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The Newtonian kernel at the intersection of two discs

Andrés Miniguano-Trujillo

TL;DR

The paper derives an exact closed-form expression for the Newtonian potential obtained by convolving the 2D Newtonian kernel with the indicator of the intersection of two discs, yielding a radially symmetric, piecewise formula that depends on the distance between disc centers. A central contribution is the explicit, closed-form representation of the intermediate function F_ε in terms of polar geometry, the dilogarithm Li_2, and a careful treatment of branch choices (monodromy) to ensure continuity across regimes. The authors develop robust small-overlap asymptotics by reparameterizing the overlap region (via λ and H(λ, ε)) and provide accurate, stable double-precision implementations, supported by extended-precision analysis and numerical experiments. The work furnishes a valuable benchmark for nonlocal convolution problems in circular geometries and informs high-order, geometry-aware numerical methods for nonlocal interactions in physics and biology.

Abstract

We present an exact, closed-form expression for the Newtonian potential of the characteristic function associated with two overlapping discs in the plane. This setting naturally arises when discretising nonlocal interaction terms present in models of phase separation, aggregation dynamics, and quantum systems. We characterise the convolution integral as a piecewise function on the distance between the disc centres, with transitions dictated by the geometry of the overlapping region. Additionally, we derive detailed asymptotic expansions for the small-overlap regime, which allows us to provide stable double-precision codes.

The Newtonian kernel at the intersection of two discs

TL;DR

The paper derives an exact closed-form expression for the Newtonian potential obtained by convolving the 2D Newtonian kernel with the indicator of the intersection of two discs, yielding a radially symmetric, piecewise formula that depends on the distance between disc centers. A central contribution is the explicit, closed-form representation of the intermediate function F_ε in terms of polar geometry, the dilogarithm Li_2, and a careful treatment of branch choices (monodromy) to ensure continuity across regimes. The authors develop robust small-overlap asymptotics by reparameterizing the overlap region (via λ and H(λ, ε)) and provide accurate, stable double-precision implementations, supported by extended-precision analysis and numerical experiments. The work furnishes a valuable benchmark for nonlocal convolution problems in circular geometries and informs high-order, geometry-aware numerical methods for nonlocal interactions in physics and biology.

Abstract

We present an exact, closed-form expression for the Newtonian potential of the characteristic function associated with two overlapping discs in the plane. This setting naturally arises when discretising nonlocal interaction terms present in models of phase separation, aggregation dynamics, and quantum systems. We characterise the convolution integral as a piecewise function on the distance between the disc centres, with transitions dictated by the geometry of the overlapping region. Additionally, we derive detailed asymptotic expansions for the small-overlap regime, which allows us to provide stable double-precision codes.

Paper Structure

This paper contains 16 sections, 7 theorems, 89 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Nested intersection
  3. Partial overlap
  4. Disc points
  5. $F_\varepsilon$ by a change in the order of integration
  6. Integration order:
  7. First primitive:
  8. Second primitive:
  9. $F_\varepsilon$ by direct integration
  10. Outer points
  11. The branch at $\phi = \pi/2$ and monodromy
  12. Asymptotics and numerical treatment
  13. Numerical experiments
  14. Proof of Lemma \ref{['lem:H_asymptotic']}
  15. Proof of Lemma \ref{['lem:H_asymptotic_at_1']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let $E_\varepsilon \coloneqq K \star \mathds{1}_{ \raisebox{.05em}{\scaleobj{0.9}{\newmoon}} }$ and let $x \in \mathbb{R}^2$ with norm $a \coloneqq \|x\|$. Then, the value of $E_\varepsilon(x)$ depends on the distance of $x$ to the origin, and it is given as follows: where the intersection angle $\varphi = \varphi_\varepsilon (a) \in [0, \pi]$ is defined by The function $F_\varepsilon: D \to \m

Figures (5)

  • Figure 1: Description of $\circleddash$.
  • Figure 2: Circular segments defined by $\Theta(a)$.
  • Figure 3: Absolute error of $H$ against the asymptotic approximation from Lemma \ref{['lem:H_asymptotic']} for decreasing values of $\varepsilon$.
  • Figure 4: Extended-precision plot of $F_\varepsilon$ scaled by $\varepsilon^2 \log \varepsilon^2$ under the parametrisation \ref{['eq:Interval_Change_of_Variables']}. The function values corresponding to $\varepsilon$ taking the values e-1 and e-30 are represented by dotted and dashed lines, respectively. The asymptotic maxima are marked with dotted-dash vertical lines.
  • Figure 5: High precision plots of $E_\varepsilon$ scaled by $\varepsilon^2 \log \varepsilon^2$ for $a$ inside the intervals $[0,\sqrt{1-\varepsilon^2}]$, $[\sqrt{1-\varepsilon^2},\sqrt{1+\varepsilon^2}]$, and $[\sqrt{1+\varepsilon^2},2]$. The function values corresponding to $\varepsilon$ taking the values e-4 and e-30 are represented by dotted and dashed lines, respectively.

Theorems & Definitions (10)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 3
  • Lemma 4
  • Lemma 5
  • Corollary 1.1