Between Uniformity and Collision: A New Phase-Transition

A New Phase Transition for Energy-Minimal Point Distributions on the Interval

Date: 2026-03-28

Author: Michael T. M. Emmerich

There are at least two very natural ways to spread $k$ points on the unit interval $[0,1]$ . One can try to maximize the sum of all pairwise distances, or one can try to maximize the smallest pairwise distance.

If one maximizes the sum of pairwise distances, then the answer is extreme clustering: the best strategy is to place roughly half the points at $0$ and the rest at $1$ . That way, as many pairs as possible contribute the full distance $1$ .

If instead one maximizes the minimum distance between points, then the answer is the opposite extreme: uniform spacing. In that case the points should be spread evenly across the interval.

A closely related singular model is given by the Riesz family, where one minimizes the sum of reciprocal distances raised to a power:

$\displaystyle \sum_{1\le i<j\le k} |x_j-x_i|^{-s}.$

Here short distances are punished very strongly, and as $s\to\infty$ the dominant contribution comes from the smallest pairwise distance. So in that limit, minimizing Riesz $s$ -energy becomes essentially the same as maximizing the minimum distance. This again leads to uniform spacing on the interval.

So one extreme favors clustering, while the singular Riesz extreme favors evenly spaced points. This raises a natural playground question: is there a smooth family of declining kernels that interpolates between these two tendencies, but is less singular than Riesz $s$ -energy?

That was my motivation for looking at the bounded family

$\displaystyle K_q(d)=e^{-d^q}, \qquad q>0,$

and the associated energy

$\displaystyle E_{k,q}(x_1,\dots,x_k)=\sum_{1\le i<j\le k} e^{-|x_j-x_i|^q}.$

Because $e^{-d^q}$ is largest when $d=0$ and decreases as points separate, minimizing the energy means trying to keep all pairs as far apart as the interval allows. This is a natural toy problem in discrete energy minimization, approximation theory, and point placement: given only room for $k$ points inside $[0,1]$ , where should they go?

That sounds innocent. But the answer depends dramatically on the exponent $q$ , and the phase diagram in the (q,k)-plane turns out to be much richer than one might expect. In fact, this phase transition appears to be a new phenomenon discovered by the author for this bounded kernel family.

The $k$ -point subset question

A helpful way to think about the problem is this: among all $k$ -point subsets of the interval – allowing repetitions if the energy minimizer wants them – which subset minimizes the total pair interaction? In other words, we are not optimizing one distance, but the whole collection of $\binom{k}{2}$ pair distances at once.

This makes the problem subtly global. Moving one point can improve some distances and worsen others. A point in the interior interacts with points on both sides. A point at an endpoint can be very far from points near the opposite endpoint, but may also encourage collisions nearby. So the real question is not simply “spread the points out evenly,” but rather:

when is it worth keeping all $k$ points distinct,
when do collisions become beneficial,
and how many points should pile up at the endpoints?

That is the $k$ -subset problem here: for each fixed $k$ and each exponent $q$ , find the energy-minimizing configuration of $k$ points in $[0,1]$ .

Why the exponent matters

The local shape near zero is controlled by

$\displaystyle e^{-d^q}=1-d^q+O(d^{2q})\qquad (d\downarrow 0).$

So $q$ decides how sharply the kernel reacts to tiny separations. For $0<q<1$ , the graph has an infinite slope at the origin: the kernel strongly dislikes near-collisions. At $q=1$ , we hit the threshold case $e^{-d}$ . For $q>1$ , the kernel becomes flat at the origin, and then a surprise appears: collisions are allowed not only in principle, but can become globally optimal.

The $(q,k)$ phase diagram in a nutshell

For $0<q<1$ , minimizers are collision-free: all points stay distinct.
At $q=1$ , endpoint clustering first appears.
For $q>1$ , there are critical exponents $q_k$ beyond which interior points disappear completely.
For odd $k$ , all tested transitions occur at one universal value.
For even $k$ , the transition values rise with $k$ in the tested range.

The easy intuition for $0<q<1$

Here the kernel has a cusp at the origin. If two points collide, opening that zero gap by a tiny amount $\varepsilon$ immediately improves the energy by order $\varepsilon^q$ . To compensate, one must steal length from somewhere else, which only costs order $\varepsilon$ . Since $\varepsilon^q\gg \varepsilon$ for $0<q<1$ , every collision can be improved away. So in this regime the minimizing $k$ -subset really is a genuine $k$ -point set with all points distinct.

The threshold $q=1$

At $q=1$ , the gain from opening a collision and the loss from adjusting the rest are both of order $\varepsilon$ . So the previous argument no longer decides the issue. Numerically, one sees partial endpoint clustering, with about one third of the points collapsing onto each boundary. A simple empirical rule is

$\displaystyle m(k)=\left\lfloor \frac{k+1}{3}\right\rfloor,$

where $m(k)$ is the number of coincident points at each endpoint.

The surprising answer just above $q=1$

The genuinely surprising part comes just above the threshold. One might guess that for every $q>1$ , however slightly above $1$ , the flatter kernel should immediately force all points to collapse to $0$ and $1$ . But that is not what happens.

