A Solution of the Diffusion Equation

Abstract

This chapter is dedicated to the solution of the diffusion equation. It gives the probability density to observe a particle with a given action at a specific time in the phase space. It is fundamental to investigate the chaotic diffusion along the phase space. We impose particular boundary conditions and concentrate all the particles leaving from an initial action and resolve analytically the probability density that provides the probability a particle can be observed with action $I\in [-I_{fisc}, I_{fisc}]$ at any time n. The knowledge of the probability density furnishes all the relevant observables, including the scaling invariance of the chaotic diffusion.