On global minimizers of
repulsive-attractive power-law
interaction energies
José Antonio Carrillo1 , Michel Chipot2 ,
rspa.royalsocietypublishing.org
Yanghong Huang1
1 Department of Mathematics, Imperial College
Research
London, London SW7 2AZ
2 Institut für Mathematik, Angewandte Mathematik,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.
Article submitted to journal
Subject Areas:
xxxxx, xxxxx, xxxx
Keywords:
interaction energy, global discrete
minimizers, swarming models
Author for correspondence:
We consider the minimisation of power-law repulsiveattractive 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.
José A. Carrillo
e-mail: [email protected]
c The Author(s) Published by the Royal Society. All rights reserved.
1. Introduction
2
Here, W is a repulsive-attractive power-law potential
W (z) = w(|z|) =
|z|α
|z|γ
−
,
γ
α
(1.2)
γ >α,
|z|η
with the understanding that η = log |z| for η = 0. Moreover, we define W (0) = +∞ if α ≤ 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 γ and α, the behaviour of the potential is
1
, the energy
depicted in Figure 1. Since this potential W is bounded from below by w(1) = γ1 − α
E[µ] always makes sense, with possibly positive infinite values.
w(r)
w(r)
1
γ
1
γ
− α1
0
1
γ>α>0
− α1
w(r)
0
1
1
γ
α ≤ 0 ≤ γ, α 6= γ
Figure 1: Three different behaviours of w(r) =
rγ
γ
− α1
−
rα
α ,
0
1
α<γ<0
γ > α.
The minimizers of the energy E[µ] are related to stationary states for the aggregation equation
ρt = ∇ · (ρ∇W ∗ ρ) studied in [8–10,14,15] with repulsive-attractive potentials [4,5,23–26,39]. 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 [5,32]. In fact, it is shown in [5] that the
dimensionality of the support of local minimizers of (1.1) depends on the strength of the repulsion
at zero of the potential W . For instance, as the repulsion at the origin gets stronger (i.e., α gets
smaller) in three dimension, the support of the local minimizer is concentrated on points, curves,
surfaces and eventually some sets of non-zero Lebesgue measure.
From the viewpoint of applications, these models with nonlocal interactions are ubiquitous
in the literature. Convex attractive potentials appear in granular media [7,14,15,33]. More
sophisticated potentials like (1.2) are included to take into account short range repulsion and
long range attraction in kinematic models of collective behaviour of animals, see [20,31,32,36,37]
and the references therein. The minimization of the interaction energy in the discrete settings is
of paramount importance for the structure of virus capsides [29], for self-assembly materials in
chemical engineering design [21,40,46], and for flock patterns in animal swarms [13,44,45].
Despite the efforts in understanding the qualitative behaviour of stationary solutions to the
aggregation equation ρt = ∇ · (ρ∇W ∗ ρ) and the structure of local minimizers of the interaction
energy E[µ], there are no general results addressing the global minimization of E[µ] in the natural
framework of probability measures. See [17] 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.
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Let µ be a probability measure on Rd . We are interested in minimizing the interaction potential
energy defined by
Z
1
E[µ] =
W (x − y)dµ(x)dµ(y) .
(1.1)
2 Rd ×Rd
ẋi = −
n
X
∇W xi − xj ,
i = 1, . . . , n ,
j6=i
lead to stability properties of the flock profiles for the second order model in swarming introduced
in [20] or with additional alignment mechanisms as the Cucker-Smale interaction [18,19], see also
[2] 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 material
science and variational calculus with hardcore potentials [1,3,22,42] in order to understand the
crystallization phenomena. However, these discrete approximations with soft potentials as (1.2)
are more difficult; apart from various properties of the minimizers [20,25,32,45], the existence
as well as the convergence of these discrete minimizers is not established 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 [6,11]. In our case, the ideas
of the proof in Section 2 will be based on convexity-type arguments in the range of exponents
γ > α ≥ 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 γ > 1 > α, the minimizers
turn out to be unique (up to translation), analogous to the simplified displacement convexity in
1D; in the special case γ = 2 and α = 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 µ is a convex combinations of Dirac Deltas, i.e.,
µ=
n
1X
δxi ,
n
x i ∈ Rd .
