Existence of Ground States of Nonlocal-Interaction Energies
EEXISTENCE OF GROUND STATES OF NONLOCAL-INTERACTIONENERGIES
ROBERT SIMIONE, DEJAN SLEP ˇCEV, AND IHSAN TOPALOGLU
Abstract.
We investigate which nonlocal-interaction energies have a ground state (global min-imizer). We consider this question over the space of probability measures and establish a sharpcondition for the existence of ground states. We show that this condition is closely related to thenotion of stability (i.e. H -stability) of pairwise interaction potentials. Our approach uses the directmethod of the calculus of variations. Introduction
We investigate the existence of ground states (global minimizers) of nonlocal-interaction energies(1.1) E ( µ ) := (cid:90) R N (cid:90) R N w ( x − y ) dµ ( x ) dµ ( y )considered over the space of probability measures P ( R N ). Nonlocal-interaction energies arise indescriptions of systems of interacting particles, as well as their continuum limits. They are im-portant to statistical mechanics [24, 35, 37], models of collective behavior of many-agent systems[6, 33], granular media [5, 18, 39], self-assembly of nanoparticles [26, 27], crystallization [1, 34, 38],and molecular dynamics simulations of matter [25].Whether the energy dissipated by a system admits a global minimizer has important consequenceson the behavior of the system. Continuum systems governed by the energy which has a groundstate typically exhibit well defined dense clumps, while the systems with no global minimizers tendto disperse indefinitely.The interaction potential w depends on the system considered. In most cases it depends only onthe distance between particles/agents. That is the interaction potential w is radially symmetric: w ( x ) = W ( | x | ) for some W : [0 , ∞ ) → ( −∞ , ∞ ]. Many potentials considered in the applicationsare repulsive at short distances ( W (cid:48) ( r ) < r small) and attractive at large distances ( W (cid:48) ( r ) > r large). Systems with finitely many particles governed by short-range-repulsive, long-range-attractive interaction potentials form well defined structures (crystals are an example [38]). Therelevance of our result is to the behavior of these systems as the number of particles grows toinfinity. Systems which have a global minimizer over the space of measures form well defined stateswhose density grows as the number of particles increases, while the systems with no ground statestypically have bounded density and increase in size indefinitely.This mirrors the considerations in classical statistical mechanics when thermodynamic limitof particle systems is considered [35]. Here we obtain mathematical results that highlight theconnection. Namely, in Theorem 3.2 (combined with Proposition 4.1), we establish that the sharpcondition for the existence of ground states of (1.1) is closely related to the notion of stability Date : October 22, 2018.1991
Mathematics Subject Classification.
Key words and phrases. ground states, global minimizers, H-stability, pair potentials, self-assembly, aggregation. a r X i v : . [ m a t h . A P ] J a n ROBERT SIMIONE, DEJAN SLEPˇCEV, AND IHSAN TOPALOGLU ( H -stability) of interacting potentials [24, 35]. More precisely we show that systems admitting aminimizer of (1.1) are (almost) precisely those for which the interaction potential is not H -stable,that is those for which the potential is catastrophic.In recent years significant interest in nonlocal-interaction energies arose from studies of dynamicalmodels. For semi-convex interaction potentials w a number of systems governed by the energy E can be interpreted as a gradient flow of the energy with respect to the Wasserstein metric andsatisfy the nonlocal-interaction equation(1.2) ∂µ∂t = 2 div ( µ ( ∇ w ∗ µ )) . Applications of this equation include models of collective behavior in biology [6, 33], granular media[5, 18, 39], and self-assembly of nanoparticles [26, 27].While purely attractive potentials lead to finite-time or infinite time blow up [7] the attractive-repulsive potentials often generate finite-sized, confined aggregations [23, 29, 31]. The study ofthe nonlocal-interaction equation (1.2) in terms of well-posedness, finite or infinite time blow-up,and long-time behavior has attracted the interest of many research groups in the recent years[3, 4, 7, 8, 9, 10, 16, 17, 21, 23, 28, 29, 30]. The energy (1.1) plays an important role in thesestudies as it governs the dynamics and as its (local) minima describe the long-time asymptotics