# Ray stretching statistics, hot spot formation, and universalities in weak random disorder

## Description

I review my three papers on ray stretching statistics, hot spot formation, and universality in motion through weak random media. In the first paper, we study the connection between stretching exponents and ray densities in weak ray scattering through a random medium. The stretching exponent is a quantitative measure that describes the degree of exponential convergence or divergence among nearby ray trajectories. In the context of non-relativistic particle motion through a correlated random potential, we show how particle densities are strongly related to the stretching exponents, where the `hot spots' in the intensity profile correspond to minima in the stretching exponents. This strong connection is expected to be valid for different random potential distributions, and is also expected to apply to other physical contexts, such as deep ocean waves. The surprising minimum in the average stretching exponent is of great interest due to the associated appearance of the first generation of hot spots, and a detailed discussion will be found in the third paper. In the second paper, we study the stretching statistics of weak ray scattering in various physical contexts and for different types of correlated disorder. The stretching exponent is mathematically linked to the monodromy matrix that evolves the phase space vector over time. From this point of view, we demonstrate analytically and numerically that the stretching statistics along the forward direction follow universal scaling relationships for different dispersion relations and in disorders of differing correlation structures. Predictions about the location of first caustics can be made using the universal evolution pattern of stretching exponents. Furthermore, we observe that the distribution of stretching exponents in 2D ray dynamics with small angular spread is equivalent to the same distribution in a simple 1D kicked model, which allows us to further explore the relation between stretching statistics and the form of the disorder. Finally, the third paper focuses on the 1D kicked model with stretching statistics that resemble 2D small-angle ray scattering. While the long time behavior of the stretching exponent displays a simple linear growth, the behavior on the scale of the Lyapunov time is mathematically nontrivial. From an analysis of the evolving monodromy matrices, we demonstrate how the stretching exponent depends on the statistics of the second derivative of the random disorder, especially the mean and standard deviation. Furthermore, the maximal Lyapunov exponent or the Lyapunov length can be expressed as nontrivial functions of the mean and standard deviation of the kicks. Lastly, we show that the higher moments of the second derivative of the disorder have small or negligible effect on the evolution of the stretching exponents or the maximal Lyapunov exponents.