# On global minimizers of repulsive-attractive power-law interaction energies

José Antonio Carrillo Author for correspondence. E-mail:
<EMAIL_ADDRESS>Department of Mathematics, Imperial College London,
London SW7 2AZ  Michel Chipot  Institut für Mathematik, Angewandte Mathematik,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland  Yanghong Huang
Department of Mathematics, Imperial College London, London SW7 2AZ

###### Abstract

We consider the minimisation of power-law repulsive-attractive interaction
energies which occur in many biological and physical situations. We show
existence of global minimizers in the discrete setting and get bounds for
their supports independently of the number of Dirac Deltas in certain range of
exponents. These global discrete minimizers correspond to the stable spatial
profiles of flock patterns in swarming models. Global minimizers of the
continuum problem are obtained by compactness. We also illustrate our results
through numerical simulations.

##  1 Introduction

Let $\mu$ be a probability measure on $\mathbb{R}^{d}$. We are interested in
minimizing the interaction potential energy defined by

$E[\mu]=\frac{1}{2}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}W(x-y)d\mu(x)d\mu(y)\,.$
(1)

Here, $W$ is a repulsive-attractive power-law potential

$W(z)=w(|z|)=\frac{|z|^{\gamma}}{\gamma}-\frac{|z|^{\alpha}}{\alpha},\quad\gamma>\alpha\,,$
(2)

with the understanding that $\frac{|z|^{\eta}}{\eta}=\log|z|$ for $\eta=0$.
Moreover, we define $W(0)=+\infty$ if $\alpha\leq 0$. This is the simplest
possible potential that is repulsive in the short range and attractive in the
long range. Depending on the signs of the exponents $\gamma$ and $\alpha$, the
behaviour of the potential is depicted in Figure 1. Since this potential $W$
is bounded from below by $w(1)=\frac{1}{\gamma}-\frac{1}{\alpha}$, the energy
$E[\mu]$ always makes sense, with possibly positive infinite values.

Figure 1: Three different behaviours of
$w(r)=\frac{r^{\gamma}}{\gamma}-\frac{r^{\alpha}}{\alpha}$, $\gamma>\alpha$.

The minimizers of the energy $E[\mu]$ are related to stationary states for the
aggregation equation $\rho_{t}=\nabla\cdot(\rho\nabla W*\rho)$ studied in [12,
13, 6, 8, 7] with repulsive-attractive potentials [20, 21, 22, 23, 35, 2, 3].
The set of local minimizers of the interaction energy, in both the discrete
setting of empirical measures (equal mass Dirac Deltas) and the continuum
setting of general probability measures, can exhibit rich complicated
structure as studied numerically in [28, 3]. In fact, it is shown in [3] that
the dimensionality of the support of local minimizers of (1) depends on the
strength of the repulsion at zero of the potential $W$. In other words, as the
repulsion at zero gets stronger (i.e., $\alpha$ gets smaller), the support of
local minimizers gets larger in Hausdorff dimension.

From the viewpoint of applications, these models with nonlocal interactions
are ubiquitous in the literature. Convex attractive potentials appear in
granular media [5, 29, 12, 13]. More sophisticated potentials like (2) are
included to take into account short range repulsion and long range attraction
in kinematic models of collective behaviour of animals, see [32, 18, 33, 28,
27] and the references therein. The minimization of the interaction energy in
the discrete settings is of paramount importance for the structure of virus
capsides [25], for self-assembly materials in chemical engineering design [19,
41, 36], and for flock patterns in animal swarms [40, 39, 11].

Despite the efforts in understanding the qualitative behaviour of stationary
solutions to the aggregation equation $\rho_{t}=\nabla\cdot(\rho\nabla
W*\rho)$ and the structure of local minimizers of the interaction energy
$E[\mu]$, there are no general results addressing the global minimization of
$E[\mu]$ in the natural framework of probability measures. See [15] for a
recent analysis of this question in the more restricted set of bounded or
binary densities. Here, we will first try to find solutions in the restricted
set of atomic measures.

The interest of understanding the global discrete minimizers of the
interaction energy is not purely mathematical. The discrete global minimizers
will give the spatial profile of typical flocking patterns obtained in
simplified models for social interaction between individuals as in [1, 28]
based on the famous 3-zones models, see for instance [26, 30]. Moreover, due
to the recent nonlinear stability results in [11], we know now that the
stability properties of the discrete global minimizer as stationary solution
of the first order ODE model

$\dot{x}_{i}=-\sum_{j\neq i}^{n}\nabla W\left(x_{i}-x_{j}\right),\quad
i=1,\dots,n\,,$

lead to stability properties of the flock profiles for the second order model
in swarming introduced in [18] or with additional alignment mechanisms as the
Cucker-Smale interaction [16, 17], see also [1] and the discussion therein.