i=1
Setting
En (x1 , · · · , xn ) =
n X
|xi − xj |γ
i6=j
γ
|xi − xj |α
−
α
,
(2.1)
1
for such a µ one has E[µ] = 2n
2 En (x1 , · · · , xn ) . In the definition of the energy, we can include the
self-interaction for non singular cases, α > 0, since both definitions coincide. Fixing W (0) = +∞
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[µ] can only be expected up to translations. Moreover, when the potential is
radially symmetric, as in our case, then any isometry in Rd will also leave invariant the interaction
energy. These invariances are also inherited by the discrete counterpart En (x1 , · · · , xn ). We
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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 [2,32] based on the famous 3-zones models, see for instance [30,34]. Moreover, due to the recent
nonlinear stability results in [13], we know now that the stability properties of the discrete global
minimizer as stationary solution of the first order ODE model
will first consider the minimizers of En (x) among all x = (x1 , · · · , xn ) ∈ (Rd )n , and then the
convergence to the global minimizers of E[µ] as n goes to infinity.
Let us consider for α < γ, the derivative of the radial potential
w0 (r) = rγ−1 − rα−1 = rα−1 rγ−α − 1 ,
which obviously vanishes for r = 1 and for r = 0 when α > 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 α < γ, three types of behaviours for w that are shown in Figure 1. In all the three cases, En is
bounded from below since
1
1
−
,
En (x) ≥ n2
γ
α
with the understanding that
|x|η
η
= log |x| for η = 0. We set
In =
inf
x∈(Rd )n
En (x).
(2.2)
Using the translational invariance of En (x1 , · · · , xn ), we can assume without loss of generality
that x1 = 0 what we do along this subsection. First we have the following lemma showing that In
is achieved, which can be proved by discussing different ranges of the exponents γ and α.
Lemma 2.1. For any finite n(≥ 2), the minimum value In is obtained for some discrete minimizers in
(Rd )n .
The case 0 < α < γ. We claim that
n2
1
1
−
γ
α
≤ In < 0.
(2.3)
Indeed consider x = (x1 , · · · , xn ) ∈ (Rd )n such that x1 , · · · , xn are aligned and |xi − xi+1 | = n1 .
Then for any i, j one has 0 < |xi − xj | ≤ 1 and w(|xi − xj |) < 0. Therefore (2.3) follows.
Let us show that the infimum In is achieved. Let x ∈ (Rd )n . Set R = maxi,j |xi − xj |. A
minimizer is sought among the points such that En (x) < 0 and one has for such a point
Rγ X |xi − xj |γ X |xi − xj |α
Rα
≤
<
≤ n2
.
γ
γ
α
α
i,j
i,j
This implies the upper bound
R≤
n2 γ
α
1
γ−α
.
(2.4)
1
γ−α
Thus, since x1 = 0, all the xi ’s have to be in the ball of center 0 and radius n2 γ/α
, i.e., x
has to be in a compact set of (Rd )n . Since En (x) is continuous, the infimum In is achieved. Note
that the bound on the radius, where all Dirac Deltas are contained, depends a priori on n.
The case α ≤ 0 ≤ γ and α 6= γ. In this case w(0+ ) = +∞ and w(∞) = +∞. We minimize among
all x such that xi 6= xj for i 6= j. Note that w and In are both positive. Since w(r) → +∞ as r → 0
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(a) Existence of minimizer: Discrete setting
4
or r → ∞, there exists an , bn > 0 such that
5
w(an ) = w(bn ) > In .
Let x ∈ (Rd )n . If for a couple i, j, one has
|xi − xj | < an
or |xi − xj | > bn
(2.5)
then one has En (x) > In . Thus the infimum (2.2) will not be achieved among the points x
satisfying (2.5) but among those in
{x ∈ (Rd )n | an ≤ |xi − xj | ≤ bn }
Since the set above is compact, being closed and contained in (B(0, bn ))n due to x1 = 0, the
infimum In is achieved.
The case α < γ < 0. In this case In < 0. Indeed it is enough to choose
|xi − xj | > 1
∀i, j
to get En (x) < 0. Since w(0+ ) = +∞, we minimize En among the points x such that xi 6= xj ,
i 6= j. Thus the summation is on n2 − n couples (i, j). Denote by xk = (xk1 , · · · , xkn ) ∈ (Rd )n a
minimizing sequence of En . Since w(r) → +∞ as r → 0, there exists a number an < 1 such that
1
1
−
> 0.
w(an ) > n(n − 1)
α
γ
If for a couple (i, j) one has |xki − xkj | < an then
E(xk ) > w(an ) + (n2 − n − 1)
1
1
−
γ
α
>
1
1
−
α
γ
>0
and xk cannot be a minimizing sequence. So without loss of generality, we may assume that
|xki − xkj | ≥ an , ∀i, j. Let us denote by y1 , · · · , yd the coordinates in Rd . Without loss of generality,
we can assume by relabelling and isometry invariance that for every k one has
xk1 = 0,
xki ∈ {y = (y1 , · · · , yd ) | yd ≥ 0} .
