Page - 72 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

72 3 LoopsandBranching Getting indicesright To implement the traversing of arrays with loops and indices, is often challengingtoget right.Youneedtounderstandthestart, stopandstep length values for the loop variable, and also how the loop variable (possibly) enters expressions inside the loop. At the same time, however, it is something that programmers do often, so it is important to develop the right skills on these matters. You are encouraged to test your understanding of the search procedure inball_max_height.pyby doing Exercise 3.9. That exercise will ask you to compare what you get “by hand” to printouts from the code. It is of fundamentalimportance toget thisprocedureasanestablishedhabitofyours, sodo theexercise rightnow! 3.3.4 Example:RandomWalkinTwoDimensions We willnowturntoanexamplewhichrepresentsthecoreofso-calledrandomwalk algorithms.Theseare used in manybranchesof science andengineering, including suchdifferentfieldsasmaterialsmanufacturingandbrain research. The procedure we will consider, is to walk a series of equally sized steps, and for each of those steps, there should be the same probability of going to the north (N), east (E), south (S), or west (W). No other directions are legal. How can we implementsuchan action in acomputerprogram? To prepare our minds for the coding, it might be useful to first reflect uponhow this could be done for real. One way, is to use a deck of cards, letting the four suits correspondto thefourdirections:clubs toN,diamondstoE,hearts toS,andspades to W, for instance. We draw a card, perform the corresponding move, and repeat the process a large numberof times. The resulting path mimics, e.g., a typical path followedbya diffusingmolecule. Inacomputerprogram,wecannotdrawcards,butwecandrawrandomnumbers. So, we may use a loop to repeatedlydraw a randomnumber,and dependingon the number, we update the coordinates of our location. There are many ways to draw random numbers and “translate” them into our four directions, and the technical detailswill typicallydependontheprogramminglanguage.However,our technique hereisuniversal:wedrawarandomnumberfromtheinterval[0,1)andlet [0,0.25) correspond to N, [0.25,0.5) to E, [0.5,0.75) to S, and [0.75,1) to W. We decide to simulate 1000 steps, each of length 1 (e.g., meter), starting from Origo in our coordinate system. To enable plotting our path, we use two arrays for storing the coordinate history, one for the x-coordinates and one for the corresponding y- coordinates. Thesuggestedcoderandom_walk_2D.py then reads import random import numpy as np import matplotlib.pyplot as plt N = 1000 # number of steps