Our objective is to show the existence of global minimizers of the interaction
energy defined on probability measures under some conditions on the exponents.
Our approach starts with the discrete setting by showing qualitative
properties about the global minimizers in the set of equal mass Dirac Deltas.
These discrete approximations are used extensively in the literature to show
various properties of the minimizers [18, 22, 28, 40], but the existence as
well as the convergence of these discrete minimizers is not justified in
general. In a certain range of exponents, we will prove that the diameter of
the support of discrete minimizers does not depend on the number of Dirac
Deltas. This result together with standard compactness arguments will result
in our desired global minimizers among probability measures.

In fact, our strategy to show the confinement of discrete minimizers is in the
same spirit as the proof of confinement of solutions of the aggregation
equation in [9, 4]. In our case, the ideas of the proof in Section 2 will be
based on convexity-type arguments in the range of exponents $\gamma>\alpha\geq
1$ to show the uniform bound in the diameter of global minimizers in the
discrete setting. Section 3 will be devoted to more refined results in one
dimension. We show that for very repulsive potentials, the bounds on the
diameter is not uniform in the number of Dirac Deltas, complemented by
numerical simulations; in the range of exponents $\gamma>1>\alpha$, the
minimizers turn out to be unique (up to translation), analogous to the
simplified displacement convexity in 1D; in the special case $\gamma=2$ and
$\alpha=1$, we can find the minimizers and show the convergence to the
continuous minimizer explicitly.

##  2 Existence of Global minimizers

We will first consider the discrete setting where $\mu$ is a convex
combinations of Dirac Deltas, i.e.,

$\mu=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}},\quad x_{i}\in\mathbb{R}^{d}.$

Setting

$E_{n}(x_{1},\cdots,x_{n})=\sum_{i\neq
j}^{n}\left(\frac{|x_{i}-x_{j}|^{\gamma}}{\gamma}-\frac{|x_{i}-x_{j}|^{\alpha}}{\alpha}\right)\,,$
(3)

for such a $\mu$ one has $E[\mu]=\frac{1}{2n^{2}}E_{n}(x_{1},\cdots,x_{n})\,.$
In the definition of the energy, we can include the self-interaction for non
singular cases, $\alpha>0$, since both definitions coincide. Fixing
$W(0)=+\infty$ for singular kernels makes $W$ upper semi-continuous, and the
self-interaction must be excluded to have finite energy configurations.

Let us remark that due to translational invariance of the interaction energy,
minimizers of the interaction energy $E[\mu]$ can only be expected up to
translations. Moreover, when the potential is radially symmetric, as in our
case, then any isometry in $\mathbb{R}^{d}$ will also leave invariant the
interaction energy. These invariances are also inherited by the discrete
counterpart $E_{n}(x_{1},\cdots,x_{n})$. We will first consider the minimizers
of $E_{n}(x)$ among all $x=(x_{1},\cdots,x_{n})\in(\mathbb{R}^{d})^{n}$, and
then the convergence to the global minimizers of $E[\mu]$ as $n$ goes to
infinity.

###  2.1 Existence of minimizer: Discrete setting

Let us consider for $\alpha<\gamma$, the derivative of the radial potential

$w^{\prime}(r)=r^{\gamma-1}-r^{\alpha-1}=r^{\alpha-1}\left(r^{\gamma-\alpha}-1\right)\,,$

which obviously vanishes for $r=1$ and for $r=0$ when $\alpha>1$. We conclude
from the sign of derivatives, that $w(r)$ attains always a global minimum at
$r=1$. There are, following the values of $\alpha<\gamma$, three types of
behaviours for $w$ that are shown in Figure 1. In all the three cases, $E_{n}$
is bounded from below since

$E_{n}(x)\geq n^{2}\left(\frac{1}{\gamma}-\frac{1}{\alpha}\right)\,,$

with the understanding that $\frac{|x|^{\eta}}{\eta}=\log|x|$ for $\eta=0$. We
set

$I_{n}=\inf_{x\in(\mathbb{R}^{d})^{n}}E_{n}(x).$ (4)

Using the translational invariance of $E_{n}(x_{1},\cdots,x_{n})$, we can
assume without loss of generality that $x_{1}=0$ what we do along this
subsection. First we have the following lemma showing that $I_{n}$ is
achieved, which can be proved by discussing different ranges of the exponents
$\gamma$ and $\alpha$.

######  Lemma 2.1.

For any finite $n(\geq 2)$, the minimum value $I_{n}$ is obtained for some
discrete minimizers in $(\mathbb{R}^{d})^{n}$ .

The case $0<\alpha<\gamma$. We claim that

$n^{2}\left(\frac{1}{\gamma}-\frac{1}{\alpha}\right)\leq I_{n}<0.$ (5)

Indeed consider $x=(x_{1},\cdots,x_{n})\in(\mathbb{R}^{d})^{n}$ such that
$x_{1},\cdots,x_{n}$ are aligned and $|x_{i}-x_{i+1}|=\frac{1}{n}$. Then for
any $i,j$ one has $0<|x_{i}-x_{j}|\leq 1$ and $w(|x_{i}-x_{j}|)<0$. Therefore
(5) follows.