Suppose that xki = (xki,1 , · · · , xki,d ) and the numbering of the points is done in such a way that
xki,d ≤ xki+1,d .
We next claim that one can assume that xki+1,d − xki,d ≤ 1, ∀i. Indeed if not, let i0 be the first
index such that
xki0 +1,d − xki0 ,d > 1.
Let us leave the first xki until i0 unchanged and replace for i > i0 , xki by
x̃ki = xki − {xki0 +1,d − xki0 ,d − 1}ed
where ed is the d-vector of the the canonical basis of Rd , i.e., we shift xki down in the direction ed
by xki0 +1,d − xki0 ,d − 1. Denote by x̃ki the new sequence obtained in this manner. One has
w(|x̃ki − x̃kj |) < w(|xki − xkj |),
∀i ≤ i0 < j,
w(|x̃ki
∀i0 < i < j, or i < j ≤ i0
− x̃kj |) = w(|xki
− xkj |),
and thus one has obtained a minimizing sequence with
x̃ki0 +1,d − x̃ki0 ,d ≤ 1,
i.e., 0 ≤ xki,d ≤ n − 1, for all i.
∀i
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an < 1 < bn ,
Repeating this process in the other directions one can assume without loss of generality that
∈ [0, n − 1]
d
(2.6)
for all k, i.e., that xk is in a compact subset of (Rd )n , and extracting a convergent subsequence,
we obtain our desired minimizer in [0, n − 1]d .
(b) Existence of minimizer: General measures
The estimates (2.4) and (2.6) give estimates on the support of a minimizer of (2.2). However,
these estimates depend on n. We will show now that the diameter of any minimizer of (2.2) can
sometimes be bounded independently of n.
Theorem 2.2. Suppose that 1 ≤ α < γ. Then the diameter of any global minimizer of En achieving the
infimum in (2.2) is bounded independently of n.
Proof. At a point x = (x1 , · · · , xn ) ∈ (Rd )n where the minimum of En is achieved one has
X |xk − xj |γ
|xk − xj |α
−
, k = 1, 2, · · · , n.
0 = ∇xk En (x1 , · · · , xn ) = ∇xk
γ
α
j6=k
Since ∇x |x|η /η = |x|η−2 x, we obtain
X
|xk − xj |γ−2 (xk − xj ) =
j6=k
X
|xk − xj |α−2 (xk − xj ),
k = 1, · · · , n.
(2.7)
j6=k
Suppose the points are labelled in such a way that
|xn − x1 | = max |xi − xj |.
i,j
Then for k = 1 and n in (2.7), we get
X
X
|x1 − xj |γ−2 (x1 − xj ) =
|x1 − xj |α−2 (x1 − xj ),
j6=1
X
j6=1
γ−2
|xn − xj |
(xn − xj ) =
j6=n
X
|xn − xj |α−2 (xn − xj ).
j6=n
By subtraction, this leads to
X |xn − xj |γ−2 (xn − xj ) − |x1 − xj |γ−2 (x1 − xj ) + 2|xn − x1 |γ−2 (xn − x1 )
j6=1,n
=
X
|xn − xj |α−2 (xn − xj ) −
j6=n
X
|xn − xj |α−2 (x1 − xj ).
j6=1
Taking the scalar product of both sides with xn − x1 we obtain
X |xn − xj |γ−2 (xn − xj ) − |x1 − xj |γ−2 (x1 − xj ), xn − x1 + 2|xn − x1 |γ
j6=1,n
=
X
j6=n
|xn − xj |α−2 (xn − xj , xn − x1 ) −
X
|xn − xj |α−2 (x1 − xj , xn − x1 ).
j6=1
For γ ≥ 2, there exists a constant Cγ > 0 such that (see [16])
|η|γ−2 η − |ξ|γ−2 ξ, η − ξ ≥ Cγ |η − ξ|γ ,
∀η, ξ ∈ Rd .