ofsolutions.It has been observed that even for quite simple repulsive–attractive potentials the ground statesare sensitive to the precise form of the potential and can exhibit a wide variety of patterns [28, 29,41]. In [2] Balagu´e, Carrillo, Laurent, and Raoul obtain conditions for the dimensionality of thesupport of local minimizers of (1.1) in terms of the repulsive strength of the potential w at theorigin. Properties of steady states for a special class of potentials which blow up approximately likethe Newtonian potential at the origin have also been studied [9, 15, 22, 23]. Particularly relevant toour study are the results obtained by Choksi, Fetecau and one of the authors [19] on the existenceof minimizers of interaction energies in a certain form. There the authors consider potentialsof the power-law form, w ( x ) := | x | a /a − | x | r /r , for − N < r < a , and prove the existence ofminimizers in the class of probability measures when the power of repulsion r is positive. When theinteraction potential has a singularity at the origin, i.e., for r <
0, on the other hand, they establishthe existence of minimizers of the interaction energy in a restrictive class of uniformly bounded,radially symmetric L -densities satisfying a given mass constraint. Carrillo, Chipot and Huang [14]also consider the minimization of nonlocal-interaction energies defined via power-law potentials andprove the existence of a global minimizer by using a discrete to continuum approach. The groundstates and their relevance to statistical mechanics were also considered in periodic setting (and onbounded sets) by S¨uto [37].1.1. Outline.
In Theorems 3.1 and 3.2 we establish criteria for the existence of minimizers of avery broad class of potentials. We employ the direct method of the calculus of variations. In Lemma2.2 we establish the weak lower-semicontinuity of the energy with respect to weak convergence ofmeasures. When the potential W grows unbounded at infinity (case treated in Theorem 3.1) thisprovides enough confinement for a minimizing sequence to ensure the existence of minimizers. If W asymptotes to a finite value (case treated in Theorem 3.2) then there is a delicate interplay betweenrepulsion at some lengths (in most applications short lengths) and attraction at other length scales(typically long) which establishes whether the repulsion wins and a minimizing sequence spreadsout indefinitely and “vanishes” or the minimizing sequence is compact and has a limit. We establish XISTENCE OF GROUND STATES OF NONLOCAL-INTERACTION ENERGIES 3 a simple, sharp condition, ( HE ) on the energy that characterizes whether a ground state exists. Toestablish compactness of a minimizing sequence we use Lions’ concentration compactness lemma.While the conditions ( H1 ) and ( H2 ) are easy-to-check conditions on the potential W itself, thecondition ( HE ) is a condition on the energy and it is not always easy to verify. Due to the aboveconnection with statistical mechanics the conditions on H -stability (or the lack thereof) can beused to verify if ( HE ) is satisfied for a particular potential. We list such conditions in Section 4.However only few general conditions are available. It is an important open problem to establish amore complete characterization of potentials W which satisfy ( HE ).We finally remark that as this manuscript was being completed we learned that Ca˜nizo, Car-rillo, and Patacchini [12] independently and concurrently obtained very similar conditions for theexistence of minimizers, which they also show to be compactly supported. The proofs however arequite different. 2. Hypotheses and Preliminaries
The interaction potentials we consider are radially symmetric, that is, w ( x ) = W ( | x | ) for somefunction W : [0 , ∞ ) → R ∪ {∞} , and they satisfy the following basic properties:( H1 ) W is lower-semicontinuous.( H2 ) The function w ( x ) is locally integrable on R N .Beyond the basic assumptions above, the behavior of the tail of W will play an important role.We consider potentials which have a limit at infinity. If the limit is finite we can add a constantto the potential, which does not affect the existence of minimizers, and assume that the limit iszero. If the limit is infinite the proof of existence of minimizers is simpler, while if the limit isfinite an additional condition is needed. Thus we split the condition on behavior at infinity intotwo conditions:( H3a ) W ( r ) → ∞ as r → ∞ .( H3b ) W ( r ) → r → ∞ . Remark . By the assumptions ( H1 ) and ( H3a ) or (
