Chapter 4 introduces phenomenologically the radiative transfer equation of the directional distribution of the energy density for a given anisotropic scattering coefficient of scalar waves in random media. We solve the radiative transfer equation analytically by using the Legendre expansion for isotropic radiation from a point source. By probabilistically interpreting the Born scattering coefficient and the Eikonal angular spectrum function and the traveling distance fluctuation for scalar waves, we construct the corresponding pseudo-random number generators, where the rejection sampling method is introduced. Then, we synthesize the space–time distribution of the energy density for isotropic radiation from a point source using the MC simulation and compare it with the analytical solution of the radiative transfer equation.

This chapter presents the theoretical framework that allows us to describe evolutions in the general case using Kraus operators as the main tool. It considers in detail the phenomenon of decoherence and gives examples of such maps. It shows that any evolution can be considered as unitary by going in a larger Hilbert space. The Lindblad equation for the evolution of the density matrix appears as a particular case of evolution in the short memory or Markov approximation. Up-jumps and down-jumps are also described within this framework using cavity damping, spontaneous emission, and the shelving technique as examples.