university of toronto department of mathematics - when do … · 2020. 7. 5. · some related...

When do interacting organisms gravitate to the verticesof a regular simplex?

Tongseok Lim & Robert McCann

University of Toronto

Slides at ‘Talk 3’ at www.math.toronto.edu/mccann

6 July 2020

Tongseok Lim & Robert McCann (UToronto) Isodiametry, variance and pattern formation 6 July 2020 1 / 20

Animals forming patterns: flocking, milling, swarming

and schooling

bottom view

Led scientists to make models...

e.g. (very incomplete)Lennard-Jones (1924) 6-12 potential for molecular interactionsParr (1927)Breder (1954) attractive-repulsive power law interaction for fish separationKeller-Segal (1971) purely attractive, 1st order (2d cell chemotaxis)..Mogilner and Edelstein-Keshet (1999) 1d attractive-repulsive + diffusionLevine, Rappel and Cohen (2000) 2nd order, preferred speedTopaz, Bertozzi and Lewis (2006)Cucker and Smale (2007) 2nd order, matched speeds..... analyze and simulatee.g. (just as incomplete) Albi, Balague, Bertozzi, Burchard, Carrillo,Choksi, Craig, Fetecau, Figalli, Frank, Huang, Kolokolnikov, Laurent, Lieb,Lopes, Pavlovski, Pattachini, Raoul, Sun, Topaloglu, Uminsky, von Brecht,...Tongseok Lim & Robert McCann (UToronto) Isodiametry, variance and pattern formation 6 July 2020 5 / 20

Attractive-repulsive pair potentials on x ∈ Rn:

Va,b(x) := Va(x)− Vb(x)

Va(x) :=1

a|x |a exponents a > b

• minimized at separation |x | = 1

First-order, interacting J particle dynamics

ODE description: xk(t) ∈ Rn for i ∈ {1, . . . , J}:

J − 1

∑i 6=k

∇Va,b(xi − xk)

PDE description:

the probability measure µ(t) :=1

J∑i=1

δxi (t) on Rn satisfies

‘aggregation / self-assembly’ equation

dt= ∇ · [µ∇(Va,b ∗ µ)]

2∇ · [µ∇(

δµ)],

which dissipates the energy

E (µ) = Ea,b(µ) :=

Va,b(x − y)dµ(x)dµ(y)

J − 1

∑i 6=k

∇Va,b(xi − xk)

PDE description:

J∑i=1

δxi (t) on Rn satisfies the

dt= ∇ · [µ∇(Va,b ∗ µ)]

2∇ · [µ∇(

δµ)],

E (µ) = Ea,b(µ) :=

J − 1

∑i 6=k

∇Va,b(xi − xk)

PDE description:

J∑i=1

δxi (t) on Rn satisfies the

dt= ∇ · [µ∇(Va,b ∗ µ)]

2∇ · [µ∇(

δµ)],

E (µ) = Ea,b(µ) :=

The continuumum J →∞ limiting dynamics

The aggregation / self-assembly equation

dt= ∇ · [µ∇(Va,b ∗ µ)] =

2∇ · [µ∇(

δµ)],

defines a flow on P(Rn), where

P(K ) := {µ ≥ 0 on Rn | µ[K ] = 1 = µ[Rn]}

denotes the space of Borel probability measures on K ⊂ Rn.

Minimizers of Ea,b(µ) on P(Rn) represent attracting fixed points of theflow (as do local minimizers in a suitable topology).

The continuumum J →∞ limiting dynamics

The aggregation / self-assembly equation

dt= ∇ · [µ∇(Va,b ∗ µ)] =

2∇ · [µ∇(

δµ)],

defines a flow on P(Rn), where

P(K ) := {µ ≥ 0 on Rn | µ[K ] = 1 = µ[Rn]}

denotes the space of Borel probability measures on K ⊂ Rn.

Minimizers of Ea,b(µ) on P(Rn) represent attracting fixed points of theflow (as do local minimizers in a suitable topology).

Piecewise linear force laws

Power law potentials

Parameter space

Some related results

M. ’94, ’05 introduced d∞-local minimization to find stable rotating stars

Balague, Carrillo, Laurent, Raoul ’13: showed d∞-local minimimizers µ ofEa,b on P(Rn) have support with Hausdorff dimension at least 2− b.

Sun, Uminsky, Bertozzi ’12: for b = 2 showed spherical shell linearly stableiff a < 4 while vertices of regular simplex are linearly stable iff a > 4.Inquired about nonlocal stability and global attraction.

• moreover, in the mildly repulsive regime b ≥ 2

Carrillo, Figalli, Patachini ’17: showed if b > 2 then d∞-localminimimizers are supported only at isolated points (and indeed onlyfinitely many such points in the case of global energy minimizers).

Kang, Kim, Lim, Seo ’19+: catalog d∞-local and global minimizers inn = 1 dimension

Global and local stability of regular simplices

Theorem (Minimizing strong attraction with mild repulsion)

1. If a>>b > 2 then E (µ) is uniquely minimized on P(Rn) (up torotations and translations) by equidistributing the mass of µ over thevertices of a regular simplex:

n∑i=0

δxi where |xi − xk | = 1 for all 0 ≤ i < k ≤ n.