There is a whole intermediate window with $q>1$ where interior points still survive. So although $q>1$ is already on the collision-friendly side of the threshold, it is still subcritical with respect to complete endpoint collapse whenever $1<q<q_k$ .

In that range, collisions may occur, but the optimal $k$ -subset is not yet supported only on the endpoints. Interior points can still lower the total energy enough to remain part of the minimizer. Only after crossing the critical value $q_k$ do the endpoints win completely.

So the answer for slightly super-threshold $q>1$ is subtle:

collisions are no longer forbidden,
but full endpoint collapse is still not yet optimal,
and only beyond a second transition at $q=q_k$ do all interior points disappear.

That two-stage story is what makes the phase diagram interesting, and it is exactly this transition structure that seems to be new here.

Odd $k$ : one exact universal transition

For odd $k=2m+1$ , the key competition is between

$\displaystyle ( \underbrace{0,\dots,0}_m,\tfrac12,\underbrace{1,\dots,1}_m )$

and the endpoint-only split

$\displaystyle ( \underbrace{0,\dots,0}_m,\underbrace{1,\dots,1}_{m+1} ).$

These two branches cross at the exact value

$\displaystyle q_{2m+1}=\frac{\log\!\Bigl(1/\bigl[-\log((1+e^{-1})/2)\bigr]\Bigr)}{\log 2}\approx 1.396363475.$

This is striking: the critical value does not depend on $m$ at all. So all odd values of $k$ line up on one vertical line in the phase diagram.

Even $k$ : a rising critical curve

For even $k=2m$ , the natural competing configuration has two symmetric interior points, and the critical exponents are numerical. For $k\le 20$ , they are

$\displaystyle q_4\approx 1.062682507,\quad q_6\approx 1.155601329,\quad q_8\approx 1.206132611,\quad q_{10}\approx 1.238523533,$

$\displaystyle q_{12}\approx 1.261308114,\quad q_{14}\approx 1.278305167,\quad q_{16}\approx 1.291510874,\quad q_{18}\approx 1.302082885,\quad q_{20}\approx 1.310744185.$

These values climb steadily upward and stay below the odd universal value in the tested range.

How to read the whole picture

The emerging $(q,k)$-diagram therefore has three visible zones:

Collision-free zone: $0<q<1$ .
Partially clustered zone: first at $q=1$ , and then continuing for $1<q<q_k$ .
Endpoint-only zone: for $q>q_k$ , where the tested minimizers live entirely on $\{0,1\}$ .

So the “surprising answer” is that the threshold $q=1$ does not immediately mark full collapse. It only marks the first moment when collisions become possible. Complete collapse comes later, at the $k$ -dependent exponent $q_k$ .

The opposite extreme: $q\to 0^+$

At the other end of the diagram, the minimizing configurations do not converge to equally spaced points. Instead, the first-order asymptotics select the classical Fekete configuration on $[0,1]$ , namely the affine Chebyshev-Lobatto points

$\displaystyle \xi_j=\frac{1-\cos\!\bigl(j\pi/(k-1)\bigr)}{2},\qquad j=0,\dots,k-1.$

So even the small- $q$ end has nontrivial structure: the optimal $k$ -subset is more concentrated near the endpoints than a uniform grid.

Detailed report

For the full technical write-up with proofs, formulas, tables, and diagrams, see the detailed report (PDF).

Why this is a nice playground problem

This is exactly the kind of mathematical toy model that rewards experimentation. The definition is short, the interval is one-dimensional, and yet the optimal configurations show symmetry, clustering, parity effects, exact formulas, and a nontrivial phase diagram. It is a small problem with unexpectedly expressive behavior.

Open questions

Is the odd critical value $1.396363475\ldots$ the true global transition for every odd $k$ ?
Do the even critical exponents keep increasing with $k$ ?
Can one describe the full large- $k$ phase diagram analytically?

References and background

S. V. Borodachov, D. P. Hardin, and E. B. Saff, Discrete Energy on Rectifiable Sets, Springer Monographs in Mathematics, 2019.
D. P. Hardin and E. B. Saff, Discretizing manifolds via minimum energy points, Notices Amer. Math. Soc. 51 (2004), 1186-1194.
D. P. Hardin and E. B. Saff, Minimal Riesz energy point configurations for rectifiable $d$ -dimensional manifolds, Adv. Math. 193 (2005), 174-204.
F. Pausinger, Greedy energy minimization can count in binary: point charges and the van der Corput sequence, Ann. Mat. Pura Appl. 200 (2021), 165-186.
S. Steinerberger, Exponential sums and Riesz energies, J. Number Theory 182 (2018), 37-56.

Read my paper with detailed derivations and closed form expressions:

Emmerich, M. T. M. (2026, March 28). Phase transitions for the one-dimensional kernel family e^(-|x-y|^q): Critical exponents, collisions, and endpoint clustering on the unit interval [PDF, CC BY 4.0]. Mathematical Playground Blog.
Link to PDF Document

Try my online python app to experiment with different q and k settings and see how a gradient flow optimizer finds the optimal energy minimal configuration starting from a near uniform distribution. https://trinket.io/library/trinkets/a02e463476c5

Between Uniformity and Collision: A New Phase-Transition

Share this:

Leave a comment Cancel reply