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ANTICIPATION BREEDS ALIGNMENT

RUIWEN SHU AND EITAN TADMOR

Abstract. We study the large-time behavior of systems driven by radial potentials, whichreact to anticipated positions, xτ ptq xptq τvptq, with anticipation increment τ ¡ 0.As a special case, such systems yield the celebrated Cucker-Smale model for alignment,coupled with pairwise interactions. Viewed from this perspective, such anticipated-drivensystems are expected to emerge into flocking due to alignment of velocities, and spatialconcentration due to confining potentials. We treat both the discrete dynamics and largecrowd hydrodynamics, proving the decisive role of anticipation in driving such systemswith attractive potentials into velocity alignment and spatial concentration. We also studythe concentration effect near equilibrium for anticipated-based dynamics of pair of agentsgoverned by attractive-repulsive potentials.

Contents

1. Introduction and statement of main results 12. A priori L8 bounds for confining potentials 93. Anticipation with convex potentials and positive kernels 134. Local vs. global weighted means 185. Anticipation dynamics with attractive potentials 226. Anticipation dynamics with repulsive-attractive potential 247. Anticipation dynamics: hydrodynamic formulation 29References 32

1. Introduction and statement of main results

Consider the dynamical system"9xiptq viptq9viptq ∇iHNpxτ1, . . . ,xτNq, xτi xτi ptq : xiptq τviptq, i 1, . . . , N.

When τ 0, this is the classicalN -particle dynamics for positions and velocities, pxiptq,viptqq PpRd,Rdq, governed by the general Hamiltonian HNp q. If we fix a small time step τ ¡ 0,

Date: July 29, 2020.1991 Mathematics Subject Classification. 82C21, 82C22, 92D25, 35Q35.Key words and phrases. anticipation, alignment, attractive-repulsive potentials, hypocoercivity, hydrody-namics, critical thresholds.Acknowledgment. Research was supported by NSF grants DMS16-13911, RNMS11-07444 (KI-Net) andONR grant N00014-1812465. ET thanks the hospitality of the Institut of Mittag-Leffler during fall 2018 visitwhich initiated this work, and of the Laboratoire Jacques-Louis Lions in Sorbonne University during spring2019, with support through ERC grant 740623 under the EU Horizon 2020, while concluding this work.We thank the anonymous reviewers for careful reading and insightful comments which help improving anearlier draft of the paper.

1

2 RUIWEN SHU AND EITAN TADMOR

then the system is not driven instantaneously but reacts to the positions xτ ptq xptqτvptq,anticipated at time t τ , where τ is the anticipation time increment. Anticipation is amain feature in social dynamics of N -agent and N -player systems, [GLL2010, MCEB2015,GTLW2017]. A key feature in the the large time behavior of such anticipated dynamics isthe dissipation of the anticipated energy

Eptq 1

2N

¸i

|vi|2 1

NHNpxτ1, . . . ,xτNq,

at a rate given by

d

dtEptq 1

N

¸i

vi 9vi 1

N

¸i

∇iHNpxτ1, . . . ,xτNq pvi τ 9viq τ

N

¸i

| 9vi|2, τ ¡ 0.

We refer to the quantity on the right, τN

°i | 9vi|2, as the enstrophy of the system.

1.1. Pairwise interactions. In this work we study the anticipation dynamics of pairwiseinteractions

(AT)

$'&'%9xiptq viptq

9viptq 1

N

N

j1

∇Up|xτi xτj |q, xτi xiptq τviptq,i 1, . . . , N,

governed by a radial interaction potential Uprq, r |x|. This corresponds to the Hamiltonian

HNpxτ1, . . . ,xτNq 1

2N

¸j,k

Up|xτj xτk|q, where the conservative N -body problem (τ 0) is

now replaced by N -agent dynamics with anticipated energy dissipation

d

dtEptq τ

N

¸i

| 9vi|2, Eptq : 1

2N

¸i

|vi|2 1

2N2

¸i,j

Up|xτi xτj |q, τ ¡ 0.(1.1)

To gain a better insight into (AT), we consider the general system

(ΦU)

$'&'%9xi vi

9vi τ

N

N

j1

Φijpvj viq 1

N

N

j1

∇Up|xi xj|q, i 1, . . . , N.

The anticipation (AT) is recovered as a special case of (ΦU) in terms of the ‘intermediate’

Hessians, Φij D2U ij, but since these intermediate Hessians are not readily available1, wewill consider (ΦU) for a general class of communication matrices, Φij P Φ, which respect the

symmetry property satisfied by the D2U ij’s,

Φ : Φp, q P Symdd | Φij : Φ

pxi,viq, pxj,vjq Φji

(.

The so-called ΦU system provides a unified framework for anticipation dynamics by couplinggeneral symmetric communication matrices, tΦ P Φu, together with pairwise interactionsinduced by the potential U . In the particular case of U 0, the general system (ΦU)

1Expanding (AT) in τ we obtain (ΦU) with Φij D2U ij :

» 1

s0

D2Up|pxi xjq τspvi vjq|qds, and

its entries can be written as pD2U ijqpq D2Up|xipt; τpqij q xjpt; τ

pqij q|q in anticipated positions, xpt; τpqij q

x τpqij v, at some intermediate times, τpqij P r0, τ s.

ANTICIPATION BREEDS ALIGNMENT 3

yields the celebrated Cucker-Smale (CS) model [CS2007a, CS2007b], 9vi τN

°j Φijpvjviq,

a prototypical model for alignment dynamics in which maxi,j |viptq vjptq| tÑ8ÝÑ 0. Thereis, however, one distinct difference: while the CS model is governed by a scalar kernelinvolving geometric distances, Φij φp|xi xj|qIdd, here (ΦU) allows for larger class ofcommunication protocols based on matrix kernels, e.g., Φij Φpxi,xjq, with a possibledependence on topological distances, [ST2018b]. The flocking behavior of such matrix-basedCS models, proved in section 3.1, is considerably more intricate than in the scalar case, dueto the lack of a maximum principle.

The main purpose of this paper is to study the decisive role of anticipation in driving theemergent behavior of (AT) and (ΦU), proving, under appropriate assumptions, flocking andspatial concentration, see (1.9),(1.18) below,

|xiptq sx0 tsv0

| |viptq sv0| tÑ8ÝÑ 0.

Viewed from the perspective of Cucker-Smale alignment dynamics, the large time flocking be-havior of (AT),(ΦU) is expected due to alignment of velocities. Moreover, our study [ST2020]shows that confinement due to external forcing, ∇V p|xi|q, leads to spatial concentration,and it is known, e.g., [ST2020, p. 351], that the dynamics with external forcing coincideswith pairwise interactions, 1

N

°j ∇Up|xi xj|q, in the special case of quadratic potential

V prq 12r2. From this perspective, here we prove spatial concentration for a (much) larger

class of attractive potentials U .We begin in section 3 with the general system (ΦU). The basic bookkeeping associated

with (ΦU) quantifies its decay rate of the (instantaneous) energy

Eptq : 1

2N

¸i

|vi|2 1

2N2

¸i,j

Up|xi xj|q,

which is given by

d

dtEptq τ

2N2

¸i,j

pvj viqJΦijpvj viq.(1.2)

This will be contrasted with the dissipation of anticipated energy (1.1) in section 5 below.To explore the enstrophy on the right of (1.2) we need to further elaborate on properties ofthe communication matrices, Φij Φp, q, and their relations to the potential U .

We start by rewriting the Hessian of the radial potential in the form

(1.3) D2Up|xixj|q U 1prijqrij

pIpxijpxJijqU2prijqpxijpxJij, rij : |xixj|, pxij : xi xjrij

,

and observe that the symmetric matrix D2Up|xixj|q has a single eigenvalue U2prijq in the

radial direction, xi xj, and d 1 multiple of the eigenvaluesU 1prijqrij

in tangential directions

pxi xjqK. We study the dynamics induced by potentials U which are at least C2. As aresult, U 1p0q 0, and we may assume Up0q 0 by adding a constant to it. We specifytwo main classes of potentials we will be working with: convex potentials2, U2prq Á xry2β,

studied in section 3, and the larger class of attractive potentials, U 1prqr

Á xry2β, studied insection 5. In either case, we postulate that the potential is bounded, in the sense that

(1.4) there exists a constant A ¡ 0 such that |U2prq| ¤ A.

2Throughout this paper, we use the notation xrys

: p1 r2qs2 for scalar r and xzy x|z|y for vectors z.


