Overdamped repulsive power-law particles spread self-similarly with radius growing as t^{1/(k+2)}, and for k<d-2 the system retains a long-lived memory of its initial pattern.
Expansion into the vacuum of stochastic gases with long-range interactions
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We study the evolution of a system of many point particles initially concentrated in a small region in $d$ dimensions. Particles undergo overdamped motion caused by pairwise interactions through the long-ranged repulsive $r^{-s}$ potential; each particle is also subject to Brownian noise. When $s<d$, the expansion is governed by non-local hydrodynamic equations. In the one-dimensional case, we deduce self-similar solutions for all $s\in (-2,1)$. The expansion of Coulomb gases remains well-defined in the infinite-particle limit: The density is spatially uniform and inversely proportional to time independent of the spatial dimension.
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Dynamical Spreading and Memory Retention Under Power Law Potential
Overdamped repulsive power-law particles spread self-similarly with radius growing as t^{1/(k+2)}, and for k<d-2 the system retains a long-lived memory of its initial pattern.