Liquid Drop Model for Nuclear Matter in the Low Density Limit

TL;DR

The paper rigorously justifies the gnocchi phase in the liquid drop model with a positive background by proving a two-term low-density expansion for the energy density in the thermodynamic limit. It shows that minimizers decompose into finitely many unit-sized droplets arranged at large separations, with their arrangement governed by the Jellium energy, and provides a precise link between microscopic droplet structure and macroscopic electron background. The upper and lower bounds are obtained by a droplet-tiling construction and a Graf–Schenker localization argument, respectively, ultimately connecting the dilute energy to the classical Jellium problem. This work thus bridges nuclear pasta phenomenology with rigorous Coulombic pattern formation, with implications for neutron star crusts and related materials.

Abstract

We consider the liquid drop model with a positive background density in the thermodynamic limit. We prove a two-term asymptotics for the ground state energy per unit volume in the dilute limit. Our proof justifies the expectation that optimal configurations consist of droplets of unit size that arrange themselves according to minimizers for the Jellium problem for point particles. In particular, we provide the first rigorous derivation of what is known as the gnocchi phase in astrophysics.

Figures (4)

• Figure 1: Schematic illustration of Theorem \ref{['thm:low_density']}. For small background density $\rho$, a minimizer $\Omega$ will consist of many droplets of size one and relative distances of order $\rho^{-1/3}$.
• Figure 2: Illustration of the proof of Proposition \ref{['prop:dipole_layer']}. The small white cubes that are full are the $\Lambda_\alpha$'s of the first type. The large cubes on the upper right are the $\Lambda_\alpha$'s of the second type. The colored large cubes are the ones merged with the partial small cubes of the same color that constitute the $\Lambda_\alpha$'s of the third type. In those cubes, $\Omega_\alpha$ is chosen to be a cube of volume $\rho|\Lambda_\alpha|$, placed at the center of mass of $\Lambda_\alpha$. The latter is at distance $O(\varepsilon/K)$ to the center of the large cube.
• Figure 3: Packing the large ball $B_K$ with smaller balls $B_j$ as in LieLeb-72LieNar-75 so that the uncovered part (in yellow in the picture) has an exponentially small fraction of the total volume $|B_K|$.
• Figure 4: Idea of the proof of the thermodynamic limit for an arbitrary sequence $\Lambda_n\nearrow\mathbb{R}^3$, following LieLeb-72LieNar-75. To get an upper bound on $E_{\Lambda_n}(\rho)$ we pack the domain $\Lambda_n$ by many small balls, using a cubic tiling as guiding principle (only the largest balls of the type $B_{K-1}$ are displayed in the picture). To get a lower bound on $E_{\Lambda_n}(\rho)$ we pack the complement of $\Lambda_n$ in a large ball $B_{K_n}$ of comparable volume.

Theorems & Definitions (18)

• Theorem 1: Thermodynamic limit
• Theorem 2: Low density
• Theorem 3: Decomposition of minimizing sequences
• Corollary 4
• proof : Proof of Corollary \ref{['cor:repulsion_ell_n']}
• Remark 5: Riesz case
• Proposition 6: Reduction to sets of finite size
• proof
• Lemma 7: Localization of the perimeter
• proof