Let us show that the infimum $I_{n}$ is achieved. Let
$x\in(\mathbb{R}^{d})^{n}$. Set $R=\max_{i,j}|x_{i}-x_{j}|$. A minimizer is
sought among the points such that $E_{n}(x)<0$ and one has for such a point

$\frac{R^{\gamma}}{\gamma}\leq\sum_{i,j}\frac{|x_{i}-x_{j}|^{\gamma}}{\gamma}<\sum_{i,j}\frac{|x_{i}-x_{j}|^{\alpha}}{\alpha}\leq
n^{2}\frac{R^{\alpha}}{\alpha}.$

This implies the upper bound

$R\leq\left(\frac{n^{2}\gamma}{\alpha}\right)^{\frac{1}{\gamma-\alpha}}.$ (6)

Thus, since $x_{1}=0$, all the $x_{i}$’s have to be in the ball of center $0$
and radius $\left(n^{2}\gamma/\alpha\right)^{\frac{1}{\gamma-\alpha}}$, i.e.,
$x$ has to be in a compact set of $(\mathbb{R}^{d})^{n}$. Since $E_{n}(x)$ is
continuous, the infimum $I_{n}$ is achieved. Note that the bound on the
radius, where all Dirac Deltas are contained, depends a priori on $n$.

The case $\alpha\leq 0\leq\gamma$ and $\alpha\neq\gamma$. In this case
$w(0^{+})=+\infty$ and $w(\infty)=+\infty$. We minimize among all $x$ such
that $x_{i}\neq x_{j}$ for $i\neq j$. Note that $w$ and $I_{n}$ are both
positive. Since $w(r)\to+\infty$ as $r\to 0$ or $r\to\infty$, there exists
$a_{n},b_{n}>0$ such that

$a_{n}<1<b_{n},\quad w(a_{n})=w(b_{n})>I_{n}.$

Let $x\in(\mathbb{R}^{d})^{n}$. If for a couple $i,j$, one has

$|x_{i}-x_{j}|<a_{n}\quad\mbox{or}\quad|x_{i}-x_{j}|>b_{n}$ (7)

then one has $E_{n}(x)>I_{n}.$ Thus the infimum (4) will not be achieved among
the points $x$ satisfying (7) but among those in

$\\{x\in(\mathbb{R}^{d})^{n}\mid a_{n}\leq|x_{i}-x_{j}|\leq b_{n}\\}$

Since the set above is compact, being closed and contained in
$(B(0,b_{n}))^{n}$ due to $x_{1}=0$, the infimum $I_{n}$ is achieved.

The case $\alpha<\gamma<0$. In this case $I_{n}<0$. Indeed it is enough to
choose

$|x_{i}-x_{j}|>1\qquad\forall i,j$

to get $E_{n}(x)<0$. Since $w(0^{+})=+\infty$, we minimize $E_{n}$ among the
points $x$ such that $x_{i}\neq x_{j}$, $i\neq j$. Thus the summation is on
$n^{2}-n$ couples $(i,j)$. Denote by
$x^{k}=(x_{1}^{k},\cdots,x_{n}^{k})\in(\mathbb{R}^{d})^{n}$ a minimizing
sequence of $E_{n}$. Since $w(r)\to+\infty$ as $r\to 0$, there exists a number
$a_{n}<1$ such that

$w(a_{n})>n(n-1)\left(\frac{1}{\alpha}-\frac{1}{\gamma}\right)>0.$

If for a couple $(i,j)$ one has $|x_{i}^{k}-x_{j}^{k}|<a_{n}$ then

$E(x^{k})>w(a_{n})+({n^{2}-n-1})\left(\frac{1}{\gamma}-\frac{1}{\alpha}\right)>\left(\frac{1}{\alpha}-\frac{1}{\gamma}\right)>0$

and $x^{k}$ cannot be a minimizing sequence. So without loss of generality, we
may assume that $|x_{i}^{k}-x_{j}^{k}|\geq a_{n},\,\forall i,j$. Let us denote
by $y_{1},\cdots,y_{d}$ the coordinates in $\mathbb{R}^{d}$. Without loss of
generality, we can assume by relabelling and isometry invariance that for
every $k$ one has

$x_{1}^{k}=0,\quad x_{i}^{k}\in\\{y=(y_{1},\cdots,y_{d})\mid y_{d}\geq
0\\}\,.$

Suppose that $x_{i}^{k}=(x^{k}_{i,1},\cdots,x^{k}_{i,d})$ and the numbering of
the points is done in such a way that $x_{i,d}^{k}\leq x_{i+1,d}^{k}.$

We next claim that one can assume that $x_{i+1,d}^{k}-x_{i,d}^{k}\leq 1$,
$\forall i$. Indeed if not, let $i_{0}$ be the first index such that