It follows that |U 1prq| ¤» r

0

|U2psq| ds ¤» r

0

A ds Ar, and hence that the communication

matrices are bounded,

AIdd D2Upq ¤ AIdd.In particular, this rules out the important class of singular kernels (in both first- and second-order dynamics), e.g., [Ja2014, Go2016, CCTT2016, PS2017, ST2017, CCP2017, Se2018,ST2018, DKRT2018, MMPZ2019], which is left for a future study. Finally, we mention thelarger class of confining potentials, (2.1), which includes the repulsive-attractive potentialsstudied in section 6.

1.2. Anticipation dynamics with convex potentials. Recall that the flocking behaviorof CS model , see (3.8) below, is guaranteed for scalar communication kernels, Φprq φprqI,which satisfy a so-called fat tail condition, [HL2009],[MT2014, Proposition 2.9],»

φprq dr 8,

or — expressed in terms of its decay rate, φprq xryβ for 0 ¤ β ¤ 1. Since the anticipationmodel (AT) can be viewed as a special case of (ΦU), it is natural to quantify the convexityof U and positivity of Φ in terms of their ‘fat tail’ decay.

Assumption 1.1 (Convex potentials). There exist constants 0 a A and β such that

(1.5) axryβ ¤ U2prq ¤ A, 0 ¤ β ¤ 1.

Typically, the upper and lower bounds associated with U and its derivatives dictate its profilenear the origin, r ! 1 and respectively, near infinity, r " 1. It particular, the lower bound in

(1.5) implies U 1prq » r

0

U2psq ds ¥» r

0

axsyβ ds ¥ axryβr, and hence D2U in (1.3) satisfies

the fat tail condition D2Up|x|q ¥ axxyβ with 0 ¤ β ¤ 1.

Assumption 1.2 (Positive kernels). There exist constants 0 φ φ and γ such that

(1.6) φpxxi xjy xvi vjyqγ ¤ Φij ¤ φ, 0 ¤ γ 1.

Observe that (ΦU) conserves momentum

(1.7)

$'''&'''%9sx sv, sx : 1

N

¸i

xi,

9sv 0, sv : 1

N

¸i

vi.

It follows that the mean velocity sv is constant in time, svptq sv0, and hence sxptq sx0 tsv0.Our first main result is expressed in terms of the energy fluctuations

δEptq : 1

2N

¸i

|vi sv|2 1

2N2

¸i,j

Up|xi xj|q.

Theorem 1 (Anticipation dynamics (ΦU) — velocity alignment and spatial con-centration). Consider the anticipation dynamics (ΦU). Assume a bounded convex potentialU with fat-tail decay of order β, (1.5), and a symmetric kernel matrix Φ with a fat-tail decay


of order γ, (1.6). If the decay parameters lie in the restricted range 3β 2 maxtβ, γu 4,then there is sub-exponential decay of the energy fluctuations

(1.8) δEptq ¤ Cet1η

, η 2 maxtβ, γu4 3β

1.

We conclude that for large time, the dynamics concentrates in space with global velocityalignment at sub-exponential rate,

(1.9) |xiptq sxptq| Ñ 0, |viptq sv0| Ñ 0, sxptq sx0 tsv0.

The proof of Theorem 1 proceeds in two steps:(i) A uniform bound, outlined in lemma 2.1 below, on maximal spread of positions |xiptq|,(1.10) max

i|xiptq| ¤ C8xty

243β , max

i|viptq| ¤ C8xty

2β43β , 0 ¤ β ¤ 1.

(ii) Observe that in view of (1.7), ddtδEptq d

dtEptq. The energy dissipation (1.2) combined

with the bounds (1.6),(1.10) imply the decay of energy fluctuations

d

dtδEptq d

dtEptq À τ

2Nxty 2γ

43β

¸i

|vi sv|2.To close the last bound we need a hypocoercivity argument carried out in section 3, whichleads to the sub-exponential decay (1.8). The conclusion of sub-exponential flocking

pxi xj,vi vjq tÑ8ÝÑ 0 follows, and naturally, pxi,viq psxptq, sv0q Ñ 0 since this is theonly minimizer of δEptq.

Since the anticipation dynamics (AT) can be viewed as a special case of ΦU system with

intermediate Hessians Φij D2U ij (outlined in footnote 1), theorem 1 applies with γ β.

Corollary 1.1 (Anticipation dynamics (AT) with convex potentials). Consider theanticipated dynamics (AT) with bounded convex potential satisfying

axryβ ¤ U2prq ¤ A, 0 ¤ β 4

5.

Then there is sub-exponential decay of the energy fluctuations


, η 2β

4 3β.

The large time flocking behavior follows: the dynamics concentrates in space with globalvelocity alignment at sub-exponential rate,

(1.12) |xiptq sxptq| Ñ 0, |viptq sv0| Ñ 0, sxptq : sx0 tsv0.

Remark 1.1 (Optimal result with improved fat-tail condition). Suppose we strengthen

assumption 1.1 with a more precise behavior of U2prq xryβ, thus replacing (1.5) with therequirement that there exist constants 0 a A and β such that

(1.13) axryβ ¤ U2prq, U 1prq ¤ Axry1β, 0 ¤ β ¤ 1.

Then the anticipation dynamics (ΦU) with a fat-tail kernel matrix Φ of order γ, (1.6),satisfies the sub-exponential decay


, η min!

1,2

4 3β

)maxtβ, γu 1.


This improved decay follows from the corresponding improvement of the uniform bound inlemma 2.1 below which reads maxi |xiptq| À xty. In the particular case of β γ, we recoveran improved corollary 1.1 for anticipated dynamics (AT), where the anticipated energy (see(1.16) below), satisfies an optimal decay of order

δEptq ¤ Cet1η

, η min! 2β

4 3β, β) 1.

1.3. Anticipation dynamics with purely attractive potential. We now turn our at-tention to the main anticipation model (AT). We already know the flocking behavior of (AT)for convex potentials, from the general considerations of the ΦU system, summarized in corol-lary 1.1. In fact, the corresponding communication matrix of (AT) prescribed in (1.3), D2U ,

has a special structure of rank-one modification of the scalar kernelU 1prqr

. This enables us

to treat the flocking behavior of (AT) for a larger class of purely attractive potentials.

Assumption 1.3 (Purely attractive potentials). There exist constants 0 a A andβ such that

(1.15) axryβ ¤ U 1prqr

, |U2prq| ¤ A, 0 ¤ β ¤ 1.

Our result is expressed in terms of fluctuations of the anticipated energy

(1.16) δEptq : 1

2N

¸i

|vi sv|2 1

2N2

¸i,j

Up|xτi xτj |q.

Theorem 2 (Anticipation dynamics (AT) with attractive potential). Consider theanticipation dynamics (AT), and assume a bounded, purely attractive potential with a fat taildecay of order β, (1.15). If the decay parameter β 1

3, then there is sub-exponential decay

of the anticipated energy fluctuations


, η 2β

1 β 1.

It follows that for large time, the anticipation dynamics concentrates in space with globalvelocity alignment at sub-exponential rate,

(1.18) |xiptq sxptq| Ñ 0, |viptq sv0| Ñ 0, sxptq sx0 tsv0.

Remark 1.2. This result is surprising if one interprets (AT) in its equivalent matrix for-mulation (ΦU), since attractive potentials do not necessarily induce communication matrixΦ D2U which is positive definite. In particular, the corresponding ‘regular’ (instantaneous)energy Eptq referred to in corollary 1.1 is not necessarily decreasing; only the anticipatedenergy is.

The proof of Theorem 2, carried out in section 5, involves two main ingredients.(i). First, we derive an a priori uniform bound on the maximal spread of anticipated positions|xτi ptq|,

(1.19) maxi|xτi ptq| ¤ C8xty

122β , 0 ¤ β 1.


(ii). A second main ingredient for the proof of Theorem 2 is based on the energy dissipation(1.1). The key step here is to relate the enstrophy in (1.1),

(1.20)τ

N

¸i

| 9vi|2 τ

N

¸i

1

N

¸j

cijpxτi xτj q2, cij

U 1p|xτi xτj |q|xτi xτj |

,

to the fluctuations of the (expected) positions. This is done by the following proposition,interesting for its own sake, which deals with the local vs. global means of arbitrary zj P Rd.

Lemma 1.1 (Local and global means are comparable). Fix 0 λ ¤ Λ and weights cij

0 λ ¤ cij ¤ Λ.

Then, there exists a constant C Cpλ,Λq ¤ 32Λ2

λ4such that for arbitrary zj P Rd there holds

(1.21)1

N2

¸i,j

|zi zj|2 ¤ Cpλ,ΛqN

¸i

1

N

¸j

cijpzi zjq2, CpΛ, λq ¤ 32

Λ2

λ4.