(2.8)
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xki
j6=n
j6=1
≤ 2(n − 1)|xn − x1 |α .
Thus if a ∧ b denotes the minimum of two numbers a and b, we derive
Cγ ∧ 1 n|xn − x1 |γ ≤ 2(n − 1)|xn − x1 |α .
That is
|xn − x1 | ≤
2 n−1
Cγ ∧ 1 n
1
γ−α
≤
2
Cγ ∧ 1
1
γ−α
,
which proves the theorem in the case γ ≥ 2. In the case where 1 < γ < 2, one can replace (2.8) with
γ−2
|η|γ−2 η − |ξ|γ−2 ξ, η − ξ ≥ cγ |η| + |ξ|
|η − ξ|2 ,
∀η, ξ ∈ Rd ,
for some constant cγ , (see [16]). We get, arguing as above,
X
γ
cγ |xn − xj | + |x1 − xj | + 2|xn − x1 |γ ≤ 2(n − 1)|xn − x1 |α .
j6=1,n
No since γ − 2 < 0, |xn − xj | < |xn − x1 | and |x1 − xj | < |xn − x1 |, we derive that
(n − 2)cγ 2γ−2 + 2 |xn − x1 |γ ≤ 2(n − 1)|xn − x1 |α .
We thus obtain the bound
|xn − x1 | <
2
(2γ−2 cγ ) ∧ 1
1
γ−α
,
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 ≤ α < γ. Then global minimizers associated to the global minimum of
En (x) with zero center of mass converge as n → ∞ toward a global minimizer among all probability
measures with bounded moments of order γ of the interaction energy E[µ] in (1.1) .
Proof. Let xn ∈ (Rd )n be a minimizer of (2.1) and
µn =
n
1X
δxnj
n
j
be the associated discrete measure. From Theorem 2.2, the radius of the supports of the measures
R
µn are bounded uniformly in n by R, provided that the center of the mass Rd xdµn is normalized
to be the origin. By Prokhorov’s theorem [38], {µn } is compact in the weak-∗ topology of measures
and also in the metric space induced by γ-Wasserstein distance dγ between probability measures
(see [28,43] for definition and basic properties). Then there is a measured µ∗ supported on B(0, R)
such that
µn * µ∗
and
dγ (µn , µ∗ ) → 0
as n goes to infinity. Note that the notion of convergence of a sequence of probability measures in
dγ is equivalent to weak convergence of the measures plus convergence of the moments of order
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Note that this is nothing else that the modulus of convexity (in the sense of [15]) of the potential
|x|γ . Thus estimating from above, we derive
X
X
(n − 2)Cγ + 2 |xn − x1 |γ ≤
|xn − xj |α−1 |xn − x1 | +
|x1 − xj |α−1 |xn − x1 |
γ, see [43, Chapter 9]. Let ν be any probability measure on Rd with bounded moment of order γ,
then E[ν] < ∞. Moreover, there is a sequence of discrete measures ν n of the form
n
1X
δyjn ,
n
j=1
n
n
such that dγ (ν , ν) → 0, and thus ν * ν, see [28,43]. By the definition of En in (2.2), we deduce
Z
Z
1
1
E[ν n ] =
W (x − y)dν n (x)dν n (y) ≥
W (x − y)dµn (x)dµn (y) = E[µn ].
2 Rd ×Rd
2 Rd ×Rd
On the other hand, since
dγ (µn ⊗ µn , µ∗ ⊗ µ∗ ) → 0,
dγ (ν n ⊗ ν n , ν ⊗ ν) → 0 ,
as n → ∞, and the function w(x − y) = |x − y|γ /γ − |x − y|α /α is Lipschitz continuous on
bounded sets in Rd × Rd with growth of order γ at infinity, then
Z
Z
1
1
E[µ∗ ] =
W (x − y)dµ∗ (x)dµ∗ (y) = lim
W (x − y)dµn (x)dµn (y)
n→∞ 2 Rd ×Rd
2 Rd ×Rd
Z
1
W (x − y)dν n (x)dν n (y) = E[ν].
≤ lim
n→∞ 2 Rd ×Rd
Therefore, µ∗ must be a global minimizer of E[µ] in the set of probability measures with bounded
moments of order γ.
Remark. The convergence of the minimizers of En can be proved also in the general framework of
Γ -convergence, a well-known technique of variational convergence of sequences of functionals.
