Universality of finite-time ray focusing statistics in two-dimensional and one-dimensional potential flows

In the investigation of extreme density or wave height statistics in a disordered medium, of special interest is the search for universal or fundamental properties shared by different types of disorder. In previous work [Chen and Kaplan, Entropy 25, 161 (2023)1099-430010.3390/e25010161] we have established a direct connection between the degree of stretching or focusing of ray trajectories and the density distribution. Here we demonstrate the universality of this connection for different physical contexts, and both analytically and numerically show a universal scaling relationship for the stretching exponent distribution in weak, small-angle scattering at finite times for different dispersion relations. We observe that the mean, skewness, and kurtosis of the stretching exponent all display universal nonmonotonic behavior on timescales comparable to the time of first caustic formation, corresponding to the first generation of hot spots in the density profile. In particular, the mean stretching exponent attains negative values before beginning its linear rise at large times. Using the correspondence between two-dimensional small-angle scattering and a one-dimensional kicked model, we show how higher moments of the distribution of the second derivative of the potential affect the statistics of the stretching exponents.