Problem $4$: Random Walk

Random walks are useful for simulating the diffusion of particles in a liquid or gas, the movement of bacteria in a solution, or even the escape of photons from the center of the Sun.

First we will program a random walk in one$-$dimension:

a$)$ Write a program that calculates the number of steps required for a particle to move a distance of $+-10$ units. The position $x$ of a particle will be computed by:

$x_{j+1}=x_j+s$

where $s$ is a random number between $-1$ and $1.$ The initial position of the particle will be $x=0.($Hint: Use rand $(1),$ which generates a random number between $0$ and $1,$ to generate a random number between $-1$ and

Google "concatenate array MATLAB" if you aren't sure how to add values to the end of an array.$)$

clear; $\%$Clear all variables in the workspace

b$)$ Copy your code from part $($a$)$ and past it below. Change that code so that it runs through a loop to calculate the number of steps needed to move a length $L$ that varies from $5$ to $100$ units in increments of $5)=(10$:$5$:$100.$ The initial posiiton of the particle will always start at $0.$

The code should store the number of steps needed to travel each length, $L.$ This can be done by initializing an empty array $($e$.$g$.$ steps$\_$per$\_$L$=[])$ at the top of the code and then concatenating new values the the end of the array everytime you run the code for a new length, $L.$

Finally, make a plot that shows the number of steps needed to move each value of L$.($e$.$g$.$ L vs$.$ number of steps$).$

clear; $8$Clear all variables in the workspace

$\%$Initialize an array to number of steps for each L distance

stepsperL$=[]$ ;