This approach was implemented sucessfully to show the rescaled configurations to the Wulff
shape [3] and general measure quantization of power repulsion-attraction potentials [27].
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 < α < γ. 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.
Remark. The range of exponents 1 ≤ α < γ in Theorem 2.3 can be extended to γ ≥ 1 and γ > α > 0,
using uniform bounds on the γ-th moments of the minimizers. First, if x = (x1 , · · · , xn ) ∈ (Rd )n
is a minimizer of En with center of mass at the origin, then by (2.3) and Hölder inequality

α/γ
X |xi − xj |γ X |xi − xj |α
X |xi − xj |γ
γ (γ−α)/γ
α

<
<
n2 γ γ−α α− γ−α
,
γ
α
γ
i,j
i,j
i,j
or equivalently
γ
α
1 X |xi − xj |γ
≤ γ γ−α α− γ−α .
2
γ
n
i,j
Since the function Φ(x) = |x − y|γ is convex for any γ ≥ 1 and y ∈ Rd , Jensen’s inequality implies
that


1 X |xi − xj |γ
1 X  1 X |xi − xj |γ  1 X 1
1X
1 X |xi |γ
γ
=
≥
|x
−
x
|
=
.
i
j
γ
n
n
γ
n
γ
n
n
γ
n2
i,j
i
j
i
j
i
As a consequence, we get a uniform bound on the γ-th moment of the discrete minimizers. As a
result, the minimizing sequence corresponding to the associated atomic measures is tight, leading
also toward a global minimizer of E[µ].
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νn =
8
3. Further Remarks in one dimension
9
(a) 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 En (x), we will always
assume that the labelling of xi s is in increasing order: x1 ≤ x2 · · · ≤ xn .
Theorem 3.1. Suppose α < γ < 0 and α < −2. If x is a minimizer of En , then there exists a constant
Cα,γ such that for n large enough
2
xn − x1 ≥ Cα,γ n1+ α
holds.
Proof. Set C =
1
α
−
1
γ
> 0. Denote by an the unique element of R such that
w(an ) = Cn2 .
(3.1)
If x is a minimizer of En , we claim that
xi+1 − xi ≥ an ,
∀i = 1, . . . , n.
(3.2)
Indeed otherwise,
En (x) ≥ w(xi+1 − xi ) − (n2 − n − 1)C > Cn2 − (n2 − n − 1)C = (n + 1)C > 0
and we know that in this case En < 0. From (3.1) we derive
an α
an α
α
an γ
an α
−
=−
,
1 − an γ−α ≥ −
Cn2 =
γ
α
α
γ
2α
for n large enough (recall that an → 0 when n → ∞). It follows that
an ≥ − 2αCn2
1
α
= − 2αC
1
α
2
nα .
Combining this with (3.2) we get
1
xn − x1 ≥ (n − 1)an ≥
1 2
(−2αC) α 1+ α2
n
− 2αC α n α =
n
2
2
1
for n large enough, proving the desired estimate with Cα,γ = (−2αC) α /2.
This property for the minimizers of this very repulsive case is similar to H-stability in statistical
mechanics [41], where the minimal distance between two particles is expected to be constant
2
when n is large, and crystallization occurs. This also suggests that the lower bound O(n1+ α ) is
not sharp, which is verified in Figure 2.
In fact, numerical experiments in [5,6] suggest that confinement happens for −1 < α < 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 α ≤ −1, our numerical simulations suggest that spreading
of the support happens for all γ with a decreasing spreading rate as γ increases. For hardcore
potentials considered in [1,3,22,42], the crystallization can be rescaled to a macroscopic cluster
with uniform density; however, the scaling relation seems to have a more delicate dependence on
the parameters when α ≤ 2.
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In this section, we concentrate on the one dimensional case (d = 1) for more refined properties.
2
10
10
2
10
=
=
=
=
−1.0
0.5
2.0
4.0
γ
γ
γ
γ
=
=
=
=
−1.0
0.5
2.0
4.0
1+2/α
)
R
R
O(n
1
10
0
1
10
0
10 0
10
1
10 0
10
2
10
10
1
10
n
2
10
n
(a) α = −2.5
(b) α = −1.5
Figure 2: The dependence of the diameter R = maxi,j |xi − xj | on the number of particles n.
