Wave diffusion and mesoscopic dynamics, towards a universal time-dependent random scattering matrix

We concern ourselves with the prediction of mesoscopic wave phenomena from statistical knowledge of classical trajectories. A diffusing particle picture for the flow of mean probability in chaotic systems is used to estimate dynamical features of mean square time-domain S matrices for waves coupled in and out through one perfectly open channel. A random process with that mean square, and with the additional constraint of unitarity, is then shown to lead to plausible S matrices with familiar mesoscopic wave dynamics. Features that are generated by this procedure include enhanced backscatter, quantum echo, power law tails, level repulsion and spectral rigidity. It is remarkable that such rich behaviours arise from such simple constraints. We conjecture that a generalization to n×n S matrices would exhibit behaviour identical to that of a Hamiltonian taken from the Gaussian Orthogonal or Unitary Ensembles (GOE or GUE) depending on its symmetries. Further constraining the S matrices to reproduce non universal aspects of classical dynamics, (known short time behaviours, periodic orbits, stable islands...) may generate mesoscopic wave features of such systems.