Mean squared displacement with a random walk

Dear,

I am working with a random walk or you can call Brownain motion in polar coordinate with Mean squared displacement (MSD). However, MSD means calculate the average of trajectory for initial and end point.

on other hand, the result must be in a smooth curve but all my results was has oscillation. What does mean wrong.

How I can change the code to have a smooth curve?

here is my code

initialX=0;
initialY=0;
X=0;
Y=0;
for i= 1:100
     theta = 2*pi*randn; %I wrote like this because I need the animals
         %choose any direction for whole circle (0.2pi). This only to way to 
         %describe the animals direction.
   
         r=abs(randn);  %I wrote the abs because I worke with the polar coordinate
         %and the result in Polar coordinate always possitive.
         X(i+1)=X(i)+r*cos(theta);  % Polar coordinate.
         Y(i+1)=Y(i)+r*sin(theta);  % Polar coordinate
    for j=1:1000
        
        d1(j)=sqrt(sum(X(i)-initialX).^2+(Y(i)-initialY).^2);
    end
    
     MSD(i)=mean(d1);%calcalute the avarage for the total displacment.
     plot(MSD);
     title('Mean squared displacement for 100 steps');
     xlabel('Number of step');
     ylabel('Avarage displacement on number of steps');
end
 figure;
    
plot(X,Y,'k');
title(' Polar coordinate 100');

Please note that I wrote the second for loop because I need calculate the realization for steps

[Jakob B. Nielsen]

With brownian motion, it is perfectly expectable that your MSD appears slightly jagged. Taking a random length step in a random direction will be a little chaotic... so I am not sure why you want / expect a smooth curve.

[Omma Alqubori]

Dear Jakob,

Thank you so much for your answer.

I expect smooth curve because I calculate the distance for each step and the distance increase by the time this means that I will have increased function.

[Sindar]

1) are you sure you want a gaussian/normal/bell-curve random angle

theta = 2*pi*randn;

and not a uniform one:

theta = 2*pi*rand;

2) your j loop appears to fill d1 with 1000 identical numbers. Why?