H3b ) the interaction potential W is boundedfrom below. Hence(2.1) C W := inf r ∈ (0 , ∞ ) W ( r ) > −∞ . If (
H3a ) holds, by adding − C W to W from now on we assume that W ( r ) (cid:62) r ∈ (0 , ∞ )As noted in the introduction the assumptions ( H1 ), ( H2 ) with ( H3a ) or (
H3b ) allow us tohandle a quite general class of interaction potentials w . Figure 1 illustrates a set of simple examplesof smooth potential profiles W that satisfy these assumptions.In order to establish the existence of ground states of E , for interaction potentials w satisfying( H1 ), ( H2 ) and ( H3b ), the following assumption on the interaction energy E is needed:( HE ) There exists a measure ¯ µ ∈ P ( R N ) such that E (¯ µ ) (cid:54) H1 ), ( H2 ) and ( H3a ) or (
H3b ) imply the lower-semicontinuityof the energy with respect to weak convergence of measures. We recall that a sequence of probability
ROBERT SIMIONE, DEJAN SLEPˇCEV, AND IHSAN TOPALOGLU (cid:160) x (cid:164) W (cid:72)(cid:160) x (cid:164)(cid:76) (cid:160) x (cid:164) W (cid:72)(cid:160) x (cid:164)(cid:76) (a) Interaction potentials satisfying ( H1 ), ( H2 ), and ( H3a ) (cid:160) x (cid:164) W (cid:72)(cid:160) x (cid:164)(cid:76) (cid:160) x (cid:164) W (cid:72)(cid:160) x (cid:164)(cid:76) (b) Interaction potentials satisfying ( H1 ), ( H2 ), and ( H3b ) Figure 1.
Generic examples of W ( | x | ).measures µ n converges weakly to measure µ , and we write µ n (cid:42) µ , if for every bounded continuousfunction φ ∈ C b ( R N , R ) (cid:90) φdµ n → (cid:90) φdµ as n → ∞ . Lemma 2.2 (Lower-semicontinuity of the energy) . Assume W : [0 , ∞ ) → ( −∞ , ∞ ] is a lower-semicontinuous function bounded from below. Then the energy E : P ( R n ) → ( −∞ , ∞ ] defined in (1.1) is weakly lower-semicontinuous with respect to weak convergence of measures.Proof. Let µ n be a sequence of probability measures such that µ n (cid:42) µ as n → ∞ . Then µ n × µ n (cid:42)µ × µ in the set of probability measures on R N × R N . If w is continuous and bounded (cid:90) R N (cid:90) R N w ( x − y ) dµ n ( x ) dµ n ( y ) −→ (cid:90) R N (cid:90) R N w ( x − y ) dµ ( x ) dµ ( y ) as n → ∞ . So, in fact, the energy is continuous with respect to weak convergence. On the other hand, if w is lower-semicontinuous and w is bounded from below then the weak lower-semicontinuity of theenergy follows from the Portmanteau Theorem [40, Theorem 1.3.4]. (cid:3) We remark that the assumption on boundedness from below is needed since if, for example, W ( r ) = − r then for µ n = (1 − n ) δ + n δ n the energy is E ( µ n ) = − n ∈ N , while µ n (cid:42) δ which has energy E ( δ ) = 0.Finally, we state Lions’ concentration compactness lemma for probability measures [32], [36,Section 4.3]. We use this lemma to verify that an energy-minimizing sequence is precompact in thesense of weak convergence of measures. Lemma 2.3 (Concentration-compactness lemma for measures) . Let { µ n } n ∈ N be a sequence ofprobability measures on R N . Then there exists a subsequence { µ n k } k ∈ N satisfying one of the threefollowing possibilities: XISTENCE OF GROUND STATES OF NONLOCAL-INTERACTION ENERGIES 5 (i) (tightness up to translation)
There exists a sequence { y k } k ∈ N ⊂ R N such that for all ε > there exists R > with the property that (cid:90) B R ( y k ) dµ n k ( x ) (cid:62) − ε for all k .(ii) (vanishing) lim k →∞ sup y ∈ R N (cid:90) B R ( y ) dµ n k ( x ) = 0 , for all R > ; (iii) (dichotomy) There exists α ∈ (0 , such that for all ε > , there exist a number R > and a sequence { x k } k ∈ N ⊂ R N with the following property:Given any R (cid:48) > R there are nonnegative measures µ k and µ k such that (cid:54) µ k + µ k (cid:54) µ n k , supp( µ k ) ⊂ B R ( x k ) , supp( µ k ) ⊂ R N \ B R (cid:48) ( x k ) , lim sup k →∞ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) α − (cid:90) R N dµ k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) (1 − α ) − (cid:90) R N dµ k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:54) ε . Existence of Minimizers
In this section we prove the existence of a global minimizer of E . We use the direct method ofthe calculus of variations and utilize Lemma 2.3 to eliminate the “vanishing” and “dichotomy” ofan energy-minimizing sequence. The techniques in our proofs, though, depends on the behaviorof the interaction potential at infinity. Thus we prove two existence theorems: one for potentialssatisfying ( H3a ) and another one for those satisfying (
H3b ). Theorem 3.1.