$x_{i_{0}+1,d}^{k}-x_{i_{0},d}^{k}>1.$

Let us leave the first $x_{i}^{k}$ until $i_{0}$ unchanged and replace for
$i>i_{0}$ , $x_{i}^{k}$ by

$\tilde{x}_{i}^{k}=x_{i}^{k}-\\{x_{i_{0}+1,d}^{k}-x_{i_{0},d}^{k}-1\\}e_{d}$

where $e_{d}$ is the $d$-vector of the the canonical basis of
$\mathbb{R}^{d}$, i.e., we shift $x_{i}^{k}$ down in the direction $e_{d}$ by
$x_{i_{0}+1,d}^{k}-x_{i_{0},d}^{k}-1$. Denote by $\tilde{x}_{i}^{k}$ the new
sequence obtained in this manner. One has

$\displaystyle w(|\tilde{x}_{i}^{k}-\tilde{x}_{j}^{k}|)$
$\displaystyle<w(|x_{i}^{k}-x_{j}^{k}|),\quad\forall i\leq i_{0}<j,$
$\displaystyle w(|\tilde{x}_{i}^{k}-\tilde{x}_{j}^{k}|)$
$\displaystyle=w(|x_{i}^{k}-x_{j}^{k}|),\quad\forall i_{0}<i<j,\mbox{or
}i<j\leq i_{0}$

and thus one has obtained a minimizing sequence with

$\tilde{x}_{i_{0}+1,d}^{k}-\tilde{x}_{i_{0},d}^{k}\leq 1,\qquad\forall i$

i.e., $0\leq x_{i,d}^{k}\leq n-1$, for all $i$.

Repeating this process in the other directions one can assume without loss of
generality that

$x_{i}^{k}\in[0,n-1]^{d}$ (8)

for all $k$, i.e., that $x^{k}$ is in a compact subset of
$(\mathbb{R}^{d})^{n}$, and extracting a convergent subsequence, we obtain our
desired minimizer in $[0,n-1]^{d}$.

###  2.2 Existence of minimizer: General measures

The estimates (6) and (8) give estimates on the support of a minimizer of (4).
However, these estimates depend on $n$. We will show now that the diameter of
any minimizer of (4) can sometimes be bounded independently of $n$.

######  Theorem 2.2.

Suppose that $1\leq\alpha<\gamma$. Then the diameter of any global minimizer
of $E_{n}$ achieving the infimum in (4) is bounded independently of $n$.

###### Proof.

At a point $x=(x_{1},\cdots,x_{n})\in(\mathbb{R}^{d})^{n}$ where the minimum
of $E_{n}$ is achieved one has

$0=\nabla_{x_{k}}E_{n}(x_{1},\cdots,x_{n})=\nabla_{x_{k}}\sum_{j\neq
k}\left(\frac{|x_{k}-x_{j}|^{\gamma}}{\gamma}-\frac{|x_{k}-x_{j}|^{\alpha}}{\alpha}\right),\quad
k=1,2,\cdots,n.$

Since $\nabla_{x}\big{(}|x|^{\eta}/\eta\big{)}=|x|^{\eta-2}x$, we obtain

$\sum_{j\neq k}|x_{k}-x_{j}|^{\gamma-2}(x_{k}-x_{j})=\sum_{j\neq
k}|x_{k}-x_{j}|^{\alpha-2}(x_{k}-x_{j}),\quad k=1,\cdots,n.$ (9)

Suppose the points are labelled in such a way that

$|x_{n}-x_{1}|=\max_{i,j}|x_{i}-x_{j}|.$

Then for $k=1$ and $n$ in (9), we get

$\displaystyle\sum_{j\neq 1}|x_{1}-x_{j}|^{\gamma-2}(x_{1}-x_{j})$
$\displaystyle=\sum_{j\neq 1}|x_{1}-x_{j}|^{\alpha-2}(x_{1}-x_{j}),$
$\displaystyle\sum_{j\neq n}|x_{n}-x_{j}|^{\gamma-2}(x_{n}-x_{j})$
$\displaystyle=\sum_{j\neq n}|x_{n}-x_{j}|^{\alpha-2}(x_{n}-x_{j}).$

By subtraction, this leads to

$\sum_{j\neq
1,n}\Big{(}|x_{n}-x_{j}|^{\gamma-2}(x_{n}-x_{j})-|x_{1}-x_{j}|^{\gamma-2}(x_{1}-x_{j})\Big{)}+2|x_{n}-x_{1}|^{\gamma-2}(x_{n}-x_{1})\\\
=\sum_{j\neq n}|x_{n}-x_{j}|^{\alpha-2}(x_{n}-x_{j})-\sum_{j\neq
1}|x_{n}-x_{j}|^{\alpha-2}(x_{1}-x_{j}).$

