Plotting the Gaussian probability distribution for a Brownian process over time

I am trying to plot the change in a Gaussian probability distribution over time, where the variance is 2Dt (as is the case for Brownian motion). I wish to plot the Gaussian probability disitrbution over the positions x, with a seperate plot for different observation times. This is my code:

time = [0.5, 1.0, 3.0, 5.0, 10.0];

figure();

for i = time

x = -20:0.01:20;

y = exp(-((x).^2)./2*i)*(1./sqrt(2*pi*i));

plot(x,y,'DisplayName',num2str(i));

hold on

end

legend

This is the final plot:

However, this is not how I expect this process to look. The curve should actually start as a relatively sharp peak and then flatten out over time, modelling the process of a freely diffusing particle. I am not sure what I am doing wrong here, as plotting the same expression in Desmos and parametrising time gives me the result I am looking for. An example for the kind of curves I should get:

Source: https://commons.wikimedia.org/wiki/File:Diffusion_of_Brownian_particles.svg

I also tried plotting it using fplot, and defining x as a symbol. I got the same results.

Any help would be appreciated.

Answer 1

[Dyuman Joshi]

The term inside exp() was not correct, as the 2*i term should be fully in the denominator, for which it needed to be enclosed in parenthesis. After the correction, the plots are as expected.

Additionally, as x is constant i.e. not varying with the loop, bring out of the loop.

I also noted that the formula in wikipedia uses 4 instead of 2 used in the formula below.

time = [0.5, 1.0, 3.0, 5.0, 10.0]; 
figure();
hold on
x = -20:0.01:20; 
    
for i = time
    %                  v   v
    y = exp(-((x).^2)./(2*i))*(1./sqrt(2*pi*i));
    plot(x,y,'DisplayName',num2str(i)); 
end
legend

[Nithila MK]

Thank you. I had overlooked that bracket.