Remark 1.3 (Why a lemma on means?). The sum on the left of (1.21) quantifies thefluctuations relative to the average sz : 1

N

°j zj,

1

N2

¸i,j

|zi zj|2 2

N

¸i

|zi sz|2 .Hence, (1.21) implies — and in fact is equivalent up to scaling, to the statement about thelocal means induced by weights θij

λ

N¤ θij ¤ Λ

N,

¸j

θij 1.

If szipθq :¸j

θijzj are the local means, then (1.21) with cij Nθij implies

(1.22)1

N

¸i

|zi sz|2 ¤ 16Λ2

λ4

1

N

¸i

|zi szipθq|2.Thus, the deviation from the local means is comparable to the deviation from the global mean.

Applying (1.21) to (1.20) with the given bounds (1.15),(1.19), yields

d

dtEptq τ

N

¸i

| 9vi|2

À τ

N2

A2

a4

maxi,j

xxτi xτj y4β¸

i,j

|xτi xτj |2 À τ

2N2xty 2β

1β

¸i,j

|xτi xτj |2.(1.23)

Observe that in this case, the enstrophy of the anticipated energy is bounded by the fluctua-tions of the anticipated positions (compared with velocity fluctuations in the ‘regular’ energydecay (1.2)). We close the last bound by hypocoercivity argument carried out in section 5which leads to the sub-exponential decay (1.17).


1.4. Anticipation dynamics with attractive-repulsive potential. For attractive-repulsivepotentials, the large time behavior of (AT) is significantly more complicated, due to the fol-lowing two reasons: The topography of the total potential energy 1

2N2

°i,j Up|xixj|q which includes multiple

local minima with different geometric configurations could be very complicated, see e.g.,[LRC2000, DCBC2006, CDP2009, KSUB2011, BCLR2013, CHM2014, Se2015, CCP2017]and the references therein. It is numerically observed in [GTLW2017] that the decay of Eptq is of order Opt1q. There-

fore, one cannot expect for sub-exponential energy dissipation rate, 9Eptq À xtyηEptq, orthat its hypocoercivity counterpart will hold.

Here we focus on the second difficulty, and give a first rigorous result in this direction.

Theorem 3 (Anticipation with repulsive-attractive potential). Consider the 2D an-ticipated dynamics (AT) of N 2 agents subject to repulsive-attractive potential which hasa local minimum at r r0 ¡ 0 where U2pr0q a ¡ 0. Then there exists a constant ε ¡ 0,such that if the initial data is close enough to equilibrium,

(1.24)|x1p0q x2p0q| r0

2 |v1p0q v2p0q|2 ¤ ε,

then the solution to (AT) satisfies the following algebraic decay:

(1.25)|x1ptq x2ptq| r0

¤ Cxty1 ln12 x1 ty, |v1ptq v2ptq| ¤ Cxty12.

The proof, based on nonlinear hypocoercivity argument for the anticipated energy is car-ried out in section 6.

Remark 1.4. The detailed description of the dynamics outlined in the proof, reveals thatthe radial component of the velocity, vr À xty1 ln12 x1 ty, decays faster than its tangential

part, vθ À xty12. Therefore, although the dynamics of (6.1) can be complicated at thebeginning, it will finally settles as a circulation around the equilibrium, provided the initialdata is close enough to equilibrium.

1.5. Anticipation hydrodynamics. The large crowd (hydro-)dynamics associated with(AT) is described by density and momentum pρ, ρuq governed by3

(1.26)

$&%ρt ∇x pρuq 0

pρuqt ∇x pρub uq »∇Up|xτ yτ |q dρpyq, xτ : x τupt,xq.

The large-time flocking behavior of (1.26) is studied in terms of lemma 4.1 — a continuumversion of the discrete lemma of means proved in section 4. That is, we obtain a sub-lineartime bound on the spread of supp ρpt, q, which in turn is used to control the enstrophy of theanticipated energy. In section 7 we outline the proof of our last main result, which states thatif (1.26) admits a global smooth solution then such smooth solution must flock, in agreementwith the general paradigm for Cucker-Smale dynamics discussed in [TT2014, HeT2017].

Theorem 4 (Anticipation hydrodynamics: smooth solutions must flock). Let pρ,uqbe a smooth solution of the anticipation hydrodynamics (1.26) with an attractive potential

3Under a simplifying assumption of a mono-kinetic closure.


subject to a fat tail decay, (1.15), of order β 13. Then there is sub-exponential decay of the

anticipated energy fluctuations

(1.27)

» » 1

2m0

|upxq su0|2 Up|xτ yτ |q

dρpxq dρpyq ¤ Cet1η

, η 2β

1 β 1.

It follows that there is large time flocking, with sub-exponential alignment

|upt,xq su0|2 dρpxq tÑ8ÝÑ 0, su0 1

m0

»pρuq0pxq dx, m0

»ρ0pxq dx.

In proposition 7.1 we verify the existence of global smooth solution (and hence flocking)of the 1D system, (1.26), provided the threshold condition, u10pxq ¥ Cpτ,m0, aq holds, fora proper negative constant depending on τ,m0 and the minimal convexity a minU2 ¡ 0.

2. A priori L8 bounds for confining potentials

In this section we prove the uniform bounds asserted in (1.10) and (1.19), correspondingto the anticipation dynamics in (ΦU) and, respectively, (AT). Due to the momentum con-servation (1.7), we may assume without loss of generality that in both cases sxptq svptq 0.This will always be assumed in the rest of this paper.

We recall the dynamics of (ΦU) assumes that U lies in the class of convex potentials,(1.5), and the dynamics of (AT) assumes a larger class of attractive potentials, (1.15). Infact, here we prove uniform bounds under a more general setup of confining potentials.

Assumption 2.1 (Confining potentials). There exist constants a ¡ 0, L ¥ 0 and β suchthat

(2.1) Uprq ¥ axry2β L

, 0 ¤ β ¤ 1.

Observe that attractive potentials (1.15) satisfy (2.1) with L 1,

(2.2) Uprq » r

0

U 1psq ds ¥» r

0

axsyβs ds a

2 β

xry2β 1.

Thus, we have the increasing hierarchy of convex, attractive and confining potentials. Theclass of confining potentials is much larger, however, and it includes repulsive-attractivepotentials (discussed in section 6 below).

Remark 2.1. General confining potentials need not be positive. But taking into account thatpurely-attractive (and likewise — convex) potentials are positive, then we can improve (2.2),

(2.3) r2 ¤ C maxtrβ, 1uUprq for some c ¡ 0.

Indeed, for r 1, (1.15) implies U 1prq Á r and since Up0q 0 then Uprq Á r2. For r ¥ 1

we have r2β ¤ Cxry2β 1

with large enough C1

e.g., C1 ¡

?2?

21

, and (2.2) implies

(2.3) with C 2βaC1.

Lemma 2.1 (Uniform bound on positions for ΦU system). Consider the anticipationdynamics (ΦU) with bounded positive communication matrix 0 ¤ Φ ¤ φIdd, and boundedconfining potential (1.4),(2.1). Then the solution tpxiptq,viptqqu satisfies the a priori estimate

(2.4) maxi|xiptq| ¤ C8xty

243β , max

i|viptq| ¤ C8xty

2β43β , 0 ¤ β ¤ 1.

Remark 2.2. Note that we require a positive communication matrix Φ but otherwise, we donot insist on any fat tail condition (1.6).


Proof. Our proof is based on the technique introduced in [ST2020, §2.2], in which we proveuniform bounds in terms of the particle energy4

(2.5) Eiptq : 1

2|vi|2 1

N

¸j

Up|xi xj|q.

We start by relating the local energy to the position of particle i: using (2.1) followed byJensen inequality for the convex mapping5 x ÞÑ xxy2β, we find

Eiptqa

¥ 1

N

¸j

xxi xjy2β L ¥ @ 1

N

¸j

pxi xjqD2β L xxiy2β L

It follows that the maximal spread of positions, maxi |xiptq|, does not exceed

(2.6) Xptq ¤E8ptqa

L

12β

, Xptq : maxi|xiptq|, E8ptq : max

iEiptq.

Next we bound the energy dissipation rate of each particle. By (1.6) the communicationmatrices Φij are non-negative and bounded6, 0 ¤ Φij ¤ φIdd, and since

°j vj 0,

d

dtEiptq vi

1

N

¸j

∇Up|xi xj|q 1

N

¸j

Φijpvj viq

1

N

¸j

∇Up|xi xj|q pvi vjq

1

N

¸j

Φijpvj viq vi 1

N

¸j

∇Up|xi xj|q vj

1

2N

¸j

Φijvi vi 1

2N

¸j

Φijpvj viq pvj viq

1

2N

¸j

Φijvj vj 1

N

¸j

∇Up|xi xj|q ∇Up|xi|q

vj

¤ φEp0q a

2Ep0q

1

N

¸j

∇Up|xi xj|q ∇Up|xi|q212

.