Suppose W satisfies the assumptions ( H1 ), ( H2 ) and ( H3a ). Then the energy (1.1) admits a global minimizer in P ( R N ) .Proof. Let { µ n } n ∈ N be a minimizing sequence, that is, lim n →∞ E ( µ n ) = inf µ ∈P ( R N ) E ( µ ).Suppose { µ k } k ∈ N has a subsequence which “vanishes”. Since that subsequence is also a minimiz-ing sequence we can assume that { µ k } k ∈ N vanishes. Then for any ε > R > K ∈ N such that for all k > K and for all x ∈ R N µ k ( R N \ B R ( x )) (cid:62) − ε. This implies that for k > K , (cid:90) (cid:90) | x − y | (cid:62) R dµ k ( x ) dµ k ( y ) = (cid:90) R N (cid:32)(cid:90) R N \ B R ( x ) dµ k ( y ) (cid:33) dµ k ( x ) (cid:62) − ε. Given M ∈ R , by condition ( H3a ) there exists
R > r (cid:62) R , W ( r ) (cid:62) M . Consider ε ∈ (0 , ) and K corresponding to ε and R . Since W (cid:62) E ( µ k ) = (cid:90) (cid:90) | x − y |
Theorem 3.2.
Suppose W satisfies the assumptions ( H1 ), ( H2 ) and ( H3b ). Then the energy E ,given by (1.1) , has a global minimizer in P ( R N ) if and only if it satisfies the condition ( HE ).Proof. Let us assume that E satisfies condition ( HE ). As before, our proof relies on the directmethod of the calculus variations for which we need to establish precompactness of a minimizingsequence.Let { µ n } n ∈ N be a minimizing sequence and let I := inf µ ∈P ( R N ) E ( µ ) . XISTENCE OF GROUND STATES OF NONLOCAL-INTERACTION ENERGIES 7
Condition ( HE ) implies that I (cid:54)
0. If I = 0 then by assumption ( HE ) there exists ¯ µ with E (¯ µ ) = 0, which is the desired minimizer. Thus, we focus on case that I <
0. Hence there exists ¯ µ for which E (¯ µ ) <
0. Also note that by Remark 2.1,
I > −∞ .Suppose the subsequence { µ n k } k ∈ N of the minimizing sequence { µ n } n ∈ N “vanishes”. Since thatsubsequence is also a minimizing sequence we can assume that { µ k } k ∈ N vanishes. That is, for any R > k →∞ sup x ∈ R N (cid:90) B R ( x ) dµ k ( y ) = 0 . Let W ( R ) = inf r (cid:62) R W ( r ) . Since W ( r ) → r → ∞ , W ( r ) → r → ∞ and W ( r ) (cid:54) r (cid:62)
0. Then we have that E ( µ k ) = (cid:90) (cid:90) | x − y | >R W ( | x − y | ) dµ k ( x ) dµ k ( y ) + (cid:90) (cid:90) | x − y | (cid:54) R W ( | x − y | ) dµ k ( x ) dµ k ( y ) (cid:62) W ( R ) + C W (cid:90) (cid:90) | x − y | (cid:54) R dµ k ( x ) dµ k ( y )= W ( R ) + C W (cid:90) R N (cid:32)(cid:90) B R ( x ) dµ k ( y ) (cid:33) dµ k ( x ) . Vanishing of the measures, (3.1), implies that lim inf k →∞ E ( µ k ) (cid:62) W ( R ) for all R >
0. Taking thelimit as R → ∞ gives lim inf k →∞ E ( µ k ) (cid:62) . This contradicts the fact that the infimum of the energy, namely I , is negative. Therefore “vanish-ing” in Lemma 2.3 does not occur.Suppose the dichotomy occurs. Let α ∈ (0 ,
1) and
R > C W be theconstant defined in (2.1). Let ε > ε < | I | | C W | min (cid:26) α − , − α − (cid:27) and let R (cid:48) be such that(3.3) | W ( R (cid:48) − R ) | = | inf r (cid:62) R (cid:48) − R W ( r ) | < | I |
