Non-spherical minimizers in the generalized liquid drop model for Yukawa and truncated Coulomb potentials

TL;DR

This work shows that screening in three-dimensional generalized liquid drop models can destroy the Gamow conjecture, producing non-spherical minimizers for both truncated Coulomb and Yukawa potentials. A key technical contribution is a slicing formula that reduces nonlocal interactions to 1D slice energies, enabling precise comparisons of energy per mass between balls and infinite cylinders. The authors prove non-sphericity by establishing regimes where the cylinder’s energy per mass is strictly smaller than that of a ball, with explicit parameter ranges (e.g., κ≈1.1 for truncated Coulomb and κ≈0.56 for Yukawa). In contrast, the unscreened Riesz case remains consistent with the conjecture, though the energy per mass of balls and cylinders is strikingly close, highlighting the delicate balance between geometry and long-range interactions in these models.

Abstract

We investigate generalized liquid drop models with screened Riesz-type interactions, focusing in particular on truncated Coulomb and Yukawa potentials in three dimensions. While the classical Gamow model with Coulomb interaction is conjectured to admit only spherical minimizers below a critical mass and no minimizer above, we show that this conjecture fails if the interaction is screened. In the case of truncated Coulomb and Yukawa potentials, we establish the existence of non-spherical minimizers for some values of the screening parameter. This gives the first evidence of such minimizers in the class of repulsive, radial, and radially nonincreasing kernels in three dimensions. Our approach relies on a comparison of the energy-per-mass ratios of balls and cylinders, in contrast with recent two-dimensional results obtained via $Γ$-convergence. We further show that in the unscreened Riesz case, the conjecture remains consistent, though delicate. Indeed we observe that the energy-per-mass ratios of balls and of cylinders are surprisingly close.

Figures (3)

• Figure 1: Comparison of the energy/mass ratios for balls and infinite cylinders with the truncated Coulomb potential for varying cut-length $\kappa$. We used $\sigma_{R_{1,\kappa}}^{3,\mathop{\mathrm{cyl}}\nolimits}(l(\kappa))$ with appropriate $l(\kappa)$ as an upper bound for $\rho_{R_{1,\kappa}}^{3,\mathop{\mathrm{cyl}}\nolimits}$. Convenient values $l(\kappa)$ are obtained using SciPy's library to optimize $l\mapsto \sigma_{R_{1,\kappa}}^{3,\mathop{\mathrm{cyl}}\nolimits}(l)$. The vertical dashed line in the right figure corresponds to $\kappa=\kappa_{\mathrm{min}}$ from \ref{['prp_eballriesztrunc']}, the threshold below which the infimum $\rho^{3,\mathrm{ball}}_{R_{1,\kappa}}$ is obtained by taking balls $B_R$ with $R\uparrow\infty$.
• Figure 2: Comparison of the energy/mass ratio for balls and infinite cylinders with the Yukawa potential for varying cut-length $\kappa$. We used $\sigma_{Y_{1,\kappa}}^{3,\mathop{\mathrm{cyl}}\nolimits}(l(\kappa))$ with appropriate $l(\kappa)$ as an upper bound for $\rho_{Y_{1,\kappa}}^{3,\mathop{\mathrm{cyl}}\nolimits}$. As in the previous section, such $l(\kappa)$ are obtained using SciPy's library to optimize $l\mapsto \sigma_{Y_{1,\kappa}}^{3,\mathop{\mathrm{cyl}}\nolimits}(l)$. The vertical dashed line corresponds to $\kappa=(2\pi)^{-\frac{1}{3}}$, the threshold below which the infimum $\rho^{3,\mathrm{ball}}_{Y_{1,\kappa}}$ is obtained by taking balls $B_R$ with $R\to\infty$.
• Figure 3: Convergence rate for the numerical integration of $F^{\mathrm{reg}}_\kappa$ on $(0,2l)$ using Simpson's quadrature formula on a regular grid. Here $\kappa=0.56$, $l=2.09$ and $e(h)$ is the error using step size $h$. For the "exact" value we use Simpson's quadrature formula on a very fine grid with step size $2l/2^{14}$.

Theorems & Definitions (68)

• Conjecture 1
• Conjecture 2
• Theorem 3
• Proposition 4
• Proposition 5
• Proposition 1.1
• Definition 1.2: Generalized minimizers
• Proposition 1.3
• proof
• Definition 1.4