It is well known that the probability density function of a point particle which undergows Brownian motion in (in 1D) is a binomial distribution Binomial Distribution -- from Wolfram MathWorld

which at the continious limit turnes into normal distribution as it is shown inthe material above.

Now assume a system of two point particles which are always at distance "a" from each other undergows Brownian motion (in 1D).

Question: what is the probability density function for one of the point particles (does not matter which one since system is simmetric) at discrete and continious limit?

Thanks.

Diffusion Equation for the rod