Taking the scalar product of both sides with $x_{n}-x_{1}$ we obtain

$\sum_{j\neq
1,n}\Big{(}|x_{n}-x_{j}|^{\gamma-2}(x_{n}-x_{j})-|x_{1}-x_{j}|^{\gamma-2}(x_{1}-x_{j}),x_{n}-x_{1}\Big{)}+2|x_{n}-x_{1}|^{\gamma}\\\
=\sum_{j\neq n}|x_{n}-x_{j}|^{\alpha-2}(x_{n}-x_{j},x_{n}-x_{1})-\sum_{j\neq
1}|x_{n}-x_{j}|^{\alpha-2}(x_{1}-x_{j},x_{n}-x_{1}).$

For $\gamma\geq 2$, there exists a constant $C_{\gamma}>0$ such that (see
[14])

$\big{(}|\eta|^{\gamma-2}\eta-|\xi|^{\gamma-2}\xi,\eta-\xi\big{)}\geq
C_{\gamma}|\eta-\xi|^{\gamma},\qquad\forall\eta,\xi\in\mathbb{R}^{d}.$ (10)

Note that this is nothing else that the modulus of convexity (in the sense of
[13]) of the potential $|x|^{\gamma}$. Thus estimating from above, we derive

$\displaystyle\big{(}(n-2)C_{\gamma}+2\big{)}|x_{n}-x_{1}|^{\gamma}$
$\displaystyle\leq\sum_{j\neq
n}|x_{n}-x_{j}|^{\alpha-1}|x_{n}-x_{1}|+\sum_{j\neq
1}|x_{1}-x_{j}|^{\alpha-1}|x_{n}-x_{1}|$ $\displaystyle\leq
2(n-1)|x_{n}-x_{1}|^{\alpha}.$

Thus if $a\wedge b$ denotes the minimum of two numbers $a$ and $b$, we derive

$\big{(}C_{\gamma}\wedge 1\big{)}n|x_{n}-x_{1}|^{\gamma}\leq
2(n-1)|x_{n}-x_{1}|^{\alpha}.$

That is

$|x_{n}-x_{1}|\leq\left(\frac{2}{C_{\gamma}\wedge
1}\frac{n-1}{n}\right)^{\frac{1}{\gamma-\alpha}}\leq\left(\frac{2}{C_{\gamma}\wedge
1}\right)^{\frac{1}{\gamma-\alpha}},$

which proves the theorem in the case $\gamma\geq 2$. In the case where
$1<\gamma<2$, one can replace (10) with

$\big{(}|\eta|^{\gamma-2}\eta-|\xi|^{\gamma-2}\xi,\eta-\xi\big{)}\geq
c_{\gamma}\big{\\{}|\eta|+|\xi|\big{\\}}^{\gamma-2}|\eta-\xi|^{2},\qquad\forall\eta,\xi\in\mathbb{R}^{d},$

for some constant $c_{\gamma}$, (see [14]). We get, arguing as above,

$\sum_{j\neq
1,n}c_{\gamma}\big{\\{}|x_{n}-x_{j}|+|x_{1}-x_{j}|\big{\\}}^{\gamma}+2|x_{n}-x_{1}|^{\gamma}\leq
2(n-1)|x_{n}-x_{1}|^{\alpha}.$

No since $\gamma-2<0$, $|x_{n}-x_{j}|<|x_{n}-x_{1}|$ and
$|x_{1}-x_{j}|<|x_{n}-x_{1}|$, we derive that

$\big{(}(n-2)c_{\gamma}2^{\gamma-2}+2\big{)}|x_{n}-x_{1}|^{\gamma}\leq
2(n-1)|x_{n}-x_{1}|^{\alpha}.$

We thus obtain the bound

$|x_{n}-x_{1}|<\left\\{\frac{2}{(2^{\gamma-2}c_{\gamma})\wedge
1}\right\\}^{\frac{1}{\gamma-\alpha}},$

which completes the proof of the theorem. ∎

As a direct consequence of this bound independent of the number of Dirac
Deltas, we can prove existence of global minimizers in the continuous setting.

######  Theorem 2.3.

Suppose that $1\leq\alpha<\gamma$. Then global minimizers associated to the
global minimum of $E_{n}(x)$ with zero center of mass converge as $n\to\infty$
toward a global minimizer among all probability measures with bounded moments
of order $\gamma$ of the interaction energy $E[\mu]$ in (1) .

