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Critical phase transitions in minimum-energy configurations for the exponential kernel family $e^{-|x-y|^q}$ on the unit interval

Michael T. M. Emmerich

Abstract

We study the optimal placement of $k$ ordered points on the unit interval for the bounded pair potential \[ K_q(d)=e^{-d^q}, \qquad q>0. \] The family interpolates between strongly cusp-like kernels for $0<q<1$, the threshold kernel $e^{-d}$, and the flatter Gaussian-type regime $q>1$. Our emphasis is on the transition from collision-free minimizers to endpoint-collapsed minimizers. We reformulate the problem in gap variables, record convexity, symmetry, and the Karush-Kuhn-Tucker conditions, and give a short proof that collisions are impossible for $0<q<1$. At the threshold $q=1$ we recover the endpoint-clustering law for $e^{-d}$, while for $q>1$ we identify critical exponents $q_k$ beyond which interior points are no longer optimal. For odd $k$ we derive the exact universal value \[ q_{2m+1} = \frac{\log(1/(-\log((1+e^{-1})/2)))}{\log 2} \approx 1.396363475, \] and for even $k=4,6,\dots,20$ we compute the numerical transition values \[ \begin{aligned} &q_4\approx 1.062682507,\quad q_6\approx 1.155601329,\quad q_8\approx 1.206132611,\quad q_{10}\approx 1.238523533,\\ &q_{12}\approx 1.261308114,\quad q_{14}\approx 1.278305167,\quad q_{16}\approx 1.291510874,\quad q_{18}\approx 1.302082885,\\ &q_{20}\approx 1.310744185. \end{aligned} \] We also include comparison tables and diagrams for the kernels $e^{-\sqrt d}$, $e^{-d}$, and $e^{-d^2}$, briefly relate the bounded family to the singular Riesz kernel $d^{-s}$, and identify the $q\to 0^+$ limit with the Fekete/Chebyshev--Lobatto configuration on $[0,1]$.

Critical phase transitions in minimum-energy configurations for the exponential kernel family $e^{-|x-y|^q}$ on the unit interval

Abstract

We study the optimal placement of ordered points on the unit interval for the bounded pair potential The family interpolates between strongly cusp-like kernels for , the threshold kernel , and the flatter Gaussian-type regime . Our emphasis is on the transition from collision-free minimizers to endpoint-collapsed minimizers. We reformulate the problem in gap variables, record convexity, symmetry, and the Karush-Kuhn-Tucker conditions, and give a short proof that collisions are impossible for . At the threshold we recover the endpoint-clustering law for , while for we identify critical exponents beyond which interior points are no longer optimal. For odd we derive the exact universal value and for even we compute the numerical transition values We also include comparison tables and diagrams for the kernels , , and , briefly relate the bounded family to the singular Riesz kernel , and identify the limit with the Fekete/Chebyshev--Lobatto configuration on .

Paper Structure

This paper contains 19 sections, 7 theorems, 56 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Gap variables, convexity, symmetry, and KKT conditions
  3. The local transition at the origin
  4. The threshold case $q=1$: endpoint clustering
  5. Critical exponents for $q>1$
  6. Odd $k$: an exact universal critical value
  7. Even $k$: a one-parameter symmetric branch
  8. The asymptotic regime $q\to 0^+$
  9. Comparison with selected kernels
  10. The cusp-like kernel $e^{-\sqrt d}$ ($q=\frac{1}{2}$)
  11. The threshold kernel $e^{-d}$ ($q=1$)
  12. The Gaussian kernel $e^{-d^2}$ ($q=2$)
  13. Comparison with Riesz energy
  14. Gradient-flow studies for different $(q,k)$ regimes
  15. Summary and conclusions
  16. ...and 4 more sections

Key Result

Lemma 2.1

For every $q>0$ and every $k\ge 2$, every minimizer of eq:defenergy satisfies $x_1=0$ and $x_k=1$. Equivalently, in gap variables,

Figures (5)

  • Figure 1: Comparison for $k=8$: the uniform grid (top) versus the Chebyshev--Lobatto/Fekete limit predicted by $q\to 0^+$ (bottom). The small-$q$ limit is more concentrated near the endpoints and is therefore not uniform.
  • Figure 2: Critical exponents $q_k$ for $k=3,\dots,20$. Odd values are exact and constant; even values are numerical.
  • Figure 3: A first attempt at a $(q,k)$ phase diagram for the kernel $e^{-|x-y|^q}$ on $[0,1]$. The green region marks the proved collision-free regime $0<q<1$. The blue vertical line at $q_{\mathrm{odd}}$ records the exact branch crossing for odd $k$, while the red points show the computed even critical values for $k=4,6,\dots,20$.
  • Figure 4: Stacked point diagrams for three representative kernels. Boundary points are drawn in dark red and stacked vertically when coincident.
  • Figure 5: Gradient flow plots for different combinations of $q,k$.

Theorems & Definitions (18)

  • Lemma 2.1: Full-span property
  • proof
  • Lemma 2.2: Convexity for $q\ge 1$
  • proof
  • Remark 2.3
  • Lemma 2.4: KKT condition
  • proof
  • Proposition 3.1: No collisions for $0<q<1$
  • proof
  • Remark 3.2: Threshold interpretation
  • ...and 8 more