Line-density and Hamiltonian desity, random bunch set-up

We assume an equilibrium bunch in a synchrotron with multi-frequency RF system and possibly energy loss or acceleration per turn; for this bunch a linear line-density \(\lambda(\tau)\) is given. We want first to determine a mathematical expression for the density function \(\rho\) in 2-dimensional longitudinal phase space having this \(\tau\)-projection \(\lambda(\tau)\) and then a randomized set of macro-particles corresponding to it.