Asymptotic hydrographs and anomalous dispersion in mass-conserving storage cascades

Sums of independent exponential random variables lead to the Erlang distribution, providing a direct probabilistic route from exponential waiting times to the integer-shape gamma law.

This paper investigates how this classical construction changes when the exponential waiting-time density is replaced by the $q$-exponential density of nonextensive statistics.

Our main result is an analytical asymptotic expression for the outflow of a mass-conserving cascade of reservoirs driven by a $q$-exponential waiting-time kernel.

In the critical case $q=5/3$, the large-cascade flow rate converges to a stable Lévy density whose time argument is shifted by a Galilean-type transformation.

This shifted Lévy law gives the asymptotic hydrograph of the cascade.

We also found that for the entire regime $1<q<2$ the macroscopic dynamics are governed by $\alpha$-stable Lévy laws.

This proves that anomalous non-Gaussian dispersion can emerge from pure mass-conserving convolutional chains without invoking fractional derivatives.