(2.7)

4In fact Ei is not a proper particle energy, since°iEi NE (the pairwise potential is counted twice).

However, it is the ratio of the kinetic energy and potential energy in (2.5) which is essential, as one wouldlike to eliminate all the positive terms with indices i in (2.5), in order to avoid exponential growth of Ei.5pxry

2βq2 βp2 βqr2xry

2β p2 βqxry

β p2 βq

p1 βqr2 1

xry

2β¡ 0 for β ¤ 1.

6Observe that we do not use the fat tail decay (1.6).


To bound the sum on the right, we use the fact that D2U is bounded, (1.4), followed by(2.6) to find,

1

N

¸j

|∇Up|xi xj|q ∇Up|xi|q|2

¤ supx|D2Up|x|q|2 1

N

¸j

|xj|2 ¤ A2

N

¸j

|xj|2

A2

2N2

¸i,j

|xi xj|2 ¤ 2βA2 maxi|xi|β 1

N2

¸i,j

|xi xj|2β

¤ 2βA2Xβ 1

2N2

¸i,j

|xi xj|2β ¤ 2βA2Xβ 1

2N2

¸i,j

Up|xi xj|q

a L

¤ 2βA2Xβ

Ep0qa

L

2

.

(2.8)

Therefore

d

dtEiptq ¤φEp0q

a2Ep0q

2βA2Xβ

Ep0qa

L

2

12

¤φEp0q a

2Ep0q

2βA2E8a

L β

2βEp0q

a L

2

12(2.9)

and taking maximum among all i’s we have7

d

dtE8ptq ¤φEp0q

a2Ep0q

2βA2

E8ptqa

L β

2βEp0q

a L

2

12.(2.10)

Set fptq : E8ptq aL, then the last inequality tells us f 1 ¤ C1 C2fα with α : β

p42βq ,and since by assumption α 12, then

f À xty 11α xty 2p2βq

43β ,

which implies the uniform bound on velocities in (2.4),

maxi|viptq| ¤ 2

aE8ptq aL À xty 2β

43β .

The uniform bound on positions, maxi |xiptq|, follows in view of (2.6).

Lemma 2.1 applies, in particular, to the anticipation dynamics (AT) with convex potential,so that D2U is positive definite. Next, we prove uniform bounds for more general confiningU ’s.

Lemma 2.2 (Uniform bound on anticipated positions). Consider the anticipationdynamics (AT) with bounded confining potential (1.4),(2.1). Then the solution of the antic-ipation dynamics (AT) satisfies the a priori estimate

(2.11) maxi|xτi ptq| ¤ C8xty

122β , 0 ¤ β 1.

7To be pedantic at this point, the time derivative on the left (2.10) exists for almost all t’s by Rademachertheorem, where it coincides with the maximal time derivatives on the left of (2.9)i.


Remark 2.3. The a priori bound (2.11) is weaker than lemma 2.1 and may not be optimalfor β close to 1. We do not pursue an improved bound since it does not provide an increasedrange of β’s for which Theorem 2 holds.

Proof of Lemma 2.2. The key quantity for proving the priori bound (2.11) is the ‘anticipatedparticle energy’ in (AT),

(2.12) Eiptq : 1

2|vi|2 1

N

¸j

Up|xτi xτj |q.

Similar to the previous proof, the confining property of U implies that the diameter ofanticipated positions, maxi |xτi ptq|, does not exceed

(2.13) X ptq ¤E8ptqa

L

12β

, X ptq : maxi|xτi ptq|, E8ptq : max

iEiptq.

Next we bound the energy dissipation rate of each particle: since°jpvj τ 9vjq 0,

d

dtEiptq vi 9vi 1

N

¸j

∇Up|xτi xτj |q pvi τ 9vi vj τ 9vjq

τ | 9vi|2 1

N

¸j

∇Up|xτi xτj |q pvj τ 9vjq

τ | 9vi|2 1

N

¸j

∇Up|xτi xτj |q ∇Up|xτi |q

pvj τ 9vjq.

(2.14)

As before, the boundedness of D2U followed by (2.13) to find imply

(2.15)1

N

¸j

∇Up|xτi xτj |q ∇Up|xτi |q2 ¤ 2βA2X β

Ep0qa

L

2

.

Inserting (2.15) into the RHS of (2.14) and adding the energy-enstrophy balance (1.1) wefind8

d

dtpEptq Eiptqq

¤ τ

N

¸j

| 9vj|2 τ | 9vi|2 c

N

¸j

|vj|2 cτ 2

N

¸j

| 9vj|2

1

4cN

¸j

∇Up|xτi xτj |q ∇Up|xτi |q2

¤ τp1 cτqN

¸j

| 9vj|2 τ | 9vi|2 2cEp0q aL

1

4c2βA2X β

Ep0qa

L

2

¤ 2

τ

Ep0q aL

τ

42βA2X β

Ep0qa

L

2

, (taking c 1τ).

8Note that a confining potential need not be positive yet U ¥ aL and hence 12N°j |vj |

2 ¤ Ep0q aL.


By taking maximum among all i,

d

dtpEptq E8ptqq ¤2

τ

Ep0q aL

τ

42βA2X β

Ep0qa

L

2

¤2

τ

Ep0q aL

τ

42βA2

E8ptqa

L

β2β

Ep0qa

L

2

The last inequality tells us that fptq : Eptq E8ptq aL satisfies f 1 ¤ C1 C2f

α withα : β

2β . Since by assumption α 1, then

f À xty 11α xty 2β

22β ,

and the uniform bound (2.11) follows in view of (2.13).

3. Anticipation with convex potentials and positive kernels

Equipped with the uniform bound (2.11), we turn to prove Theorem 1 by hypocoercivityargument. In [ST2020] we use hypocoercivity to prove the flocking with quadratic potentials.Here, we make a judicious use of the assumed fat tail conditions, (1.6),(1.5), to extend thesearguments for general convex potentials.

Proof of Theorem 1. We introduce the modified energy, pEptq, by adding a multiple of thecross term 1N°

i xi vi, pEptq : Eptq εptqN

¸i

xiptq viptq.

We claim that with a proper choice εptq, the modified energy is positive definite. Indeed, theconvex (hence attractive) potential satisfies the pointwise bound (2.3), and together withthe uniform bound (1.10) they imply

|εptq 1

N

¸i

xi vi| ¤ 1

4N

¸i

|vi|2 ε2ptqN

¸i

|xi|2 1

4N

¸i

|vi|2 ε2ptq2N2

¸i,j

|xi xj|2

¤ 1

4N

¸i

|vi|2 ε2ptqC max p2Xptqqβ, 1( 1

2N2

¸i,j

Up|xi xj|q

¤ 1

4N

¸i

|vi|2 ε2ptqCp2C8qβ 1xty 2β

43β1

2N2

¸i,j

Up|xi xj|q.

Therefore it suffices to choose

(3.1) εptq ε0xtyα, α ¡ β

4 3β,

with small enough ε0 ¡ 0 and any α ¡ β43β

which is to be determined later, to guarantee

|εptqN°i xi vi| ¤ Eptq2, hence the positivity of pEptq ¥ Eptq2 ¡ 0.

Next, we turn to verify the coercivity of pEptq. First notice that Lemma 2.1 implies thefollowing L8 bound on xτi :

|xτi | ¤ |xi| τ |vi| ¤ p1 τqC8xty2

43β


This together with the assumed fat-tails of Φ and D2U , imply their lower-bounds: by (1.6)Φijptq are bounded from below by,

(3.2) Φijptq ¥ φptq : cxty 2γ43β ,

and integrating (1.5) U2prq ¥ axryβ twice, implies U has the lower bound (2.2)

(3.3) Up|xi xj|q ¥ c|xi xj|2x|xi xj|yβ ¥ |xi xj|2ψptq, ψptq : cxty 2β43β .

Now, we turn to conduct hypocoercivity argument based on the energy estimate (1.2).To this end, we append to Eptq, a proper multiple of the cross term

°xi vi, consult e.g.,

[ST2020, DS2020]. Using the symmetry of Φij, the time derivative of this cross term is givenby

d

dt

1

N

¸i

xi vi

1

N

¸i

|vi|2 1

N

¸i

xi 1

N

¸j

∇Up|xi xj|q 1

N

¸j

Φijpvj viq

1

2N2

¸i,j

|vi vj|2 1

2N2

¸i,j

pxi xjq ∇Up|xi xj|q

1

2N2

¸i,j

Φijpvj viq pxi xjq.

