I don’t know much about the history of this subject. All I know (or what I think I know) is that Robert Brown was a botanist who observed the random movement of pollen grains in water. The continuous and jerky movement of such a particle is what we describe as Brownian motion and the mathematical representation of this is called the Wiener process in honour of Norbert Wiener.
The cool thing about the Wiener process is that it is everywhere continuous but nowhere differentiable. Although a computer could never totally simulate brownian motion I like to create pictures using the idea of Brownian motion.
The trajectories of 10 particles in two dimensions starting from the origin:
t = 2.0;
dt = 0.005;
n = floor(t/dt);
ntraj = 20;
T = linspace(0,2*pi,100);
circle_x = 10*sin(T)
x = zeros(ntraj,n+1);
y = zeros(ntraj,n+1);
tspan = [0:dt:t];
for i = 1:n
x(:,i+1) = x(:,i) + sqrt(dt)*randn(ntraj,1);
y(:,i+1) = y(:,i) + sqrt(dt)*randn(ntraj,1);
set(gca, ‘ColorOrder’, rand(1,3));
I will be posting an animation of 10 particles of a two dimension Wiener process.