###### Proof.

Let $x^{n}\in(\mathbb{R}^{d})^{n}$ be a minimizer of (3) and

$\mu^{n}=\frac{1}{n}\sum_{j}^{n}\delta_{x^{n}_{j}}$

be the associated discrete measure. From Theorem 2.2, the radius of the
supports of the measures $\mu^{n}$ are bounded uniformly in $n$ by $R$,
provided that the center of the mass $\int_{\mathbb{R}^{d}}xd\mu^{n}$ is
normalized to be the origin. By Prokhorov’s theorem [34], $\\{\mu^{n}\\}$ is
compact in the weak-$*$ topology of measures and also in the metric space
induced by $\gamma$-Wasserstein distance $d_{\gamma}$ between probability
measures (see [24, 38] for definition and basic properties). Then there is a
measured $\mu^{*}$ supported on $B(0,R)$ such that

$\mu^{n}\rightharpoonup\mu^{*}\qquad\mbox{and}\qquad
d_{\gamma}(\mu^{n},\mu^{*})\to 0$

as $n$ goes to infinity. Note that the notion of convergence of a sequence of
probability measures in $d_{\gamma}$ is equivalent to weak convergence of the
measures plus convergence of the moments of order $\gamma$, see [38, Chapter
9]. Let $\nu$ be any probability measure on $\mathbb{R}^{d}$ with bounded
moment of order $\gamma$, then $E[\nu]<\infty$. Moreover, there is a sequence
of discrete measures $\nu^{n}$ of the form

$\nu^{n}=\frac{1}{n}\sum_{j=1}^{n}\delta_{y_{j}^{n}},$

such that $d_{\gamma}(\nu^{n},\nu)\to 0$, and thus
$\nu^{n}\rightharpoonup\nu$, see [24, 38]. By the definition of $E_{n}$ in
(4), we deduce

$E[\nu^{n}]=\frac{1}{2}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}W(x-y)d\nu^{n}(x)d\nu^{n}(y)\geq\frac{1}{2}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}W(x-y)d\mu^{n}(x)d\mu^{n}(y)=E[\mu^{n}].$

On the other hand, since

$d_{\gamma}(\mu^{n}\otimes\mu^{n},\mu^{*}\otimes\mu^{*})\to 0,\qquad
d_{\gamma}(\nu^{n}\otimes\nu^{n},\nu\otimes\nu)\to 0\,,$

as $n\to\infty$, and the function
$w(x-y)=|x-y|^{\gamma}/\gamma-|x-y|^{\alpha}/\alpha$ is Lipschitz continuous
on bounded sets in $\mathbb{R}^{d}\times\mathbb{R}^{d}$ with growth of order
$\gamma$ at infinity, then

$\displaystyle E[\mu^{*}]$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}W(x-y)d\mu^{*}(x)d\mu^{*}(y)=\lim_{n\to\infty}\frac{1}{2}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}W(x-y)d\mu^{n}(x)d\mu^{n}(y)$
$\displaystyle\leq\lim_{n\to\infty}\frac{1}{2}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}W(x-y)d\nu^{n}(x)d\nu^{n}(y)=E[\nu].$

Therefore, $\mu^{*}$ must be a global minimizer of $E[\mu]$ in the set of
probability measures with bounded moments of order $\gamma$. ∎

######  Remark.

Global minimizers of the energy in the continuum setting might be a convex
combination of a finite number of Dirac Deltas. Numerical experiments suggest
that it is always the case in the range $2<\alpha<\gamma$. It is an open
problem in this range to show that global minimizers in the discrete case do
not change (except symmetries) for $n$ large enough and coincide with global
minimizers of the continuum setting.

##  3 Further Remarks in one dimension

In this section, we concentrate on the one dimensional case ($d=1$) for more
refined properties.

###  3.1 Confinement of Discrete Global Minimizers

We check first how sharp are the conditions on the exponents of the potential
to get the confinement of global discrete minimizers independently of $n$. In
fact, when the potential is very repulsive at the origin, we can show that a
uniform bound in $n$ of the diameter of global minimizers of the discrete
setting does not hold. If $x$ is a minimizer of $E_{n}(x)$, we will always
assume that the labelling of $x_{i}$s is in increasing order: $x_{1}\leq
x_{2}\cdots\leq x_{n}.$

######  Theorem 3.1.

Suppose $\alpha<\gamma<0$ and $\alpha<-2$. If $x$ is a minimizer of $E_{n}$,
then there exists a constant $C_{\alpha,\gamma}$ such that for $n$ large
enough

$x_{n}-x_{1}\geq C_{\alpha,\gamma}\,n^{1+\frac{2}{\alpha}}$

holds.