(3.4)

We prepare the following three bounds. Noticing that since U is convex U 1prq is increasing,

hence Uprq » r

0

U 1psq ds ¤ rU 1prq implies

1

2N2

¸i,j

pxi xjq ∇Up|xi xj|q 1

2N2

¸i,j

U 1p|xi xj|q|xi xj|

¥ 1

2N2

¸i,j

Up|xi xj|q.(3.5a)

Using the weighted Cauchy-Schwarz twice — weighted by the positive definite 0 Φij ¤ φ,and then by the yet-to-be determined κptq ¡ 0, 1

2N2

¸i,j

Φijpvj viq pxi xjq

¤ κ

4N2

¸i,j

Φijpvi vjq pvi vjq 1

4κN2

¸i,j

Φijpxi xjq pxi xjq

¤ κptq4N2

¸i,j

Φijpvi vjq pvi vjq φ4κptqN2

¸i,j

|xi xj|2

(3.5b)


Now set α ¥ 2γ

4 3βso that φptq decays no faster than εptq; moreover, φptq decays no

faster than xty2 since 6β 2γ ¤ 8, and hence, with small enough ε0, δ0 ¡ 0, the first

pre-factor on the right of (3.7) does not exceed τφptq4. Next, let α ¥ 2β

4 3βso that

εptqψptq is bounded: hence, with small enough ε0 ! δ0, the second pre-factor on the rightof (3.7) does not exceed εptq2. We conclude

d

dtpEptq À φptq

N

¸i

|vi|2 εptq2N2

¸i,j

Up|xi xj|q À xtyη pEptq, η 2 maxtβ, γu4 3β

.

This implies the sub-exponential decay of pE, and thus that of the comparable E.

3.1. Flocking of matrix-based Cucker-Smale dynamics. The Cucker-Smale model[CS2007a, CS2007b],

(3.8)

$'&'%9xi vi

9vi τ

N

N

j1

Φijpvj viq,

is a special case of (ΦU) with no external potential U 0, which formally corresponds to β 0, in which case theorem 1 would yield flocking for γ 12. Here we justify these formalitiesand prove the velocity alignment of (3.8) (no spatial concentration effect, however), under aslightly larger threshold.

Proposition 3.1 (Alignment of (3.8) model with positive kernels). Consider theCucker-Smale dynamics (3.8) with symmetric matrix kernel Φ satisfying (compared to (1.6)with v 0) for some constants 0 φ φ and γ,

(3.9) φxxi xjyγ ¤ Φpxi,xjq ¤ φ, 0 ¤ γ 23.Then there is sub-exponential decay of the energy fluctuations


, η 3γ

2, δEptq : 1

2N

¸i

|vi sv|2.It follows that there is a flock formation around the mean sxptq with large time velocity align-ment at sub-exponential rate:

(3.11) viptq Ñ sv0, xiptq sxptq Ñ x8i , sxptq : sx0 tsv0,

for some constants x8i .

The proof is similar but follows a slightly different strategy from that of theorem 1: westart by a priori estimate for the particle energy Ei, and then proceed to controlling theposition xxiy, which in turn gives enough energy dissipation.

Proof. Define the particle energy

(3.12) Eiptq : 1

2|vi|2, E8ptq max

iEiptq.


Observe that it satisfies

d

dtEiptq 1

N

N

j1

Φijpvj viq vi

1

2N

N

j1

Φijpvj viq pvj viq 1

2N

N

j1

Φijvi vi 1

2N

N

j1

Φijvj vj

¤ φptq |vi|2

2 φEptq,

(3.13)

where φptq, is a time-dependent lower-bound of the symmetric Φpxiptq,xjptqq which can betaken, in view of (3.9),

(3.14) Φpxiptq,xjptqq ¥ φptq, φptq : φ x2Xptqyγ, Xptq maxi|xiptq|.

Taking i as the particle with the largest Ei, then

(3.15)d

dtE8ptq ¤ φptqE8ptq φEptq ¤ φptqE8ptq φEp0q.

This implies

(3.16) E8ptq ¤ E8p0q φEp0qt.Next, we notice that

(3.17)d

dtXptq ¤ max

i|vi| ¤

a2E8ptq ¤

a2pE8p0q φEp0qtq.

This yields Xptq ¤ Cxty32, and in view of the fat tail (1.6), we end with the lower bound

φptq ¥ cxty 3γ2 with c p2Cqγ.

Therefore the energy dissipation (1.2) gives

(3.18)d

dtEptq ¤ φptqEptq ¤ cxty 3γ

2 Eptq,

which implies the sub-exponential decay (3.10), Eptq ¤ Ep0qecxty1η with η 3γ2 1.

Equipped with this sub-exponential decay of Eptq, we revisit (3.15): this time it implies

E8ptq ¤ eΦptqE8p0q φ

» t

0

eΦptsqEpsq ds

¤ Cxtyecxty1η , Φptq :» t

0

φpsq ds ¥ cxty1η(3.19)

This shows the sub-exponential decay of the kinetic energy of each agent, E8ptq, indepen-

dently of N , |viptq sv0| Ñ 0. As a result, xiptq xip0q » t

0

vipsq ds, converges as t Ñ 8since the last integral converges absolutely in view of |viptq| ¤

a2E8ptq.


4. Local vs. global weighted means

In this section we prove lemma 1.1 about discrete means, which in turn will be used inproving the hypocoercivity of the discrete anticipation dynamics (AT). We also treat thecorresponding continuum lemma of means in lemma 4.1, which is used in the hypocoercivityof the hydrodynamic anticipation model (1.26).

We begin with the proof of the Lemma of means 1.1:

Proof of Lemma 1.1. We first treat the scalar setup, in which case we may assume, withoutloss of generality that the zi’s are rearranged in a decreasing order, z1 ¥ z2 ¥ ¥ zN , andhave a zero mean

°j zj 0, and we need to bound the fluctuations on the left of (1.21)

which amount to1

N2

°i,j |zi zj|2 2

N

°i |zi|2. Let i0 be the smallest index i such that

(4.1)1

N

i1

j1

zj ¥ λ

2pΛ λqzi.

Noticing that if i is the maximal index of the positive entries, zi¤i ¥ 0, then (4.1) clearlyholds for i ¡ i (where LHS ¡ 0 ¡ RHS), hence i0 ¤ i, and since LHS is increasing (fori ¤ i) and RHS is decreasing, see figure 4.1 below, (4.1) holds for all i ¥ i0

(4.2)1

N

i1

j1

zj ¥ λ

2pΛ λqzi, i ¥ i0.

For i i0 we have zi ¥ 0, hence

1

N

¸j

cijpzi zjq 1

N

i

j1

cijpzi zjq 1

N

N

ji1

cijpzi zjq

¥ Λ

N

i

j1

pzi zjq λ

N

N

ji1

pzi zjq

Λ

N

i

j1

zj λ

N

i

j1

zj Λ

N

i

j1

zi λ

N

N

ji1

zi

¥ Λ λ

N

i1

j1

zj λzi, i i0,

(4.3)

and therefore, by the minimality of i0 in (4.2)

1

N

¸j

cijpzi zjq ¥ Λ λ

2pΛ λqλzi λzi λ

2zi ¥ 0, i i0.

It follows that

(4.4)1

N

i01

i1

1

N

¸j

cijpzi zjq2 ¥ λ2

4

1

N

i01

i1

z2i .


0 10 20 30 40 50 60 70 80 90 100i

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

i0

i1

i+

Figure 4.1. Comparison of means

Else, for i ¥ i0, (4.2) implies

zi ¤ zi0 ¤2pΛ λq

λ

1

N

i01

j1

zj, i ¥ i0.

It follows that for all positive entries, 0 ¤ zi ¤ zi0 ,

(4.5)1

N

i

ii0z2i ¤ z2

i0¤ 4pΛ λq2

λ2

1

N2

i01

j1

zj

2

¤ 4pΛ λq2λ2

1

N

i01

j1

z2j

Therefore, by (4.5),(4.4),

1

N

¸zi¥0

z2i

1

N

i01

i1

z2i

1

N

i

ii0z2i

¤

1 4pΛ λq2λ2

1

N

i01

j1

z2j

¤ 4

λ2

1 4

Λ

λ 1

2

1

N

¸i

1

N

¸j

cijpzi zjq2.

