Dynamical Spreading Under Power Law Potential

We examine the dynamic spreading of a dense overdamped suspension of particles under power law repulsive potentials, often called Riesz gases. That is, potentials that decay with distance as 1/r^k where k\in (-2,\infty]. Depending on the value of k relative to the system's spatial dimension $D$, the potentials are categorized as short-ranged for k > D, and long-ranged when $k \leq D$. Such systems naturally occur in contexts involving particle suspensions, granular media, and charged systems, where interactions can be influenced by physical fields that decrease over distance. Our analytical findings reveal that the particles spread in a self-similar form, with the radius growing with time as t^1/(k+2). The theoretical predictions derived for a general dimension D, are verified by numerical simulations involving thousands of particles in free space, in both one and two dimensions. Furthermore, the simulations not only confirm our analytical results but also reveal a rich diversity of behaviors depending on the value of k. We demonstrate that the density profiles differ significantly depending on whether k is larger than, smaller than, or equal to D-2, where D is the dimension. For k>D-2 the density is centered in the middle and we also notice a Wigner lattice emerging as a result of the repulsive interactions, for k = D-2, density is uniform and for k<D-2, density is centered at the edges. This new classification indicates that the long/short-range classification is insufficient for predicting the density profile of the suspension. When k<D-2, we observed an interesting phenomenon when two or more suspensions are placed near each other: a particle-free zone is formed where the two populations meet, resembling structures of bubbles.