Theorem (Many local minima throughout the mildly repulsive regime)

2. If a>b > 2 and 0 < m0 ≤ · · · ≤ mn with 1 =∑n

i=0 mi then

µ =n∑

miδxi where |xi − xk | = 1 as above

is a d∞-local minimizer of E (µ) (mininimizing strictly up to rigid motions).

Global and local stability of regular simplices

Theorem (Minimizing strong attraction with mild repulsion)

1. If a>>b > 2 then E (µ) is uniquely minimized on P(Rn) (up torotations and translations) by equidistributing the mass of µ over thevertices of a regular simplex:

n∑i=0

δxi where |xi − xk | = 1 for all 0 ≤ i < k ≤ n.

Theorem (Many local minima throughout the mildly repulsive regime)

2. If a>b > 2 and 0 < m0 ≤ · · · ≤ mn with 1 =∑n

i=0 mi then

µ =n∑

miδxi where |xi − xk | = 1 as above

is a d∞-local minimizer of E (µ) (mininimizing strictly up to rigid motions).

Parameter space

Kantorovich-Rubinstein-Wasserstein Lp-transport metric dp

Given p > 1 and K ⊂ Rn compact,

dp(µ, ν) :=

γ∈Γ(µ,ν)

∫Rn×Rn

|x − y |pdγ(x , y)

metrizes the weak topology on P(K ), where

Γ(µ, ν) :=

{γ ∈ P(R2n) | µ[U] = γ[U × Rn],

γ[Rn × U] = ν[U]∀U ⊂ Rn

}denotes the set of joint measures with given marginals.

d∞(µ, ν) := limp→∞

dp(µ, ν)

metrizes a much finer topology.

Kantorovich-Rubinstein-Wasserstein Lp-transport metric dp

Given p > 1 and K ⊂ Rn compact,

dp(µ, ν) :=

γ∈Γ(µ,ν)

∫Rn×Rn

|x − y |pdγ(x , y)

metrizes the weak topology on P(K ), where

Γ(µ, ν) :=

{γ ∈ P(R2n) | µ[U] = γ[U × Rn],

γ[Rn × U] = ν[U]∀U ⊂ Rn

}denotes the set of joint measures with given marginals.

d∞(µ, ν) := limp→∞

dp(µ, ν)

metrizes a much finer topology.

What distinguishes b = 2? (and for that matter a = 2)?

Introducing the mean

x̄(µ) :=

xdµ(x),

the quadratic contribution

|x − y |2dµ(x)dµ(y) = − |x̄(µ)|2 +

|x |2dµ(x)

=: Var(µ)

to the energy of µ becomes linear on the centered measures

P0(Rn) := {µ ∈ P(Rn) | d2(µ, δ0) <∞, x̄(µ) = 0}

What distinguishes b = 2? (and for that matter a = 2)?

Introducing the mean

x̄(µ) :=

xdµ(x),

the quadratic contribution

|x − y |2dµ(x)dµ(y) = − |x̄(µ)|2 +

|x |2dµ(x)

=: Var(µ)

to the energy of µ becomes linear on the centered measures

P0(Rn) := {µ ∈ P(Rn) | d2(µ, δ0) <∞, x̄(µ) = 0}

Variance maximization under diameter constraint

Thus− minµ∈P0(Rn)

E∞,2(µ) = maxdiam sptµ≤1

Var(µ)

• although the diameter constraint is non-convex (unless n = 1), thevariance maximizers can be found by solving an infinite-dimensional linearprogram for each compact set K ⊂ Rn of unit diameter. As described in acompanion work, they turn out to be precisely the measures from THM 1.

• by comparison, the same measures minimize E∞,b(µ) for all b > 2.

• Γ-convergence: since Ea,bΓ→E∞,b on (P0(Rn), d2), subsequences of

minimizers µa,b converge weakly to minimizers µ∞,b of the limitingproblem.

Variance maximization under diameter constraint

Thus− minµ∈P0(Rn)

E∞,2(µ) = maxdiam sptµ≤1

Var(µ)

• although the diameter constraint is non-convex (unless n = 1), thevariance maximizers can be found by solving an infinite-dimensional linearprogram for each compact set K ⊂ Rn of unit diameter. As described in acompanion work, they turn out to be precisely the measures from THM 1.

• by comparison, the same measures minimize E∞,b(µ) for all b > 2.

• Γ-convergence: since Ea,bΓ→E∞,b on (P0(Rn), d2), subsequences of

minimizers µa,b converge weakly to minimizers µ∞,b of the limitingproblem.

First and second variations

• this convergence can be strengthened using the Euler-Lagrange equation

Va,b ∗ µa,b(x)≥2Ea,b(µa,b) with equality holding µa,b-a.e.

• for large a, the mass of µa,b is thus localized to a union of small ballsaround the vertices of a regular simplex; a 2nd variation calculation(exploiting the uniform radial convexity of Va,b at its minimum and its lackof uniform concavity at the origin) then shows the energy is reduced byconcentrating all of the mass in each ball at a single point

• the regular simplex is the only shape which succeeds in placing all n + 1points at unit distance from each other, hence is energetically optimal

• the same 2nd variation calculation establishes THM 2; the d∞-closenessto a measure supported on a regular simplex replaces the Γ-convergenceargument to provide the required localization in this case

Thank you

university of toronto department of mathematics - when do … · 2020. 7. 5. · some related...

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