(4.6)

Now apply (4.6) to zi replaced by zi, to find the same upper-bound on the negative entries


1

N

¸zi¤0

z2i ¤

4

λ2

1 4

Λ

λ 1

2

1

N

¸i

1

N

¸j

cijpzi zjq2.(4.7)

The scalar result follows from (4.6),(4.7). For the d-dimensional case, notice given that°i zi 0 ¸

i,j

|zi zj|2 2¸i

|zi|2 d

k1

¸i

|zki |2,

¸i

1

N

¸j

cijpzi zjq2 d

k1

¸i

1

N

¸j

cijpzki zkj q2

where superscript stands for component. Therefore the conclusion follows by applying thescalar result to the components of zi tzki uk for each fixed k, ending with the same constantCpΛ, λq which is independent of d.

Next, we extend the result from the discrete framework to the continuum.

Lemma 4.1 (Local and global means are comparable). Let pΩ,F , µq be a probability

measure, and X : Ω Ñ Rd be a random variable with finite second moment,

»|Xpω1q|2 dµpω1q

8. Then, for any measurable c cpω, ω1q : Ω Ω ÞÑ R satisfying

0 λ ¤ cpω, ω1q ¤ Λ,

there holds¼|Xpωq Xpω1q|2 dµpωq dµpω1q ¤ 32

Λ2

λ4

» » cpω, ω1qXpωq Xpω1q dµpω1q2 dµpωq.

Observe that the quantity on the left can be equally expressed as the amount of fluctuationsrelative to the mean sX¼

|Xpωq Xpω1q|2 dµpωq dµpω1q 2

»|Xpωq sX|2 dµpωq, sX :

»Xpω1q dµpω1q,

and in particular, Dirac measure dµ 1

N

¸j

δpx zjq recovers the discrete case of lemma

1.1.

Proof. We first prove the 1D case, for which the map X, denoted by X, may assume a zeromean, sX 0, without loss of generality. Take ω with Xpωq : x ¥ 0, then»

cpω, ω1qpxXpω1qq dµpω1q

»ω1:Xpω1q¡x

cpω, ω1qpXpω1q xq dµpω1q »ω1:Xpω1q¤x

cpω, ω1qpxXpω1qq dµpω1q

¥ Λ

»ω1:Xpω1q¡x

pXpω1q xq dµpω1q λ

»ω1:Xpω1q¤x

pxXpω1qq dµpω1q

pΛ λq»ω1:Xpω1q¡x

pXpω1q xq dµpω1q λx

¥ pΛ λq»ω1:Xpω1q¡x

Xpω1q dµpω1q λx


Let

(4.8) x0 : sup tx : Y pxq ¥ 0u , Y pxq »ω1:Xpω1q¡x

Xpω1q dµpω1q λ

2pΛ λqx.

Since limxÑ8 Y pxq 8 and Y p0q ¥ 0, x0 is finite and non-negative. It is clear that Y pxqis decreasing and right-continuous. Therefore Y pxq ¥ 0 for any x x0, and Y pxq ¤ 0 forany x ¥ x0.

If x ¥ x0, then»cpω, ω1qpxXpω1qq dµpω1q ¥ pΛ λq λ

2pΛ λqx λx λ

2x(4.9)

Thus taking square and integrating in ω with x Xpωq ¥ x0 ¥ 0 gives

(4.10)

»ω:Xpωq¥x0

»cpω, ω1qpXpωq Xpω1qq dµpω1q

2

dµpωq ¥ λ2

4

»ω:Xpωq¥x0

X2pωq dµpωq.

Then we claim that the above integral on tω : Xpωq ¥ x0u is enough to get the conclusion.Notice that for any ε ¡ 0, one has Y px0 εq ¥ 0, i.e.,

(4.11) x0 ε ¤ 2pΛ λqλ

»ω:Xpωq¡x0ε

Xpω1q dµpω1q

and therefore

px0 εq2 ¤ 4pΛ λq2λ2

»ω:Xpωq¡x0ε

Xpω1q dµpω1q2

¤ 4pΛ λq2λ2

»ω:Xpωq¡x0ε

Xpω1q2 dµpω1q»

ω:Xpωq¡x0εdµpω1q

¤ 4pΛ λq2

λ2

»ω:Xpωq¡x0ε

Xpω1q2 dµpω1q

Taking ε Ñ 0, noticing that the RHS integral domain tω : Xpωq ¡ x0 εu converges totω : Xpωq ¥ x0u, we get

(4.12) x20 ¤

4pΛ λq2λ2

»ω:Xpωq¥x0

Xpω1q2 dµpω1q

Thus, using (4.12) and (4.10) we find»ω1:Xpω1q¥0

Xpω1q2 dµpω1q »ω1:0¤Xpω1q x0

Xpω1q2 dµpω1q »ω1:Xpω1q¥x0

Xpω1q2 dµpω1q

¤

4pΛ λq2λ2

1

»ω1:Xpω1q¥x0

Xpω1q2 dµpω1q

¤

4pΛ λq2λ2

1

4

λ2

»rx0,8q

»cpω, ω1qpXpωq Xpω1qq dµpω1q

2

dµpωq.

Apply the last bound with Xpq replaced by Xpq to find that the

»ω1:Xpω1q¤0

Xpω1q2 dµpω1qsatisfies the same bound on the right, which completes the scalar part of the proof. For the


d-dimensional case with X pX1, . . . , Xdq, notice that»|Xpω1q|2 dµpω1q

d

k1

»|Xkpω1q|2 dµpω1q

and similarly,» » cpω, ω1qpXpωq Xpω1qq dµpω1q2 dµpωq

d

k1

» » cpω, ω1qpXkpωq Xkpω1qq dµpω1q2 dµpωq

Applying the 1D result to the random variable Xk gives

(4.13)k

»|Xkpω1q|2 dµpω1q ¤ Cpλ,Λq

» » cpω, ω1qpXkpωq Xkpω1qq dµpω1q2 dµpωq.

Summing (4.13)k recovers the desired result with the constant CpΛ, λq independent of d.

5. Anticipation dynamics with attractive potentials

In this section we prove the flocking behavior of (AT) asserted in Theorem 2. Here, wetreat the larger class of attractive potentials, thus extending the case of convex potentials ofTheorem 1. The starting point is the anticipated energy balance (1.1)

d

dtEptq τ

N

¸i

| 9vi|2.

Remark 5.1. We note in passing that the first-order system

9xi 1

N

N

j1

∇Up|xi xj|q,

satisfies an energy estimate, reminiscent of the energy-enstrophy balance in the anticipationdynamics (AT),

d

dt

1

2N2

¸i,j

Up|xi xj|q 1

N

¸i

| 9xi|2.

Proof of Theorem 2. We aim to conduct a hypocoercivity argument to complement the an-ticipated energy estimate (1.1). To this end, we use the ‘anticipated’ cross term,

d

dtp 1

N

¸i

xτi viq 1

N

¸i

ppvi τ 9viq vi xτi 9viq

¤ 1

N

¸i

|vi|2 τpτ

2| 9vi|2 1

2τ|vi|2q 1

2p|xτi |2 | 9vi|2q

¤ 1

2

1

N

¸i

|vi|2 1

2

1

N

¸i

|xτi |2 τ 2 1

2

1

N

¸i

| 9vi|2

(5.1)

Consider the modified anticipated energy pEptq : Eptq εptq 1N

°i x

τi vi, where εptq ¡ 0 is

small, decreasing, and is yet to be chosen. We first need to guarantee that this modifiedenergy is positive definite, and in fact — comparable to Eptq,

(5.2)εptq 1

N

¸i

xτi vi ¤ Eptq

2 1

4N

¸i

|vi|2 1

4N2

¸i,j

Up|xτi xτj |q.


Indeed, notice that

|εptq 1

N

¸i

xτi vi| ¤1

4N

¸i

|vi|2 ε2ptq 1

N

¸i

|xτi |2

¤ 1

4N

¸i

|vi|2 εptq2C max p2X qβ, 1( 1

2N2

¸i,j

Up|xτi xτj |q

¤ 1

4N

¸i

|vi|2 ε2ptqCp2C8qβ 1xty β

22β1

2N2

¸i,j

Up|xτi xτj |q.

The second inequality is obtained similarly to (2.15) and using (2.3), and the third inequalityuses Lemma 2.2. Therefore it suffices to choose

(5.3) εptq ε0p10 tqα, α ¥ β

4 4β

with small enough ε0 to guarantee (5.2).

We now turn to verify the (hypo-)coercivity of pEptq,d

dt

Eptq εptq 1

N

¸i

xτi vi

¤ τ

N

¸i

| 9vi|2 εptq2

1

N

¸i

|vi|2

εptq

1

2

1

N

¸i

|xτi |2 τ 2 1

2

1

N

¸i

| 9vi|2 | 9εptq| 1

N

¸i

|xτi vi|

¤ τ εptqτ

2 1

2

1

N

¸i

| 9vi|2

εptq | 9εptq|2

1

N

¸i

|vi|2 εptq | 9εptq|2

1

N

¸i

|xτi |2

(5.4)

The first pre–factor on the right of (5.4) is less than τ2

for small enough ε0. The secondpre-factor is negative since

| 9εptq| αε0p10 tqα1 ¤ α

10εptq.

