Symmetry breaking in the liquid drop model with screened interactions

Technical Summary: Symmetry Breaking in the Liquid Drop Model with Screened Interactions

Problem Statement
This paper investigates the generalized Gamow liquid drop model, defined by the energy functional:
$$E_G(E) = P(E) + \int_{E \times E} G(x-y) \, dx \, dy$$
where $P(E)$ is the perimeter of a set $E \subset \mathbb{R}^n$ with fixed mass $|E|=m$, and $G$ is a radial, nonnegative, locally integrable interaction kernel. The study focuses on the effect of "screening" the interaction, specifically comparing the classical Coulomb case ($G(x) = |x|^{-1}$ in $\mathbb{R}^3$) against screened variants:

1. Truncated Coulomb potentials: $R_{1,\kappa}(x) = \mathbb{1}_{|x|<\kappa}|x|^{-1}$.
2. Yukawa potentials: $Y_{1,\kappa}(x) = e^{-|x|/\kappa}|x|^{-1}$.

The central question is whether the minimizers of this energy remain balls (spheres) for small masses, as established for the classical Coulomb case by Chodosh and Gianocca (2026), or if screening induces symmetry breaking, leading to non-ball minimizers.

Methodology
The authors employ a variational approach comparing the "energy-per-mass" ratios ($\rho$) of different geometric shapes. The core strategy involves:

1. Generalized Minimizers: The authors consider generalized minimizers, which are sequences of disjoint sets $(E_i)$ with total mass $m$. They establish that if the global infimum of the energy-per-mass ratio is achieved by a shape other than a ball (or a union of balls), then a true minimizer exists that is not a ball.
2. Shape Comparison: The paper defines and computes the optimal energy-per-mass ratios for three specific geometries:
   • Balls ($\rho_{ball}$): The standard candidate for minimizers.
   • Infinite Cylinders ($\rho_{cyl}$): Used as a test case to detect symmetry breaking. The authors calculate the limit of the energy-per-mass ratio as the cylinder length $L \to \infty$.
   • Core-shells ($\rho_{shell}$): Annular regions $B_R \setminus B_r$, used to test for radial symmetry breaking.
3. Slicing Formula: A key technical tool is a slicing formula (Proposition 1.12) that expresses the non-local interaction energy of a convex set as an integral over its one-dimensional slices. This allows the authors to reduce the $n$-dimensional interaction integrals to 1D problems, facilitating the computation of energies for balls and cylinders.
4. Numerical and Symbolic Verification: For the screened potentials, the resulting energy expressions involve special functions (incomplete Beta functions for truncated Coulomb, modified Bessel functions $K_0$ for Yukawa). The authors use symbolic computation libraries (SymPy) and high-precision numerical integration (Simpson's rule with singularity subtraction) to rigorously compare the computed ratios.

Key Contributions and Results

• Existence of Non-Ball Minimizers (Theorem B):
  The authors prove that for specific parameters in the screened models, the global minimizer is not a ball.

  • Truncated Coulomb: For $n=3$ and $\kappa = 11/10$, there exists a mass $m$ such that the minimizer is connected but not a ball. Furthermore, the global minimizer is neither a ball nor a disjoint union of balls.
  • Yukawa: For $n=3$ and $\kappa = 28/5$, a similar result holds: a connected, non-ball minimizer exists.
  • Mechanism: This is demonstrated by showing that for these parameters, the energy-per-mass ratio of long cylinders is strictly lower than that of balls ($\rho_{cyl} < \rho_{ball}$).

• Existence of Non-Radial Minimizers (Theorem C):
  Focusing on the truncated Coulomb potential with $\kappa = 11/10$, the authors prove a stronger result: the minimizer is not only non-ball but also non-radial.

  • They show that the energy-per-mass ratio of cylinders is strictly lower than that of core-shells ($\rho_{cyl} < \rho_{shell}$).
  • Since the optimal core-shell ratio for this parameter coincides with the ball ratio ($\rho_{shell} = \rho_{ball}$), and cylinders are non-radial, the minimizer cannot be a ball, a core-shell, or any disjoint union of such radial sets.
  • Significance: The authors claim this provides the first evidence of non-radial minimizers in the generalized liquid drop model for any radial kernel (previously, non-radial minimizers were only known for anisotropic kernels or non-monotonic kernels in 2D).

• Contrast with Riesz Potentials (Proposition G):
  The paper contrasts these findings with the classical Riesz case ($G(x) = |x|^{-\alpha}$). For integer $\alpha \in (1, n)$, the authors prove $\rho_{ball} < \rho_{cyl}$. This supports the conjecture that balls are the unique minimizers for Riesz interactions, highlighting that the symmetry breaking is a specific consequence of the rapid decay (screening) of the interaction kernel.

Significance and Claims
The paper claims to provide the first rigorous evidence of symmetry breaking in the liquid drop model for radial kernels in three dimensions. Specifically:

1. It demonstrates that screening the Coulomb interaction destabilizes the spherical shape, allowing for elongated (cylindrical-like) structures to become energetically favorable.
2. It establishes the existence of non-radial minimizers for a radial, radially non-increasing kernel, challenging the intuition that radial symmetry is preserved in such isotropic problems.
3. The results are specific to the screened potentials; the authors note that for the unscreened Coulomb case, balls remain the unique minimizers (per Chodosh–Gianocca).

The authors remain modest regarding the Yukawa case for non-radial minimizers (Conjecture D). While they expect the phenomenon to hold, they state that the computations involving Bessel functions are "untractable" for a rigorous analytical proof at this time, leaving it as an open conjecture. The paper relies on numerical evidence and symbolic computation to support the existence of non-ball minimizers in the Yukawa case, but the non-radial result is currently proven only for the truncated Coulomb potential.