32 min (cid:26) α − , − α − (cid:27) . As in the proof of Theorem 3.1, we can assume that dichotomy occurs along the whole sequence.Let µ k and µ k be measures described in Lemma 2.3. Let ν k = µ k − ( µ k + µ k ). Note that ν k is anonnegative measure with | ν k | < ε , where | ν k | = ν k ( R N ).Let B [ · , · ] denote the symmetric bilinear form B [ µ, ν ] := 2 (cid:90) R N (cid:90) R N W ( | x − y | ) dµ ( x ) dν ( y ) . By the definition of energy E ( µ k ) = E ( µ k ) + E ( µ k ) + B ( µ k , µ k ) + B ( µ k + µ k , ν k ) + E ( ν k ) (cid:62) E ( µ k ) + E ( µ k ) − | W ( R (cid:48) − R ) | − | C W | ε (3.4) ROBERT SIMIONE, DEJAN SLEPˇCEV, AND IHSAN TOPALOGLU where we used that the supports of µ k and µ k are at least R (cid:48) − R apart. We can also assume,without loss of generality, that E ( µ k ) < I for all k . Let α k = | µ k | , β k = | µ k | .Let us first consider the case that α k E ( µ k ) (cid:54) β k E ( µ k ). Note that the energy has the followingscaling property: E ( cσ ) = c E ( σ )for any constant c > σ . Our goal is to show that for some λ >
0, for all large enough k , E ( α k µ k ) < E ( µ k ) − λ | I | which contradicts the fact that µ k is a minimizing sequence.Let us consider first the subcase that E ( µ k ) (cid:62) α k E ( µ k ) < I k . Using the estimates again, we obtain E ( µ k ) − E (cid:18) α k µ k (cid:19) (3.4) (cid:62) (cid:18) − α k (cid:19) E (cid:0) µ k (cid:1) − | W ( R (cid:48) − R ) | − | C W | ε (3.5) (cid:62) (cid:18) α k − (cid:19) | I | − | W ( R (cid:48) − R ) | − | C W | ε (3.2) , (3.3) (cid:62) (cid:18) α − (cid:19) | I | . Thus µ k is not a minimizing sequence. Contradiction.Let us now consider the subcase E ( µ k ) (cid:54) k . Using (3.4) and β k α k E ( µ k ) (cid:54) E ( µ k ) weobtain I (cid:62) E ( µ k ) (cid:62) (cid:18) β k α k (cid:19) E ( µ k ) − | W ( R (cid:48) − R ) | − | C W | ε. From (3.2) and (3.3) follows that for all k α k E ( µ k ) (cid:54) I . Combining with above inequalities gives E ( µ k ) − E (cid:18) α k µ k (cid:19) (cid:62) (cid:18) β k α k − α k (cid:19) E ( µ k ) − | W ( R (cid:48) − R ) | − | C W | ε (cid:62) (cid:18) α k − α k − β k (cid:19) | I | − (cid:18) α − (cid:19) (cid:18) | I |
32 + | I | (cid:19) (cid:62) | I | (cid:18) α − (cid:19) for k large enough. This contradicts the assumption that µ k is a minimizing sequence.The case α k E ( µ k ) > β k E ( µ k ) is analogous. In conclusion the dichotomy does not occur. There-fore “tightness up to translation” is the only possibility. As in the proof of Theorem 3.1, we cantranslate measures µ n k to obtain a tight, energy-minimizing sequence ˜ µ n k .By Prokhorov’s theorem, there exists a further subsequence of { ˜ µ n k } k ∈ N , still indexed by k , suchthat µ n k (cid:42) µ as k → ∞ XISTENCE OF GROUND STATES OF NONLOCAL-INTERACTION ENERGIES 9 for some measure µ ∈ P ( R N ) in P ( R N ) as k → ∞ . Therefore, by lower-semicontinuity of theenergy, µ is a minimizer of E in the class P ( R N ).We now show the necessity of condition ( HE ). Assume that E ( µ ) > µ ∈ P ( R N ). Toshow that the energy E does not have a minimizer consider a sequence of measures which “vanishes”in the sense of Lemma 2.3(ii). Let ρ ( x ) = 1 ω N χ B (0) ( x ) , where ω N denotes the volume of the unit ball in R N and χ B R (0) denotes the characteristic functionof B R (0), the ball of radius R centered at the origin. Consider the sequence ρ n ( x ) = 1 n N ρ (cid:16) xn (cid:17) for n (cid:62)
