We study equilibrium measures for Riesz gases in dimension $d$ with pairwise interaction kernel $|x-y|^{-s}$, subject to radially symmetric external fields. We characterise broad classes of confining potentials for which the equilibrium measure is supported on the unit ball and admits an explicit density. Our main contribution is a converse construction: starting from a prescribed radially symmetric equilibrium density given as a power series in the squared radius, we determine the associated external potential and establish the corresponding Euler-Lagrange variational conditions. A key ingredient in the proof is an identity between two ${}_3F_2$ hypergeometric functions evaluated at unit argument, which is of independent interest. As applications, we identify the external potentials corresponding to equilibrium densities proportional to $(1-|x|^2)^α$, $α>-1$, and show that these potentials can be expressed in terms of Gauss hypergeometric functions ${}_2F_1$, reducing to polynomials for special values of $α$. We also determine the equilibrium measure associated with purely power-type external potentials, often referred to as Freud or Mittag--Leffler potentials in the context of log gases, for which the equilibrium density admits an explicit ${}_2F_1$ representation. Furthermore, we apply our framework to a Coulomb gas in dimension $d+1$ confined by a harmonic potential to the half-space. We derive a necessary condition under which the equilibrium measure is fully supported on the boundary hyperplane of dimension $d$, with the induced density corresponding to that of a Riesz gas with exponent $s=d-1$.
Alex: Welcome to another episode of ResearchPod. Sam, you've been digging into some work on particle systems—walk us through what caught your eye.
Sam: This paper studies systems called Riesz gases. The central puzzle is finding the right confining force—or "bowl"—to hold particles in a specific steady crowd shape. Imagine many identical particles in a multi-dimensional space, like a room. Each one repels the others, but the push weakens with distance. An outside bowl shape pulls them toward the center.
Alex: So they flip the usual approach—starting from the crowd shape instead of the bowl?
Sam: Yes. Normally, you pick the bowl first and see where particles settle. Here, they start with a desired steady density inside a sphere of radius one—thicker or thinner in spots. Then they calculate the exact bowl that holds them there stably. This models real things like trapped ions or electrons in quantum dots.
Alex: Give me an example—how does that play out for one of their crowd shapes?
Sam: Take their power-type crowd: density proportional to (one minus distance squared) raised to alpha, for alpha greater than minus one. Particles pile higher near the center if alpha is large, tapering to the edges—like stacking books thicker in the middle of a shelf. The matching bowl is a hypergeometric function, or sometimes a simple polynomial for certain values.
Alex: Does that connect to limits we know, like electric repulsion?
Sam: Yes—in the Coulomb limit, it matches known cases from Poisson's equation. As the repulsion parameter s nears the dimension d minus two, or approaches d, it aligns with expected bowl shapes.
Alex: Those checks make sense. What about power-law bowls, like distance to an even power?
Sam: They consider bowls proportional to distance to the power 2p. Particles feel a steady push everywhere—like a gentle slope everywhere in a dish. The steady density is a hypergeometric function of distance squared inside the sphere, zero outside. For p equals one, it's a quadratic bowl, like a parabolic dish.
Alex: Near the edge of that sphere, how does the density drop off?
Sam: For s between d minus two and d, it drops to zero like (one minus distance squared) to a specific power. In the Coulomb case, it jumps instead.
Alex: They also look at a half-space with a hard wall—how does that change things?
Sam: Yes—particles face a wall, with a quadratic bowl only on one side and infinite push on the other, like marbles blocked in half a dish. Without the wall, the crowd fills the sphere uniformly. As the wall moves in, particles first ignore it, then pile up against it partially, and finally flatten fully onto it past a critical position.
Alex: The wall position reshapes the crowd completely. For the general case—picking a power series density—how do they prove the forces balance?
Sam: Radial symmetry simplifies multi-dimensional integrals to one dimension—like focusing on rings of particles at different distances. They rewrite the repulsion at any point as an integral over that radial density. Expanding into series, a key identity cancels most terms—pairing high and low powers to zero—leaving a clean series inside the sphere. They set the bowl to balance that exactly, with positivity ensuring no leakage outside.
Alex: Like summarizing the total push from particles at every distance, then tuning the bowl to cancel it perfectly?
Sam: Precisely. Proved by induction and gamma function symmetries, it works for any safe sequence of coefficients that builds a non-negative density. This gives the first explicit pairs of crowd shapes and bowls for higher dimensions and general s.
Alex: Overall, what does this mean?
Sam: It provides exact solutions beyond special cases, skipping heavy simulations. This could help predict ion arrangements in plasma traps or electron spreads in quantum dots, like in trapped-ion quantum computing. Positivity holds for their shapes, but generally remains open. Fully characterizing safe densities is hard too.
Alex: A meaningful step for these systems. Well put, Sam—thanks for breaking it down. That's our look at these particle equilibria. Thanks for listening to ResearchPod.