It remains to control the last term on the right of (5.4). To this end we recall that U isassumed attractive, U 1prqr ¥ xryβ, hence, by Lemma 2.2

A ¥ U 1prτijqrτij

¥ axrτijyβ ¥ cxty β22β , rτij |xτi xτj |.

We now invoke Lemma 1.1: it implies

1

N

¸i

| 9vi|2 1

N

¸i

1

N

¸j

U 1prτijqrτij

pxτi xτj q2

¥ cxtyη 1

N2

¸i,j

|xτi xτj |2, η 2β

1 β(5.5)


Therefore, the last term on the right of (5.4) does not exceed9 À εptqxtyη 1N

°i | 9vi|2 and

choosing εptq as in (5.3) with α η yields

(5.6)d

dtpEptq ¤ τ

4

1

N

¸i

| 9vi|2 ε0p10 tqη4

1

N

¸i

|vi|2.

As before, since U is bounded, it has at most quadratic growth,

1

2N2

¸i,j

Up|xτi xτj |q ¤A1

2N2

¸i,j

|xτi xτj |2 ¤ Cxty 2β1β

1

N

¸i

| 9vi|2 Cxtyη 1

N

¸i

| 9vi|2,

and we conclude the sub-exponential decay

(5.7)d

dtpEptq ¤ cxtyη pEptq ; pEptq ¤ Cet

1η

,

which implies the same decay rate of Eptq.

6. Anticipation dynamics with repulsive-attractive potential

In this section we prove Theorem 3. The assumption°i xi

°i vi 0 amounts to saying

that x : x1 x2, v : v1 v2. Replacing Up|x|q by Up2|x|q and r0 by r02, (AT)becomes

(6.1)

"9x v

9v ∇Up|xτ |qwhere Uprq has a local minimum at r r0 ¡ 0 with U2pr0q a ¡ 0.

We use polar coordinates

(6.2)

"xτ1 r cos θ

xτ2 r sin θ

and

(6.3)

"vr v1 cos θ v2 sin θ

vθ v1 sin θ v2 cos θ

Then (6.1) becomes

(6.4)

$'''''''&'''''''%

9r vr τU 1prq9θ vθ

r

9vr U 1prq v2θ

r

9vθ vrvθr

We will focus on perturbative solutions near r r0, vr vθ 0. Write r : r0 δr, andthere hold the approximations

(6.5) Uprq a

2δ2r , U 1prq aδr, U2prq a

9We may assume without loss of generality, that the two time invariant moments vanish,°

xi °

vi 0,and hence 1

N

°i |x

τi |

2 12N2

°i,j |x

τi xτj |

2.


Observe that our assumed initial configuration in (1.24) implies, and in fact is equivalent tothe assumption of smallness on the anticipated energy, Ep0q ¤ 2p1 τqε. Theorem 3 is aconsequence of the following proposition on the polar system (6.4),

Proposition 6.1 (polar coordinates). There exists a constant ε ¡ 0, such that if theinitial data is small enough:

(6.6) E0 :Uprq 1

2v2r

1

2v2θ

|t0

¤ ε

then the solution to (6.4) decays to zero at the following algebraic rates:

(6.7) δr ¤ Cxty1 ln12 xty, vr ¤ Cxty1 ln12 xty, vθ ¤ Cxty12

Proof. Fix 0 ζ ¤ mint r02, 1u as a small number such that

(6.8)a

2¤ U2prq ¤ 2a, @δ2

r ¤ ζ,

and as a result,

(6.9)a

2|δr| ¤ |U 1prq| ¤ 2a|δr|, a

4δ2r ¤ Uprq ¤ aδ2

r , @δ2r ¤ ζ.

We start from the energy estimate for the anticipated energy Eptq : Uprq 1

2v2r

1

2v2θ

d

dtEptq U 1prq pvr τU 1prqq vr

U 1prq v2

θ

r

vθ vrvθ

r τU 1prq2

Therefore, for any positive ε ¤ a

4ζ to be chosen later, if E0 ¤ ε, then

(6.10) δ2r ¤

4

aUprq ¤ 4

aε ζ, v2

r ¤ ε,

hold for all time which in turn implies that (6.9) holds. Next we consider the cross terms

d

dtpvrv2

θq U 1prq v2

θ

r

v2θ 2vrvθ

vrvθr

v4θ

r U 1prqv2

θ 2v2rv

2θ

r,(6.11)

and

d

dtpU 1prqvrq U2prqvr pvr τU 1prqq U 1prq

U 1prq v2

θ

r

U2prqv2

r τU2prqvrU 1prq U 1prq2 U 1prqv2θ

r

(6.12)

We now introduce the modified energy,

pEptq : Uprq 1

2v2r

1

2v2θ cvrv

2θ cU 1prqvr,


depending on a small c ¡ 0 which is yet to be determined. A straightforward calculation,based on (6.9) shows its decay rate does not exceed

d

dtpEptq τU 1prq2 c

rv4θ cU2prqv2

r

c

U 1prqv2

θ 2v2rv

2θ

r

c

τU2prqvrU 1prq U 1prq2 U 1prqv

2θ

r

¤ τ a

2

4δ2r

c

2r0

v4θ c

a

2v2r

c

2a|δr|v2

θ 4v2rv

2θ

r0

c

4τa2|vrδr| 4a2δ2

r 4a|δr|v2θ

r0

,

and by Cauchy-Schwarz

d

dtpEptq ¤ τ a

2

4δ2r

c

2r0

v4θ c

a

2v2r

c

1

κaδ2

r κav4θ κ

2

r0

v4θ

1

κ

2

r0

v4r

c

2κτa2v2

r 1

κ2τa2δ2

r 4a2δ2r

1

κ

2a

r0

δ2r κ

2a

r0

v4θ

τa2

4 c

κ

a 2τa2 4κa2 2a

r0

δ2r

c

1

2r0

κpa 2

r0

2a

r0

qv4θ c

a

2 1

κ

2

r0

v2r 2κτa2

v2r ,

(6.13)

with κ P p0, 1q which is yet to be determined. We want to guarantee that the three pre-factorson the right are positive. To this end, we first fix the ratio

(6.14)1c

κ τ a

2

8

a 2τa2 4a2 2ar0

so that the first pre-factor is lower-bounded by τ a2

8. Then we choose

(6.14)2 κ ¤ min

#1,

14r0

a 2r0 2a

r0

,a4

2τa2

+so that the second pre-factor, the coefficient of v4

θ , becomes larger thanc

4r0

. Finally, the

third pre-factor is also positive because (i) a small enough κ was chosen in (6.14)2, and (ii)a key aspect in which vr can be made small enough to compensate for small κ, so that the

negative contribution of 1

κ

2

r0

v2r can be absorbed into the rest: indeed, if

(6.14)3 v2r ¤

a8

1κ

2r0

ar0

16κ

then

c

a

2 1

κ

2

r0

v2r 2κτa2

¤ c

a

2 1

κ

2

r0

a8

1κ

2r0

a

4

ca

8.


Therefore, (6.13) implies the decay rate

(6.15)d

dtpEptq ¤ η1pδ2

r v4θ v2

rq, η1 min

"τa2

8,c

4r0

,ca

8

*,

provided (6.9) and (6.14)1–(6.14)3 are satisfied.

Moreover, we claim that pE is comparable to the original anticipated energy E . Indeed, ifin addition

(6.16)1 c

car0

16κ ¤ 1

4

holds, then in view of (6.14)3, c|vrv2θ | ¤

1

4v2θ , and if

(6.16)2 c ¤ min!1

8,

1

4a

),

holds, then in view of (6.9),

c|U 1prqvr| ¤ capδ2r v2

rq ¤ ca4

aUprq v2

r

¤ 1

2

Uprq 1

2v2r

.

It follows that

(6.17)1

2Eptq ¤ pEptq ¤ 2Eptq,

provided (6.16)1–(6.16)2 are satisfied. These last two conditions are clearly met for smallenough κ: recall that the ratio cκ was fixed in (6.14)1 then

(6.18)1 κ ¤ a 2τa2 4a2 2ar0

τ a2

8

mint?ar0,1

8,

1

4au

suffices to guarantee (6.16)1–(6.16)2. Thus, we finally choose small enough κ to satisfy both

(6.14)2,(6.18)1, and small enough ε min a

4ζ,ar0

16κ(

so that (6.10) and (6.14)3 hold. By

now we proved (6.15) and (6.17). Finally, notice that for small enough δr, vr we have

δ2r v4

θ v2r ¥ δ4

r v4θ v4

r ¥1

3pδ2r v2

θ v2rq2 ¥

1

3min

! 1

a2, 1)E2ptq.

