Answers: random motion from coin tosses and dice rolls.

a): We run a shorter simulation (\(10^4\) dice rolls)…

Random walk

Mean height

Mean squared height

and a longer one (\(3\cdot 10^6\) dice rolls).

Random walk

Mean height

Mean squared height


A sample MATLAB code for doing this is:

clc; clf; clear all; close all

maxIter = 10000000;
walkY = zeros(maxIter,1);
walkX = zeros(maxIter,1);
hMean = zeros(maxIter,1);
h2Mean = zeros(maxIter,1);
for i = 2:maxIter
    if rand >= 0.5
        walkY(i) = walkY(i-1) + 1;
    else
        walkY(i) = walkY(i-1) - 1;
    end
    walkX(i) = walkX(i-1) + 1;
    hMean(i) = mean(walkY(1:i));
    h2Mean(i) = mean(walkY(1:i).^2);
end

figure(1)
plot(walkX(1:3122344), walkY(1:3122344))
xlabel('Time $t$', 'FontSize', 15, 'interpreter', 'latex')
ylabel('$h$', 'FontSize', 15, 'interpreter', 'latex')
set(gcf, 'PaperPositionMode', 'auto');
print('walkLong.png', '-dpng', '-r300')

figure(2)
plot(walkX(1:3122344), hMean(1:3122344))
xlabel('Time $t$', 'FontSize', 15, 'interpreter', 'latex')
ylabel('Mean of $h$', 'FontSize', 15, 'interpreter', 'latex')
set(gcf, 'PaperPositionMode', 'auto');
print('meanHLong.png', '-dpng', '-r300')

figure(3)
plot(walkX(1:3122344), h2Mean(1:3122344))
xlabel('Time $t$', 'FontSize', 15, 'interpreter', 'latex')
ylabel('Mean of $h^2$', 'FontSize', 15, 'interpreter', 'latex')
set(gcf, 'PaperPositionMode', 'auto');
print('meanH2Long.png', '-dpng', '-r300')

b): The average of \(h\) will go to zero, for very large times - each step is just as likely to increase the height by one, as it is to decrease it!
c): The average of \(h^2\) up to time t will be proportional to t for very large times. Don't worry if your data showed different trends - it can sometimes take very large times to see this!
d): Since the average of the height goes to zero for long walks, it is not very useful. The average of the square of the height is more useful since there are no cancellations - taking the square root of this quantity gives a good measure of how much the particle has moved.
e): See the animation following the exercise for a large-scale example of this!