English: One million steps in a two-dimensional random walk, at 1500 steps per second. At each step, the particle randomly moves to any one of the eight positions in its Moore neighbourhood. The grid is 400x400 square with toroidal boundary conditions (i.e. going off one edge reappears on the opposite edge). Generated with MATLAB using my own script - video has 20000 frames. The steps slowly fade out exponentially at 0.9999^t (a version showing all the steps taken is here). Encoded with Ogg Theora at 600 kbps: video has noticeable compression artifacts, but these do not appear to significantly detract from quality. MATLAB source code shown below:

tmax=20000;%the maximum time
x=1;       %the number of particles
n=400;     %the size of the square grid
pos=randi(n/2,2,x)+n/4; %initial position of particles.
vel=randi(3,2,x)-2;     %initial velocity of particles.
img=zeros(n);
r=1;
for t=1:tmax
    for q=1:50 %how many steps per frame
        img=0.9999*img;%reset the image but with a shadow or trail.
        for i=1:x
            img(pos(1,i),pos(2,i))=img(pos(1,i),pos(2,i))+10;
        end
        vel=randi(3,2,x)-2; %velocity of particles.
        pos=pos+vel;
        pos=mod(pos-1,n)+1;
    end
    imwrite(img./11,[num2str(t) '.png'],'png');
end