We conclude, in view of (6.15) and (6.17),

d

dtpEptq ¤ η pE2ptq ; pEptq ¤ 1

ηt 1pEp0q , η η1

12min

"1

a2, 1

*.

It follows that |vθ| ¤ Cxty12.To get a better decay rate for δr and vr, we use yet another modified energy functional,

qEptq : Uprq 1

2v2r c1U

1prqvr,


for which we find

d

dtqEptq τU 1prq2 c1U

2prqv2r

vrv2θ

r c1

τU2prqvrU 1prq U 1prq2 U 1prqv

2θ

r

¤ τ

a2

4δ2r c1

a

2v2r

2vrv2θ

r0

c1

4τa2|vrδr| 4a2δ2

r 4a|δr|v2θ

r0

¤

τa2

4 c1

κ1

2τa2 4κ1a

2 2a

r0

δ2r c1

a

2 κ1

1

r0

κ1 2τa2

v2r

1

c1κ1r0

c1κ12a

r0

v4θ .

By similar choices of c1 and κ1, one can guarantee that qEptq is equivalent to δ2r v2

r , and thecoefficients of δ2

r and v2r are positive. Therefore

(6.19)d

dtqEptq ¤ η2

qEptq Cv4θ ¤ η2

qEptq Cxty2

This gives

(6.20) qEptq eη2t qEp0q C

» t

0

eη2ptsqp1 sq2 ds

We estimate the last integral for large enough t,» t

0

eη2ptsqp1 sq2 ds ¤» t 1

η2ln xty

0

» t

t 1µ

ln xty

eη2ptsqp1 sq2 ds

¤ xty1

» t

0

p1 sq2 ds

1 pt 1

η2

ln xtyq2 1

η2

ln xty

¤ xty2 2

η2

xty2 ln xty.

(6.21)

This shows that qEptq ¤ Cxty2 ln x1 ty, and therefore |vr| |δr| ¤ Cxty1 ln12 x1 ty.

Finally, we conclude by noting that the last bound on δr tells us that|xτ1ptq xτ2ptq| r0

¤ Cxty1 ln12 x1 ty, |v1ptq v2ptq| ¤ Cxty12.

Observe that this bound on relative anticipated positions is in fact equivalent to the claimedstatement of the current positions,

|x1ptqx2ptq| r0

À xty1 ln12 x1 ty, which concludesthe proof of theorem 3.

Remark 6.1. Numerical examples [GTLW2017, sec. 1] show that the rate vθ Opt12q isoptimal. Therefore,

θptq θ0 » t

0

1

rpsqvθpsq ds Op?tq,

which means that θ needs not converge to any point, even for near equilibrium initial data.Thus, although we trace the dynamics of δr, vr, vθ using essentially perturbative arguments,the dynamics of (6.1) is not.


Remark 6.2. The optimal algebraic rate of vθ in Proposition 6.1 can also be derived viacentre manifold reduction. To this end, rewriting the system (6.4) in terms of perturbationvariables δr, vr and vθ, 9δr

9vr9vθ

τa 1 0a 0 00 0 0

δrvrvθ

τ pU 1pr0 δrq aδrqU 1pr0 δrq aδr

v2θr0δr vrvθ

r0δr

.Since the 2 2 leading minor is stable, the centre manifold near the equilibrium, W c, can beparametrized by vθ,

W c "pδr, vr, vθq

δr v2θ

r0

Opv3θq, vr v3

θ

r0

Opv3θq*.

It follows that vθ is governed by 9vθ vrvθr

τ v3θ

r0

Opv4θq, which implies its algebraic

decay of order xty12.

7. Anticipation dynamics: hydrodynamic formulation

The large crowd dynamics associated with (AT) is captured by the macroscopic densityρpt,xq : RRd ÞÑ R and momentum ρupt,xq : RRd ÞÑ Rd, which are governed by thehydrodynamic description (1.26)$&%

ρt ∇x pρuq 0

pρuqt ∇x pρub uq »∇Up|xτ yτ |qρpt,xq dρpt,yq, xτ : x τupt,xq.

The flux on the left involves additional second-order moment fluctuations, P , which can bedictated by proper closure relations, e.g., [HT2008, CFRT2010, Go2016, CCP2017, FK2019].As in [HT2008], we will focus on the mono-kinetic case, in which case P 0.

To study the large time behavior we appeal, as in the discrete case, to the basic balancebetween energy and enstrophy: here we consider the anticipated energy

(7.1) Eptq :»

1

2|upt,xq|2ρpt,xq dx 1

2

» »Up|xτ ptq yτ ptq|q dρpt,xq dρpt,yq.

Away from vacuum, the velocity field upxq upt,xq satisfies the transport equation

(7.2a) utpt,xq u ∇xupt,xq Apρ,uqpt,xq,

where Apρ,uq denotes the anticipated interaction term

(7.2b) Apρ,uqpt,xq : »∇Up|xτ ptq yτ ptq|q dρpt,yq, xτ ptq x τupt,xq.


where Λ A and λ are the upper- and respectively, lower-bounds ofU 1p|xτ yτ |q|xτ yτ | ,

d

dtEptq τ


2 dρpxq

À

minU 1p|xτ yτ |q|xτ yτ |

4¼ xτ yτ

2 dρpyq dρpxq(7.5)

Step (ii). A bound on the spread of the anticipated positions supported on non-vacuousstates

(7.6) maxxτPsupp ρpt,q

|xτ | ¤ cxtyη.

Arguing along the lines of lemma 2.2 one finds that (7.6) holds with η 1

2p1 βq , hence

axty β2p1βq À U 1p|xτ yτ |q

|xτ yτ | ¤ A, xτ ,yτ P supp ρpt, q,

and (7.5) implies

d

dtEptq τ


2 dρpxq

À τxty 2β1β

¼|xτ yτ |2 dρpxq dρpyq.

(7.7)

We are now exactly at the point we had with the discrete anticipation dynamics, in which thedecay of anticipated energy is controlled by the fluctuations of anticipated position, (1.23).

Step (iii). To close the decay rate (7.7) one invokes hypocoercivity argument on themodified energy, pEptq : Eptq εptq

»xτ upxq dρpxq.

Arguing along the lines of section 5, one can find a suitable εptq ¡ 0 which leads to the

sub-exponential decay of pEptq and hence of the comparable Eptq, thus completing the proofof theorem 4.

7.2. Existence of smooth solutions – the 1D case. We study the existence of smoothsolutions of the 1D anticipated hydrodynamic system

(7.8)

$&%Btρ Bxpρuq 0

Btu uBxu »U 1p|xτ yτ |qsgnpxτ yτ qρpyq dy, xτ x τupt, xq,

subject to uniformly convex potential U2pq ¥ a ¡ 0. Let d : Bxu. Then

Btd uBxd d2 p1 τdq»U2p|xτ yτ |qρpyq dy(7.9)

or

d1 d2 cp1 τdq, 1 : Bt uBx,(7.10)


where by uniform convexity c cpt, xq : ³U2p|xτ yτ |qρpyq dy P rm0a,m0As. The discrim-

inant of RHS, given by pτcq2 4c cpτ 2c 4q is non-negative, provided τ 2m0a ¥ 4. In thiscase, the smaller root of (7.10) is given by

(7.11)1

2pτc

acpτ 2c 4qq ¤ 1

2pτm0a

am0apτ 2m0a 4qq,

and the region to its right is an invariant of the dynamics (7.9). We conclude the following.

Proposition 7.1 (Existence of global smooth solution). Consider the 1D anticipationhydrodynamic system (7.8) with uniformly convex potential 0 a ¤ U2 ¤ A. It admits aglobal smooth solution for sub-critical initial data, pρ0, u0q, satisfying

minxu10pxq ¥ 1

2pτm0a

am0apτ 2m0a 4qq, τ ¥ 2?

m0a.

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Ruiwen ShuDepartment of MathematicsandCenter for Scientific Computation and Mathematical Modeling (CSCAMM)University of Maryland, College Park MD 20742

Email address: [email protected]

Eitan TadmorDepartment of Mathematics, Institute for Physical Sciences & Technology (IPST)andCenter for Scientific Computation and Mathematical Modeling (CSCAMM)University of Maryland, College Park MD 20742

Email address: [email protected]

home.cscamm.umd.edu · 2020-07-30 · anticipation breeds alignment ruiwen shu and eitan tadmor...

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