Dynamical Spreading and Memory Retention Under Power Law Potential

Power law potentials dictate interactions across scales and matter, controlling the structure and dynamics of inanimate, and living systems. Though the equilibrium distributions of particles with a power law repulsion were extensively studied, their unconfined dynamical evolution gained far less attention -- Yet, living matter is inherently out of equilibrium and is seldom static. Here, we investigate the overdamped dynamic spreading of a dense suspension of particles under repulsive pair-potential of the form $1/r^k$. Coarse graining the pair interactions, we predict that the suspension spreads in a self-similar form, with its radius growing in time as $t^{1/(k+2)}$, independent of the spatial dimension ($d$). We confirm this prediction experimentally in quasi-two dimensions using perpendicularly magnetized colloids with dipolar repulsion ($k=3$). Numerical simulations corroborate the experiments for the $k=3$ case and further predict a categorically different behavior at a critical power law: when $k<d-2$, the initial distribution is no longer concentrated at the origin. Instead, particles accumulate at the perimeter and retain a long-lived memory of their original pattern. We demonstrate that below this threshold, the initial distribution seeds the resulting pattern, encoding the future structure of an unconfined, and dynamically evolving system.