Brownians motion and the perfect sphere

We see this equation whenever the ensemble knows Pi exactly.  The particles are perfectly divergent and so the third derivative of the mapping from t to x goes to zero. Time is just the integrating resolution on the number line. There is a one half implied the diffusion constant, D.  It makes the spheres separable. When the system knows pi exactly then there is always room on the number line to separate the integrands and sum then up.