Expansion into the vacuum of stochastic gases with long-range interactions

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 white Gaussian noise. When s<d, the expansion is governed by nonlocal hydrodynamic equations. In the one-dimensional case, we deduce self-similar solutions for all s∈(-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.