###### Proof.

Set $C=\frac{1}{\alpha}-\frac{1}{\gamma}>0$. Denote by $a_{n}$ the unique
element of $\mathbb{R}$ such that

$w(a_{n})=Cn^{2}.$ (11)

If $x$ is a minimizer of $E_{n}$, we claim that

$x_{i+1}-x_{i}\geq a_{n},\qquad\forall i=1,\dots,n.$ (12)

Indeed otherwise,

$E_{n}(x)\geq w(x_{i+1}-x_{i})-(n^{2}-n-1)C>Cn^{2}-(n^{2}-n-1)C=(n+1)C>0$

and we know that in this case $E_{n}<0$. From (11) we derive

$Cn^{2}=\frac{{a_{n}}^{\gamma}}{\gamma}-\frac{{a_{n}}^{\alpha}}{\alpha}=-\frac{{a_{n}}^{\alpha}}{\alpha}\left(1-\frac{\alpha}{\gamma}{a_{n}}^{\gamma-\alpha}\right)\geq-\frac{{a_{n}}^{\alpha}}{2\alpha},$

for $n$ large enough (recall that $a_{n}\to 0$ when $n\to\infty$). It follows
that

$a_{n}\geq\big{(}-2\alpha Cn^{2}\big{)}^{\frac{1}{\alpha}}=\big{(}-2\alpha
C\big{)}^{\frac{1}{\alpha}}n^{\frac{2}{\alpha}}.$

Combining this with (12) we get

$x_{n}-x_{1}\geq(n-1)a_{n}\geq\frac{n}{2}\big{(}-2\alpha
C\big{)}^{\frac{1}{\alpha}}n^{\frac{2}{\alpha}}=\frac{(-2\alpha
C)^{\frac{1}{\alpha}}}{2}n^{1+\frac{2}{\alpha}}$

for $n$ large enough, proving the desired estimate with
$C_{\alpha,\gamma}=(-2\alpha C)^{\frac{1}{\alpha}}/2$. ∎

(a) $\alpha=-2.5$ (b) $\alpha=-1.5$

Figure 2: The dependence of the diameter $R=\max_{i,j}|x_{i}-x_{j}|$ on the
number of particles $n$.

This property for the minimizers of this very repulsive case is similar to
H-stability in statistical mechanics [37], where the minimal distance between
two particles is expected to be constant when $n$ is large, and
crystallization occurs. This also suggests that the lower bound
$O(n^{1+\frac{2}{\alpha}})$ is not sharp, which is verified in Figure 2.

In fact, numerical experiments in [3, 4] suggest that confinement happens for
$-1<\alpha<1$. It is an open problem to get a uniform bound in the support of
the discrete minimizers as in Section 2 in this range. In the range
$\alpha\leq-1$, our numerical simulations suggest that spreading of the
support happens for all $\gamma$ with a decreasing spreading rate as $\gamma$
increases.

###  3.2 Uniqueness of global minimizers

We turn now to the issue of uniqueness (up to isometry) of global discrete and
continuum minimizers. If $x$ is a minimizer of $E_{n}(x)$, we can always
assume at the expense of a translation that the center of mass is zero, that
is $\frac{x_{1}+\cdots+x_{n}}{n}=0$. Let us recall that

$E_{n}(x)=\sum_{i\neq j}^{n}w(|x_{i}-x_{j}|)$

with the convention that $x_{i}\neq x_{j}$ when $i\neq j$, $\alpha<0$.

######  Lemma 3.2.

Suppose that $\alpha\leq 1$, $\gamma\geq 1$, and $\alpha<\gamma$. Let $x,y$ be
two points of $\mathbb{R}^{n}$ such that

$\frac{x_{1}+\cdots+x_{n}}{n}=0,\ x_{1}\leq x_{2}\leq\cdots\leq x_{n},$ (13a)
$\frac{y_{1}+\cdots+y_{n}}{n}=0,\ y_{1}\leq y_{2}\leq\cdots\leq y_{n}.$ (13b)

then

$E_{n}\left(\frac{x+y}{2}\right)<\frac{E_{n}(x)+E_{n}(y)}{2}$

unless $x=y$.

###### Proof.

One has $w^{\prime\prime}(r)=(\gamma-1)r^{\gamma-2}-(\alpha-1)r^{\alpha-2}>0$,
for all $r>0$. Thus $w$ is strictly convex. Then one has by the strict
convexity of $w$,

$\displaystyle E_{n}\left(\frac{x+y}{2}\right)$ $\displaystyle=\sum_{i\neq
j}^{n}w\left(\Big{|}\frac{x_{i}+y_{i}}{2}-\frac{x_{j}+y_{j}}{2}\Big{|}\right)$
$\displaystyle=\sum_{i\neq
j}^{n}w\left(\Big{|}\frac{x_{i}-x_{j}}{2}+\frac{y_{i}-y_{j}}{2}\Big{|}\right)$
$\displaystyle\leq\frac{1}{2}\left(\sum_{i\neq
j}^{n}w(|x_{i}-x_{j}|)+\sum_{i\neq
j}^{n}w(|y_{i}-y_{j}|)\right)=\frac{E_{n}(x)+E_{n}(y)}{2}$

The equality above is strict unless $x_{i}-x_{j}=y_{i}-y_{j}$ for all $i,j$,
that is $x_{i+1}-x_{i}=y_{i+1}-y_{i}$. Therefore $x=y$. ∎

As a consequence, we can now state the following result regarding the
uniqueness of global discrete minimizers.