1. Note that ρ n are in P ( R N ). We estimate0 < E ( ρ n ) = 1 ω N n N (cid:90) B n (0) (cid:90) B n (0) W ( | x − y | ) dxdy (cid:54) ω N n N (cid:90) B n (0) (cid:32)(cid:90) B n ( y ) | W ( | x | ) | dx (cid:33) dy (cid:54) ω N n N (cid:32)(cid:90) B R (0) | W ( | x | ) | dx + (cid:90) B n (0) \ B R (0) | W ( | x | ) | dx (cid:33) (cid:54) C ( R ) ω N n N + 2 N ω N sup r (cid:62) R | W ( r ) | . Since sup r (cid:62) R | W ( r ) | → R → ∞ , for any ε > R so that N ω N sup r (cid:62) R | W ( r ) | < ε .We can then choose n large enough for C ( R ) ω N n N < ε to hold. Therefore lim n →∞ E ( ρ n ) = 0, that is,inf µ ∈P ( R N ) E ( µ ) = 0. However, since E ( · ) is positive for any measure in P ( R N ) the energy doesnot have a minimizer. (cid:3) Stability and Condition (HE)
The interaction energies of the form (1.1) have been an important object of study in statisticalmechanics. For a system of interacting particles to have a macroscopic thermodynamic behaviorit is needed that it does not accumulate mass on bounded regions as the number of particles goesto infinity. Ruelle called such potentials stable (a.k.a. H -stable). More precisely, a potential W : [0 , ∞ ) → ( −∞ , ∞ ] is defined to be stable if there exists B ∈ R such that for all n and for allsets of n distinct points { x , . . . , x n } in R N (4.1) 1 n (cid:88) (cid:54) i Proposition 4.1 (Stability conditions) . Let W : [0 , ∞ ) → R be an upper-semicontinuous functionsuch that W is bounded from above or there exists R such that W is nondecreasing on [ R, ∞ ) . Thenthe conditions ( S1 ) w is a stable potential as defined by (4.1) , ( S2 ) for any probability measure µ ∈ P ( R N ) , E ( µ ) (cid:62) are equivalent. Note that all potentials considered in the proposition are finite at 0. We expect that the conditioncan be extended to a class of potentials which converge to infinity at zero. Doing so is an openproblem. We also note that the condition ( S2 ) is not exactly the complement of ( HE ), as thenonnegative potentials whose minimum is zero satisfy both conditions. Such potentials indeedexist: for example consider any smooth nonnegative W such that W (0) = 0. Then the associatedenergy is nonnegative and E ( δ ) = 0 so any singleton is an energy minimizer. Note that E satisfiesboth ( HE ) and stability. To further remark on connections with statistical mechanics we note thatsuch potentials W are not super-stable , but are tempered if W decays at infinity (both notions aredefined in [35, Chapter 3]). Proof. To show that ( S2 ) implies ( S1 ) consider µ = n (cid:80) ni =1 δ x i . Then from E ( µ ) (cid:62) n (cid:80) (cid:54) i 0. We can assume that ε < . There exists R > Q R = [ − R, R ] N , µ X ( R N \ Q R ) < ε . For integer l such that √ N Rl < ε divide Q R into l N disjoint cubes Q i , i =1 , . . . , l N with sides of length 2 R/l . While cubes have the same interiors, they are not requiredto be identical, namely some may contain different parts of their boundaries, as needed to makethem disjoint. Note that the diameter of each cube, √ N Rl , is less than ε . Let n > n be suchthat l N n < ε . Let p = n . For i = 1 , . . . , l N let p i = µ ( Q i ), n i = (cid:98) p i n (cid:99) , and q i = n i p . Note that0 (cid:54) p i − q i (cid:54) p and thus s q = (cid:80) i q i (cid:62) (cid:80) i p i − l N p > − ε . In each cube Q i place n i distinctpoints and let ˜ X be the set of all such points. Note that ˜ n = (cid:80) i n i = s q n > (1 − ε ) n . Let ˆ X bean arbitrary set of n − ˜ n distinct points in Q R \ Q R . Let X = ˜ X ∪ ˆ X . Note that X is a set of n distinct points. Then for any Borel set Aµ ( A ) (cid:54) (cid:88) i : µ ( A ∩ Q i ) > µ ( Q i ) + ε (cid:54) (cid:88) i : µ ( A ∩ Q i ) > ( µ X ( Q i ) + p ) + ε (cid:54) µ X ( A + ε ) + ε. Similarly µ X ( A ) (cid:54) µ ( A + ε ) + ε. Therefore d LP ( µ, µ X ) (cid:54) ε .Consequently there exists a sequence of sets X m with n ( m ) points satisfying n ( m ) → ∞ as m → ∞ for which the empirical measure µ m = µ X m converges weakly µ m (cid:42) µ as m → ∞ . By XISTENCE OF GROUND STATES OF NONLOCAL-INTERACTION ENERGIES 11 assumption (S1) (cid:90) (cid:90) x (cid:54) = y W ( x − y ) dµ m ( x ) dµ X m ( y ) (cid:62) − n ( m ) B. Let us first consider the case that W is an upper-semicontinuous function bounded from above. Itfollows from Lemma 2.2 that the energy E is an upper-semicontinuous functional. Therefore E ( µ ) (cid:62) lim sup m →∞ E ( µ m ) (cid:62) lim sup m →∞ − n ( m ) ( B − W (0)) = 0as desired.If W is an upper-semicontinuous function such that there exists R such that W is nondecreasingon [ R, ∞ ) we first note that we can assume that W ( r ) → ∞ as r → ∞ , since otherwise W is boundedfrom above which is covered by the case above. If µ is a compactly supported probability measurethen there exists L such that for all m , supp µ m ⊆ [ − L, L ] N . Since W is upper-semicontinuous itis bounded from above on compact sets and thus upper-semicontinuity of the energy holds. Thatis E ( µ ) (cid:62) lim sup m →∞ E ( µ m ) (cid:62) µ is not compactly supported it suffices to show that there exists a compactly supportedmeasure ˜ µ such that E ( µ ) (cid:62) E (˜ µ ), since by above we know that E (˜ µ ) (cid:62) 0. Note that since E ( ( δ x + δ )) (cid:62) W ( | x | ) (cid:62) − W (0). Therefore W is bounded from below by − W (0) and W (0) (cid:62) W ( r ) → ∞ as r → ∞ there exists R (cid:62) R such that W ( R ) (cid:62) max { , max r (cid:54) R W ( r ) } and m = µ ( B R (0)) > . Let R be such that W ( R ) > W ( R ), and define the constants m = µ ( B R (0) \ B R (0)) and m = µ ( R N \ B R (0)). Note that m + m + m = 1. Consider themapping P ( x ) = (cid:40) x if | x | (cid:54) R | x | > R . Let ˜ µ = P (cid:93) µ . Estimating the interaction of particles between the regions provides: E (˜ µ ) (cid:54) E ( µ ) + 2 W (0) m + 2( W ( R ) + W (0)) m m − W ( R ) − W ( R )) m m (cid:54) E ( µ ) + W ( R ) m ( m + 4 m − m ) < E ( µ ) . (cid:3) As we showed in Theorem 3.2 the property ( HE ) is necessary and sufficient for the existenceof ground states when E is defined via an interaction potential satisfying ( H1 ), ( H2 ) and ( H3b ).The property ( HE ) is posed as a condition directly on the energy E , and can be difficult to verifyfor a given W . It is then natural to ask what conditions the interaction potential W needs to satisfyso that the energy E has the property ( HE ). In other words, how can one characterize interactionpotentials w for which E admits a global minimizer?We do not address that question in detail, but just comment on the partial results established inthe context of H -stability of statistical mechanics and how they apply to the minimization of thenonlocal-interaction energy.Perhaps the first condition which appeared in the statistical mechanics literature states thatabsolutely integrable potentials