Exercise $1.(50$ points$)$ Diffusion

Homework #$3$

In this exercise, we use a $3$D random walk to simulate a diffusion process. Imagine a particle starting at

the origin $(0,0,0)$ that has equal probabilities to go in $6$ possible directions$-$ left, right, backward, forward,

down, and up$.$ For example, when the particle is at $($x$,$y$,$z$),$ with equal probability $1/6,$ its next location is

at $($x $1,$y$,$z$),($x $+1,$y$,$z$),($x$,$y $1,$z$),($x$,$y $+1,$z$),($x$,$y$,$z $1)$ or $($x$,$y$,$z $+1).$

The particle will conduct the random walk for n steps. We are interested in the distribution of the final

locations of particles after each takes n steps. Specifically, we would like to know the distribution of the

distance between the final location and the origin. In order to obtain this distribution, we simulate m such

particles, and check the proportion of the particles that lies within rn distance from the origin, where r is a

real number between $0$ and $1.$ Note all the particles will be within a sphere with radius n since particles only

move n steps and the furthest they can go is a distance n from the origin. In our simulation, we will calculate

the proportion of particles that are within rn from the origin for r $=0\mathrm{.}05,0\mathrm{.}10,0\mathrm{.}15,...,0\mathrm{.}90,0\mathrm{.}95,1\mathrm{.}00.$

Below is the main$()$ function for this program. Note how we use command line arguments.

int main$($int argc, char $*$argv$[])$

$\{$

if$($argc $!=3)$

$\{$

printf$("$Usage: $\%$s n m

$",$ argv$[0])$;

return $0$;

$\}$

int n $=$ atoi$($argv$[1])$;

int m $=$ atoi$($argv$[2])$;

srand$(12345)$;

diffusion$($n$,$ m$)$;

return $0$;

$\}$

We need to implement the function

void diffusion$($int n$,$ int m$)$

$\{$

$\}$

In this function, we need to dynamically allocate memory to represent a $3$D grid where x$,$y$,$z coordinates

range from $$n to n$.$ For all the possible x$,$y$,$z coordinates within the range, we save the number of particles

that end up at the location $($x$,$y$,$z$).$ During the simulation, if the final location of a particle is $($x$,$y$,$z$),$ the

corresponding value will be incremented by $1.$

Generate a random number using rand$()\%6.$ If this number is $0$ or $1,$ you should move left or right

on the x$-$axis, respectively. If it is $2$ or $3,$ you should move up or down on the y$-$axis, respectively. If it is $4$

or $5,$ move up or down on the z$-$axis, respectively.

In the implementation of main, we use the below function to print out results.

$1$

void print$\_$result$($int $*$grid$,$ int n$)$

$\{$

printf$("$radius density

$")$;

for$($int k $=1$; k $<=20$; k$++)$

$\{$

printf$("\%\mathrm{.}2$lf $\%$lf

$",0\mathrm{.}05*$k$,$ density$($grid$,$ n$,0\mathrm{.}05*$k$))$;

$\}$

$\}$

The function

double density$($int $*$grid$,$ int n$,$ double r$)$

$\{$

$\}$

returns the proportion of particles in the grid that are within the sphere of rn radius from the origin. Note

that to implement the above function, we need to calculate the Euclidean distance between $($x$,$y$,$z$)$ and the

origin $(0,0,0).$ The formula is x$2+$y$2+$z$2.$ Think about how we can avoid using the square root in our

code.

Below are outputs from a few sample runs. We can use these outputs to check our code.

$ $./$diffusion $101000000$

radius density

$0\mathrm{.}050\mathrm{.}019415$

$0\mathrm{.}100\mathrm{.}019415$

$0\mathrm{.}150\mathrm{.}196635$

$0\mathrm{.}200\mathrm{.}263900$

$0\mathrm{.}250\mathrm{.}465962$

$0\mathrm{.}300\mathrm{.}542348$

$0\mathrm{.}350\mathrm{.}686846$

$0\mathrm{.}400\mathrm{.}826187$

$0\mathrm{.}450\mathrm{.}908716$

$0\mathrm{.}500\mathrm{.}938628$

$0\mathrm{.}550\mathrm{.}982609$

$0\mathrm{.}600\mathrm{.}991070$

$0\mathrm{.}650\mathrm{.}997860$

$0\mathrm{.}700\mathrm{.}999031$

$0\mathrm{.}750\mathrm{.}999883$

$0\mathrm{.}800\mathrm{.}999959$

$0\mathrm{.}850\mathrm{.}999996$

$0\mathrm{.}900\mathrm{.}999996$

$0\mathrm{.}951\mathrm{.}000000$

$1\mathrm{.}001\mathrm{.}000000$

$ $./$diffusion $201000000$

radius density

$0\mathrm{.}050\mathrm{.}007120$

$0\mathrm{.}100\mathrm{.}113322$

$0\mathrm{.}150\mathrm{.}272127$

$0\mathrm{.}200\mathrm{.}517046$

$0\mathrm{.}250\mathrm{.}690763$

$0\mathrm{.}300\mathrm{.}857808$

$2$

$0\mathrm{.}350\mathrm{.}934197$

$0\mathrm{.}400\mathrm{.}979449$

$0\mathrm{.}450\mathrm{.}994164$

$0\mathrm{.}500\mathrm{.}998872$

$0\mathrm{.}550\mathrm{.}999802$

$0\mathrm{.}600\mathrm{.}999970$

$0\mathrm{.}650\mathrm{.}999999$

$0\mathrm{.}701\mathrm{.}000000$

$0\mathrm{.}751\mathrm{.}000000$

$0\mathrm{.}801\mathrm{.}000000$

$0\mathrm{.}851\mathrm{.}000000$

$0\mathrm{.}901\mathrm{.}000000$

$0\mathrm{.}951\mathrm{.}000000$

$1\mathrm{.}001\mathrm{.}000000$