(b) Uniqueness of global minimizers
We turn now to the issue of uniqueness (up to isometry) of global discrete and continuum
minimizers. In general, a large amount of discrete minimizers (partially due to symmetries) are
expected, and the uniqueness can be shown only in the macroscopic limit [3]. If x is a minimizer
of En (x), we can always assume at the expense of a translation that the center of mass is zero,
n
that is x1 +···+x
= 0. Let us recall that
n
En (x) =
n
X
w(|xi − xj |)
i6=j
with the convention that xi 6= xj when i 6= j, α < 0.
Lemma 3.2. Suppose that α ≤ 1, γ ≥ 1, and α < γ. Let x, y be two points of Rn such that
x1 + · · · + xn
= 0, x1 ≤ x2 ≤ · · · ≤ xn ,
n
y1 + · · · + yn
= 0, y1 ≤ y2 ≤ · · · ≤ yn .
n
then
En
x + y
2
<
(3.3a)
(3.3b)
En (x) + En (y)
2
unless x = y.
Proof. One has w00 (r) = (γ − 1)rγ−2 − (α − 1)rα−2 > 0, for all r > 0. Thus w is strictly convex.
Then one has by the strict convexity of w,
n
x + y X
xj + yj x + yi
En
=
w i
−
2
2
2
i6=j
=
n
X
i6=j
yi − yj xi − xj
w +
2
2


n
n
X
En (x) + En (y)
1 X
w(|xi − xj |) +
w(|yi − yj |) =
≤
2
2
i6=j
i6=j
The equality above is strict unless xi − xj = yi − yj for all i, j, that is xi+1 − xi = yi+1 − yi .
Therefore x = y.
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..........................................................
γ
γ
γ
γ
As a consequence, we can now state the following result regarding the uniqueness of global
discrete minimizers.
Proof. Let x, y be two minimizers of En satisfying (3.3). If x 6= y, by Lemma 3.2, one has
x + y E (x) + E (y)
n
n
<
= In
En
2
2
and a contradiction. This shows the uniqueness of a minimizer satisfying (3.3a). Denote now by
s the symmetry defined by s(ξ) = −ξ, ξ ∈ R. If x is a minimizer of En (x) satisfying (3.3a) then y
defined by
yi = s(xn+1−i )
i = 1, · · · , n
is also a minimizer satisfying (3.3b). Thus by uniqueness
xi = −xn+1−i
i = 1, · · · , 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 (1.1).
In the seminal work of R. McCann [35] that introduces the notion of displacement convexity,
he already dealt with the uniqueness (up to translation) of the interaction energy functional
(1.1) using the theory of optimal transportation: if W is strictly convex in Rd , then the global
minimizer is unique among probability measures by fixing the center of mass, as the energy E[µ]
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 [12], to check the displacement
convexity of the energy E[µ] 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, ∞), then the energy functional (1.1) is
strictly displacement convex for probability measures with zero center of mass. As a consequence,
the global minimizer of (1.1) in the set of probability measures is unique up to translations.
Lemma 3.2 shows that this condition is equivalent to α ≤ 1, γ ≥ 1, and α < γ, for power-law
potentials. Finally, the convexity of En in Lemma 3.2 is just the displacement convexity of the
energy functional (1.1) 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 γ = 2, α = 1, which corresponds to quadratic attraction and Newtonian repulsion in one
dimension (see [23]). When x is a minimizer of En (x), we have by (2.7) that
X
X
(xk − xj ) =
sign(xk − xj ) = k − n − 1, ∀ k = 1, · · · , n.
j6=k
j6=k
Replacing the index k by k + 1, the equation becomes
X
X
(xk+1 − xj ) =
sign(xk+1 − xj ) = 2k − n + 1,
j6=k+1
∀ k = 1, · · · , n − 1.
j6=k+1
Subtracting the two equations above, we get
n(xk+1 − xk ) = 2, ∀ k = 1, · · · , n − 1,
that is xk+1 − xk = n2 , for all k = 1, · · · , n − 1.
This shows that in the case γ = 2 and α = 1, the points xi are uniformly distributed; as n goes to
P
infinity, the corresponding discrete measure µ = n1 n
i=1 δxi converges to the uniform probability
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Theorem 3.3. Suppose α ≤ 1, γ ≥ 1, and α < γ. Up to translations, the minimizer x of En is unique and
symmetric with respect to its center of mass.
11
measure on the interval [−1, 1]. This uniform density is known to be the global minimizer of the
energy E[µ] in the continuum setting, see [17,23].
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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