Diffusion in phase space

In order to study diffusion in any region of phase space containing nested closed curves we choose action-angle variables, {gamma}, J. the action J labels each closed phase curve and is equal to its area divided by 2{pi}. We can introduce rectangular variables Q,P by the equations Q=(2J){sup 1/2}sin{gamma}, P=(2J){sup 1/2}cos{gamma}, where the angle variable {gamma} is measured clockwise from the P-axis. The phase curves are circles in the Q,P plane with radius (2J){sup 1/2}. We assume that the motion consists of a Hamiltonian motion along a curve of fixed J (in the original coordinate system and in the system Q,P) plus a diffusion and a damping which can change the value of J. Now consider a system of particles described by a density {rho}(J,t), so that the number of particles between the curves J and J+dJ is dN={rho}(J,t)dJ. These cN particles are distributed uniformly in the phase space between the curves J and J+dJ