######  Theorem 3.3.

Suppose $\alpha\leq 1$, $\gamma\geq 1$, and $\alpha<\gamma$. Up to
translations, the minimizer $x$ of $E_{n}$ is unique and symmetric with
respect to its center of mass.

###### Proof.

Let $x$, $y$ be two minimizers of $E_{n}$ satisfying (13). If $x\neq y$, by
Lemma 3.2, one has

$E_{n}\left(\frac{x+y}{2}\right)<\frac{E_{n}(x)+E_{n}(y)}{2}=I_{n}$

and a contradiction. This shows the uniqueness of a minimizer satisfying
(13a). Denote now by $s$ the symmetry defined by $s(\xi)=-\xi$,
$\xi\in\mathbb{R}.$ If $x$ is a minimizer of $E_{n}(x)$ satisfying (13a) then
$y$ defined by

$y_{i}=s(x_{n+1-i})\qquad i=1,\cdots,n$

is also a minimizer satisfying (13b). Thus by uniqueness

$x_{i}=-x_{n+1-i}\qquad i=1,\cdots,n,$

and this completes the proof of the theorem. ∎

######  Remark (Uniqueness and displacement convexity in one dimension).

Lemma 3.2 and Theorem 3.3 are just discrete versions of uniqueness results for
the continuum interaction functional (1). In the seminal work of R. McCann
[31] that introduces the notion of displacement convexity, he already dealt
with the uniqueness (up to translation) of the interaction energy functional
(1) using the theory of optimal transportation: if $W$ is strictly convex in
$\mathbb{R}^{d}$, then the global minimizer is unique among probability
measures by fixing the center of mass, as the energy $E[\mu]$ is (strictly)
displacement convex. However, the displacement convexity of a functional is
less strict in one dimension than that in higher dimensions. As proven in
[10], to check the displacement convexity of the energy $E[\mu]$ in one
dimension, it is enough to check the convexity of the function $w(r)$ for
$r>0$. Therefore, if $w(r)$ is strictly convex in $(0,\infty)$, then the
energy functional (1) is strictly displacement convex for probability measures
with zero center of mass. As a consequence, the global minimizer of (1) in the
set of probability measures is unique up to translations. Lemma 3.2 shows that
this condition is equivalent to $\alpha\leq 1$, $\gamma\geq 1$, and
$\alpha<\gamma$, for power-law potentials. Finally, the convexity of $E_{n}$
in Lemma 3.2 is just the displacement convexity of the energy functional (1)
restricted to discrete measures. We included the proofs of the convexity and
uniqueness because they are quite straightforward in this case, without
appealing to more involved concepts in optimal transportation.

######  Remark (Explicit convergence to uniform density).

As a final example we consider the case where $\gamma=2$, $\alpha=1$, which
corresponds to quadratic attraction and Newtonian repulsion in one dimension
(see [20]). When $x$ is a minimizer of $E_{n}(x)$, we have by (9) that

$\sum_{j\neq k}(x_{k}-x_{j})=\sum_{j\neq
k}\mbox{sign}(x_{k}-x_{j})=k-n-1,\quad\forall\ k=1,\cdots,n.$

Replacing the index $k$ by $k+1$, the equation becomes

$\sum_{j\neq{k+1}}(x_{k+1}-x_{j})=\sum_{j\neq{k+1}}\mbox{sign}(x_{k+1}-x_{j})=2k-n+1,\quad\forall\
k=1,\cdots,n-1.$

Subtracting the two equations above, we get

$n(x_{k+1}-x_{k})=2,\ \forall\ k=1,\cdots,n-1,$

that is $x_{k+1}-x_{k}=\frac{2}{n}$, for all $k=1,\cdots,n-1$.

This shows that in the case $\gamma=2$ and $\alpha=1$, the points $x_{i}$ are
uniformly distributed; as $n$ goes to infinity, the corresponding discrete
measure $\mu=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}$ converges to the uniform
probability measure on the interval $[-1,1]$. This uniform density is known to
be the global minimizer of the energy $E[\mu]$ in the continuum setting, see
[20, 15].

## Acknowledgement

JAC acknowledges support from projects MTM2011-27739-C04-02, 2009-SGR-345 from
Agència de Gestió d’Ajuts Universitaris i de Recerca-Generalitat de Catalunya,
and the Royal Society through a Wolfson Research Merit Award. JAC and YH
acknowledges support from the Engineering and Physical Sciences Research
Council (UK) grant number EP/K008404/1. MC acknowledges the support of a J.
Nelder fellowship from Imperial College where this work was initiated. The
research leading to these results has received funding from Lithuanian-Swiss
cooperation programme to reduce economic and social disparities within the
enlarged European Union under project agreement No CH-3-SMM-01/0. The research
of MC was also supported by the Swiss National Science Foundation under the
contract $\\#$ 200021-146620. Finally, the paper was completed during a visit
of MC at the Newton Institute in Cambridge. The nice atmosphere of the
institute is greatly acknowledged.

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