbT5TE1dQ/cW9Rg5sDtttAqtyEEpmZZSl7p6K8PGp7Byyy7uOrMdQ89pryJDEZEY96/0Tdw3cS4btudyz7kd+MUZbamivlvkkJRMS6kpKHT+7x/LGPnJjxxbuzrjb+1LHxUZiojEtJy8Ap7/cDF/+dcK2jSqxZSfn0T3FvXCDkskbiiZllKRuXU3w5NT+e6nLVzcrSnPXKYiQxGRWDc/M5thyamkb9jBDf1aMuLCEziqmhZgETkcSqbliE1PW8PDU+ZRUOi8MKA7V/ZspkIVEZEYll9QyB++XM7IT36kQa1qvHlzb07v0CjssETikpJpKbEdufk8OnUBk+dkkNiiHqMHJ9KyoYoMRURi2crNOxmenMqcVVu5uGtTnr68C/VqVgs7LJG4pWRaSiRlVRbDklNZvWUXd5/Vjl+erSJDEZFY5u6M/341T05fSOVKxqhBifRPPE6fJIocISXTclgKCp3Xv0hn5KdLaVKnBuNv60fv1g3CDktERA5i845cHpw8j08Xreektg15YUB3jqt3VNhhiZQLSqal2DK37mb4+FS+W7GFn3U/jqcu60Ldo1RkKCISy2Yu3sD9k9LYtjuPRy4+gZtPbk2lSrobLVJalExLsbw3dw0PvzMPd3hpYHcu76EiQxGRWLZ7TwHPfrCIt75ZScfGtfn7Lb05vkmdsMMSKXeUTMtBFS0y7JFQj9GDepDQsGbYYYmIyEEUnfLu5pNb88AFHalRVVPeiUSDkmk5oJRVWQwdn0pGlooMRUTiQUGh88evlvPix0uoX7MafxvSm1Pba8o7kWhSMi3/o6DQ+f3MdEZ9FikyTL69Hye2UpGhiEgsy9y6m3snpPLt8i1c0LkJz17Rlfq1NOWdSLRFLZk2sz8DlwAb3L1L0NYASAZaASuAge6eZZHBt6OBi4BdwI3uPidascmBZWTt4p7kuXy3YguXdj+OJ1VkKCIS86bNXcOv3plHYaHz26u6MaBXc9W1iJSRaH5m/1fggn3aRgCfuXt74LNgG+BCoH3wuA14PYpxyQFMm7uGC0d/xcK123hpYHdGD05UIi0iEsO25eQxPDmVu8el0O7Yo5kx9FQGJrVQIi1ShqJ2Z9rdvzSzVvs09wfOCJ6/CXwBPBi0v+XuDnxrZvXMrKm7r41WfPIfO3Lz+c3U+UyZk6kiQxGROPHdT1sYnpzKum05DDunPXed2Y4qqmsRKXNlPWa6cZEEeR3QOHjeDFhdZL+MoO1/kmkzu43I3WsSEhKiF2kFMWdVFsP2Fhme3Z67z1JnLFIRmVkL4C0i/bIDY9x9tIbnxZ68gkJGffojr3+xjOb1azLh9n70alk/7LBEKqzQChDd3c3MS3DcGGAMQFJS0mEfLxEFhc5rM9MZrSJDEYnIB+519zlmVhuYbWafADcSGZ73nJmNIDI870H+e3heHyLD8/qEEnkFsmzjDoYnp5KWkc2AXs159NLOHF1dcwmIhKmsfwPX7x2+YWZNgQ1BeybQosh+zYM2iYKMrF0MT07l+xVZKjIUEQCCTw3XBs+3m9kiIp8QanheDHB3xn63iqemL6J61Uq8fm1PLuzaNOywRISyT6anATcAzwVfpxZpv8vMxhO5s5GtDjk69lZ8u8PIQd25vEfzsEMSkRgT1Lv0AGZxhMPzNDTvyG3ekcuDk9P4dNEGTml3DC8M6E6TujXCDktEAtGcGm8ckbsZx5hZBvAokSR6gpkNAVYCA4PdZxAZd5dOZOzdTdGKq6LanpPHo9MWMGVOJj0T6jFKRYYish9mdjQwGRjm7tuKzgpRkuF5Gpp3ZGYu3sD9k9LYlpPHry/pxE0ntaJSJc3UIRJLojmbx9UHeOns/ezrwJ3RiqWiU5GhiBSHmVUlkki/7e5TgmYNzwvB7j0FPPvBIt76ZiUdG9fm77f05vgmdcIOS0T2Q1UL5di+RYYTbu9HkooMRWQ/gtk53gAWuftLRV7S8LwyNj8zm2HJqaRv2MGQU1pz//kdqVG1cthhicgBKJkup1Zv2cU9EyJFhv0TI0WGdWqoyFBEDuhk4DpgnpmlBm0Po+F5Zaag0PnjV8t58eMlNKhVjb8N6c2p7RuFHZaIHIKS6XJoamomj7wzH0dFhiJSPO7+T+BAg3E1PC/KMrfu5t4JqXy7fAsXdmnCM5d3pX6tamGHJSLFoGS6HNmek8ejUxcwJUVFhiIi8WLvLEuFhc5vr+rGgF7NtRy4SBxRMl1OzF6ZxbDkFDKzdqvIUEQkDmwLboC8k5JJj4R6jBqUSMuGtcIOS0QOk5LpOJdfUMhrM5fx8ucqMhQRiRezlm/mnglzWbcth2HntOeuM3UDRCReKZmOY6u3RFYy/GGligxFROLBnvxCXvrkR/7w5TJaNqjJpDv60SOhfthhicgRUDIdp4oWGY4alMhlPZqFHZKIiBzE0vXbGZacyoI127i6dwseubgTtarrz7BIvNNvcZzZnpPHb4Ixdj0T6jF6cA9aNFCRoYhIrHJ33vx6Bc9+sJha1avwx+uTOLdT40MfKCJxQcl0HClaZKgxdiIisW/9thzun5TGlz9u5MyOjXj+qm4cW7tG2GGJSClSMh0HihYZNq1bg4l39KNXSxUZiojEsg/nr2XElHnk5BXw5GVd+H99EjTlnUg5pGQ6xhUtMrws8TieUJGhiEhM256Tx+PvLWTS7Ay6NqvLqMGJtG10dNhhiUiUKJmOYSoyFBGJLz+s2MLwCalkZu3mrjPbMfSc9lTVcDyRck3JdAwqWmTYq2V9Rg1KVJGhiEgMyyso5OXPlvLazHSa1T9Kc/6LVCBKpmOMigxFROLLso07GJ6cSlpGNgN6Nec3P+tEbQ3HE6kwlEzHCBUZiojEF3fn77NW8fT7C6lRtTKvX9uTC7s2DTssESljoSTTZrYC2A4UAPnunmRmDYBkoBWwAhjo7llhxFfWihYZXt6jGY/376wiQxGRGLZxey4PTk7j88UbOLX9MbwwoDuN62jKO5GKKMw702e6+6Yi2yOAz9z9OTMbEWw/GE5oZWdvkSHA6MGJ9E9UkaGISCz7ZOF6RkxOY0duPo/9rBPX92tFpUqa8k6kooqlYR79gTOC528CX1COk2kVGYqIxJedufk89f5Cxn23mk5N6zB+cCLtG9cOOywRCVlYybQDH5uZA39w9zFAY3dfG7y+DtjvWqtmdhtwG0BCQkJZxFrqZq/cwrDkVBUZiojEiZRVWQxPTmXlll3ccXpb7jm3A9WqqN8WkfCS6VPcPdPMjgU+MbPFRV90dw8S7f8RJN5jAJKSkva7T6zKLyjk1ZnpvPJ5OsfVU5GhiEisK9pvN6lTg3G39qVvm4ZhhyUiMSSUZNrdM4OvG8zsHaA3sN7Mmrr7WjNrCmwII7Zo2bfI8In+nTV1kohIDFuxaSfDklNJXb1VxeEickBlnkybWS2gkrtvD56fBzwBTANuAJ4Lvk4t69iiRUWGIiLxw91J/n41T0xfSJVKxitX9+Bn3Y8LOywRiVFh3JluDLxjZnu//1h3/9DMvgcmmNkQYCUwMITYSlXRIsOklvUZqSJDEZGYtnlHLg9Onseni9ZzUtuGvDiwO03rHhV2WCISw8o8mXb35UD3/bRvBs4u63iiZW+R4ZqtOQw/pwN3ntlWRYYiIjHs88XreWDSPLbtzuORi0/g5pNba8o7ETmkWJoar1zYt8hwwu396NWyfthhiYjIAezak89T7y9i7KxVHN+kNn8b0psTmtYJOywRiRNKpkuRigxFROJL0SnvbjutDfee14HqVSqHHZaIxBEl06VERYYiIvEjr6CQVz9P59WZkSnvxt7Sl35tNeWdiBw+JdNHSEWGIiLxZfnGHQxPTmVuRjZX9GjGY5ryTkSOgJLpIzB7ZRbDklO0kqGISBxwd/4+axVPv7+QGlUr89o1Pbm4W9OwwxKROKdkugTyCwp5beYyXv58KU3raiVDEZFYt2FbDg9MTuOLJRs5tf0xvDCgO43r1Ag7LBEpB5RMH6Z9iwy1IpaISGz7cP5aHpoyj117Cnj80s5c368lwVoHIiJHTMn0YVCRoYhI/Niek8fj7y1k0uwMujary8hBibQ79uiwwxKRckbJdDFsz8nj0akLmJKSSa+W9RmlIkMRkZj23U9buGdCKmu27uauM9tx99ntqVZFNS0iUvqUTB+CigxFROLHnvxCRn76I//3j2W0qF9TNS0iEnVKpg+goNB5bWY6oz9TkaGISDz4cf12ho1PZeHabQw+sQWPXNKJo6vrz5yIRJd6mf3IyIoUGX6/IovLEo/jicu6qMhQRCRGFRY6f/l6Bc9/uJja1avwx+uTOLdT47DDEpEKQsn0PqbNXcOv3pmHO4walMhlPVRkKCISq9Zm7+a+iXP5V/pmzj7+WJ67shuNalcPOywRqUCUTAeKFhn2TKjH6ME9VGQoIhLDpqZm8ut355Nf6Dx7RVcGn9hCU96JSJlTMg3MWZXFsPGpZGTtYujZ7fnlWSoyFBGJVdm78vj11PlMm7uGHgn1GDkwkVbH1Ao7LBGpoCp0Mr1vkeGE2/uR1EpFhiIisepf6Zu4b+JcNmzP5Z5zO/CLM9rq5oeIhCrmkmkzuwAYDVQG/uTuz0Xj+6jIUETkyJVVn52TV8DvPlrCG//8iTaNajHl5yfRvUW9aHwrEZHDElPJtJlVBl4DzgUygO/NbJq7LyzN71O0yHDkoO5c3qN5ab69iEiFUFZ99oI12QxPTuXH9Tu4vl9LHrrwBI6qVrk0v4WISInFVDIN9AbS3X05gJmNB/oDpdIxuzsjJs8j+YfV9Eyox6hBPUhoqCJDEZESimqfDTDuu1X8Zup86tesxl9vOpEzOh5bWm8tIlIqYi2ZbgasLrKdAfQpuoOZ3QbcBpCQkHBYb25mtG5Ui7vPbs/dKjIUETlSUe2zAVofU4vzOjXhqcu6UL9WtSMIVUQkOmItmT4kdx8DjAFISkrywz3+jtPblnpMIiKyf0faZ/dt05C+bRqWelwiIqUl1m7NZgItimw3D9pERCT2qM8WkQov1pLp74H2ZtbazKoBg4FpIcckIiL7pz5bRCq8mBrm4e75ZnYX8BGRaZb+7O4LQg5LRET2Q322iEiMJdMA7j4DmBF2HCIicmjqs0Wkoou1YR4iIiIiInFDybSIiIiISAkpmRYRERERKSEl0yIiIiIiJWTuhz2Hfswws43AyhIcegywqZTDiRXl+dygfJ+fzi1+lfT8Wrp7o9IOJlapzz6g8nx+Orf4VZ7Pr1T77LhOpkvKzH5w96Sw44iG8nxuUL7PT+cWv8r7+YWtvP/7lufz07nFr/J8fqV9bhrmISIiIiJSQkqmRURERERKqKIm02PCDiCKyvO5Qfk+P51b/Crv5xe28v7vW57PT+cWv8rz+ZXquVXIMdMiIiIiIqWhot6ZFhERERE5YkqmRURERERKqMIl02Z2gZktMbN0MxsRdjyHy8xamNlMM1toZgvMbGjQ3sDMPjGzpcHX+kG7mdnLwfmmmVnPcM/g0MysspmlmNn0YLu1mc0KziHZzKoF7dWD7fTg9VahBn4IZlbPzCaZ2WIzW2Rm/crZdRse/EzON7NxZlYjXq+dmf3ZzDaY2fwibYd9rczshmD/pWZ2QxjnEu/UZ8fF73657LOhfPfb5anPhnD77QqVTJtZZeA14EKgE3C1mXUKN6rDlg/c6+6dgL7AncE5jAA+c/f2wGfBNkTOtX3wuA14vexDPmxDgUVFtp8HRrp7OyALGBK0DwGygvaRwX6xbDTwobsfD3Qnco7l4rqZWTPgbiDJ3bsAlYHBxO+1+ytwwT5th3WtzKwB8CjQB+gNPLq3I5fiUZ8d+7/7gfLaZ0M57bfLYZ8NYfbb7l5hHkA/4KMi2w8BD4Ud1xGe01TgXGAJ0DRoawosCZ7/Abi6yP7/3i8WH0Dz4Af+LGA6YERWKaqy7zUEPgL6Bc+rBPtZ2OdwgPOqC/y0b3zl6Lo1A1YDDYJrMR04P56vHdAKmF/SawVcDfyhSPt/7adHsa6B+uzY/90vl312EGO57bfLY58dxBZKv12h7kzznx+evTKCtrgUfMzSA5gFNHb3tcFL64DGwfN4O+dRwANAYbDdENjq7vnBdtH4/31uwevZwf6xqDWwEfhL8HHon8ysFuXkurl7JvACsApYS+RazKZ8XLu9DvdaxdU1jFHl6t9QfXbc/d6X2367gvTZUEb9dkVLpssNMzsamAwMc/dtRV/zyH+n4m7OQzO7BNjg7rPDjiUKqgA9gdfdvQewk/983ATE73UDCD4G60/kj89xQC3+9+O2ciOer5WEQ312XCq3/XZF67MhuteqoiXTmUCLItvNg7a4YmZViXTKb7v7lKB5vZk1DV5vCmwI2uPpnE8GLjWzFcB4Ih8bjgbqmVmVYJ+i8f/73ILX6wKbyzLgw5ABZLj7rGB7EpFOujxcN4BzgJ/cfaO75wFTiFzP8nDt9jrcaxVv1zAWlYt/Q/XZcft7X5777YrQZ0MZ9dsVLZn+HmgfVKtWIzLYflrIMR0WMzPgDWCRu79U5KVpwN6q0xuIjMvb2359ULnaF8gu8pFHTHH3h9y9ubu3InJtPnf3a4GZwFXBbvue295zvirYPybvELj7OmC1mXUMms4GFlIOrltgFdDXzGoGP6N7zy/ur10Rh3utPgLOM7P6wV2g84I2KT712TH8u1+e+2wo9/12Reizoaz67bAHi5f1A7gI+BFYBvwq7HhKEP8pRD6mSANSg8dFRMYufQYsBT4FGgT7G5Fq+GXAPCKVu6GfRzHO8wxgevC8DfAdkA5MBKoH7TWC7fTg9TZhx32Ic0oEfgiu3btA/fJ03YDHgcXAfOBvQPV4vXbAOCLjCPOI3J0aUpJrBdwcnGM6cFPY5xWPD/XZsf+7H8Rd7vrsIOZy22+Xpz47iDG0flvLiYuIiIiIlFBFG+YhIiIiIlJqlEyLiIiIiJSQkmkRERERkRJSMi0iIiIiUkJKpkVERERESkjJtFQoZlbPzH4RPD/OzCaFHZOIiByY+m2JdZoaTyoUM2tFZC7ULmHHIiIih6Z+W2JdlUPvIlKuPAe0NbNUIpO4n+DuXczsRuAyoBbQHngBqAZcB+QCF7n7FjNrS2Si90bALuBWd19c1ichIlKBqN+WmKZhHlLRjACWuXsicP8+r3UBrgBOBJ4Gdrl7D+Ab4PpgnzHAL929F3Af8PuyCFpEpAJTvy0xTXemRf5jprtvB7abWTbwXtA+D+hmZkcDJwETzWzvMdXLPkwREQmo35bQKZkW+Y/cIs8Li2wXEvldqQRsDe6OiIhI+NRvS+g0zEMqmu1A7ZIc6O7bgJ/MbACARXQvzeBEROR/qN+WmKZkWioUd98M/MvM5gO/K8FbXAsMMbO5wAKgf2nGJyIi/039tsQ6TY0nIiIiIlJCujMtIiIiIlJCSqZFREREREpIybSIiIiISAkpmRYRERERKSEl0yIiIiIiJaRkWkRERESkhJRMi4iIiIiU0P8HXrJp1locVNIAAAAASUVORK5CYII=\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots(1,2, figsize=(12,4)) # make 2 subplots\n", "ax[0].plot(T_list, xmean_list) # ax[0] is the first subplot\n", "ax[0].set_xlabel('time')\n", "ax[0].set_ylabel('mean')\n", "ax[1].plot(T_list, xvar_list) # ax[1] is the second subplot\n", "ax[1].set_xlabel('time')\n", "ax[1].set_ylabel('variance')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "source": [ "We see that now the mean also increases linearly with time, meaning that the particles are \"drifting\" to the right. Notice that the slope of the variance has changed compared to the unbiased case above. Can you mathematically derive the expected values of the slopes of both mean and variance?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "Let us plot the temporal changes of the distribution in this case." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.figure()\n", "for i in range(1,len(T_list)): # skip the first time point T=0\n", " xmean = xmean_list[i] # get the mean of distribution at given time point\n", " xvar = xvar_list[i] # get the variance of distribution at given time point\n", " x_array = np.linspace(xmean-100, xmean+100, 201) # choose sample points to draw the curve\n", " g_array = 1./np.sqrt(2*np.pi*xvar) * np.exp(-0.5*(x_array-xmean)**2/xvar) # calculate Gaussian curve\n", " plt.plot(x_array, g_array, label=f'T={T_list[i]}') # label curves by time point\n", "plt.xlabel('x')\n", "plt.ylabel('distribution')\n", "plt.legend(ncol=2)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "source": [ "It is clear that the distribution is moving to the right while becoming wider." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "If we denote the drift velocity by $v$ so that the mean position changes as $\\mu = v t$, and the variance changes as $\\sigma^2 = 2Dt$ as before, then the Gaussian distribution can be written as:\n", "\\begin{equation}\n", "P(x,t) = \\frac{1}{\\sqrt{4\\pi Dt}} \\, \\mathrm{e}^{-\\frac{(x-vt)^2}{4Dt}}\n", "\\end{equation}\n", "This is a solution to the **drift-diffusion equation**:\n", "\\begin{equation}\n", "\\frac{\\partial}{\\partial t} P(x,t) = D \\frac{\\partial^2}{\\partial x^2} P(x,t) - v \\frac{\\partial}{\\partial x} P(x,t)\n", "\\end{equation}\n", "as one may check by inserting the solution into the equation." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 4 }