which integrate to a negative number over the ambient space arenot stable (cf. [20, Theorem 2] or [35, Proposition 3.2.4]). In our language these results translateto the following proposition. Proposition 4.2. Consider an interaction potential w ( x ) = W ( | x | ) where W satisfies the hypothe-ses ( H1 ), ( H2 ) and ( H3b ). If w is absolutely integrable on R N and (cid:90) R N W ( | x | ) dx < , then the energy E defined by (1.1) satisfies the condition ( HE ).Proof. By rescaling, we can assume that (cid:82) R N W ( | x | ) dx = − 1. Let M = (cid:82) R N | W ( x ) | dx . Let R besuch that (cid:82) B R (0) W ( | x | ) dx < − and (cid:82) B R (0) c | W ( | x | ) | dx < . Consider n large, to be set later,and let ρ ( x ) := ω N ( nR ) N χ B nR (0) ( x ), i.e., the scaled characteristic function of the ball of radius nR .Using the fact that B R (0) ⊂ B nR ( y ) for | y | < ( n − R , we obtain ω N ( nR ) N E ( ρ ) = (cid:90) B nR (0) (cid:90) B nR (0) W ( | x − y | ) dxdy = (cid:90) B nR (0) (cid:32)(cid:90) B nR ( y ) W ( | x | ) dx (cid:33) dy (cid:54) (cid:90) B ( n − R (0) (cid:32)(cid:90) B R (0) W ( | x | ) dx + 14 (cid:33) dy + (cid:90) B nR (0) \ B ( n − R (0) M dy (cid:54) − 12 ( n − N R N ω N + N ω N n N − R N M < n is large enough. This shows that the energy E satisfies ( HE ). (cid:3) An alternative condition for instability of interaction potentials is given in [13, Section II]. Thiscondition, which we state and prove in the following proposition, extends the result of Proposition4.2 to interaction potentials which are not necessarily absolutely integrable. Proposition 4.3. Suppose the interaction potential W satisfies the hypotheses ( H1 ), ( H2 ) and( H3b ). If there exists p (cid:62) for which (4.2) (cid:90) R N W ( | x | ) e − p | x | dx < , then the energy E defined by (1.1) satisfies the condition ( HE ).Proof. Let p (cid:62) p = 0 has beenconsidered in Proposition 4.2, we can assume p > 0. Consider the function ρ ( x ) = p N π N/ e − p | x | . Clearly ρ ∈ L ( R N ) and (cid:107) ρ (cid:107) L ( R N ) = 1; hence, it defines a probability measure on R N . Considerthe linear transformation on R N given by u = x − y, v = x + y. XISTENCE OF GROUND STATES OF NONLOCAL-INTERACTION ENERGIES 13 We note that the Jacobian of the transformation is 2. Thus E ( ρ ) = (cid:90) R N (cid:90) R N W ( | x − y | ) e − p | x | e − p | y | dxdy = 12 (cid:90) R N (cid:90) R N W ( | u | ) e − p | u + v | / e − p | u − v | / dudv = 12 (cid:90) R N (cid:90) R N W ( | u | ) e − p ( | u | + | v | ) dudv = 12 (cid:90) R N (cid:18)(cid:90) R N W ( | u | ) e − p | u | du (cid:19) e − p | v | dv < . Hence, the energy E satisfies ( HE ). (cid:3) Remark . Another useful criterion can be obtained by using the Fourier transform, as also notedin [35]. Namely if w ∈ L ( R N ), for measure µ that has a density ρ ∈ L ( R N ), by Plancharel’stheorem E ( µ ) = (cid:90) R N (cid:90) R N w ( x − y ) dµ ( x ) dµ ( y ) = (cid:90) R N ˆ w ( ξ ) | ˆ ρ ( ξ ) | dξ. So if the real part of ˆ w is positive, the energy does not have a minimizer.This criterion can be refined. By Bochner’s theorem the Fourier transforms of finite nonnegativemeasures are precisely the positive definite functions. Thus we know which family of functions,ˆ ρ belongs to. 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