Abstract: Consider a pressureless gas interacting through an attractive-repulsive potential given as a dierence of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild repulsion and
strong attraction, we show that the minimum energy conguration of gas is uniquely attained - apart from translations and rota-
tions - by equidistributing the particles of gas over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in
two dimensions and regular tetrahedron in three).
If the attraction is not assumed to be strong, we show these congurations are at least local energy minimizers in the relevant d metric from optimal transportation, as are all of the other uncountably many unbalanced congurations with the same support.
We infer the existence of phase transitions.
An ingredient in the proof which may have independent interest is the establishment of a simple isodiametric variance bound which generalizes Popoviciu’s inequality from one to higher dimensions and characterizes regular simplices: it shows that among probability measures on Rn whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass 1/(n + 1) to each vertex of a (unit-diameter) regular simplex.
Based on preprint with Tongseok Lim at https://arxiv.org/abs/1907.13593