The road to catastrophe:

stability and collapsein 2D many particle systems

Maria R. D’Orsogna, Y.L. Chuang, A. Bertozzi, L. Chayes Department of Mathematics, UCLA

Why?

flocks

army ants

bacteria

herds

barracuda

jack, tuna

But also:

Unmanned Vehicle Operations

Exploration :

Space, Underwater

Dangerous missions:

Land-mine removal, Earthquake recovery

Military missions

Individuals: limited capabilities

Teams: new, better propertieswithout leaders

…to artificial systems?

From Nature …

?

Interactions:

Mediated by background: Gradients of chemical or physical fields

food, light concentrations

temperature

electromagnetic fields

Nucleation agents: External agents as triggers

tuna fish under floating objects

Direct information exchange between particles:

fish, birds

Bacteria, plankton

Interactions: design challenges

Pattern formation

Dispersive behavior vs. Convergence to a site

Complex behavior without crashing

One example of an emergent behavior that I was not anticipating: I was trying to get the robots to spread evenly throughout their environment, trying to have them move themselves so that there were robots everywhere in the whole room, leaving no empty spaces. And I made an error in the program; I flipped some signs in the equations. And when I ran the software, the robots formed into little clumps. Essentially they made polka dots on the floor, which was very entertaining after the fact.

James McLarkin, Nova-PBS December 2004

A First Study:

Vicsek algorithm CVA (PRL,1995):

Constant speed

Velocity direction adjusts

according to neighbor directions

+ noise

Phase transition to finite velocity

|v| ~ (c- )0.45

t=0 (a)

High density – high noise (c)

Low density – high noise (b)

High density – low noise (d)

Simple discrete model:

2( )

( )i j i j

a r

ii i i i i j

j

x x x x

l li j a r

vm v v U x x

t

U x x C e C e

Morse potential

Rayleigh friction 2

iv

Levine et al 2000

Schweitzer et al 2000

Mogilner et al 2003

Self propulsion:

Self-acceleration+

Friction

Optimal speed

2

iv

Attractive-Repulsive potential:

Ca, Cr, la, lr

Parameter choice

A few examples:

0.5

1.0, 40.0

0.6, 0.1a r

a r

C C

l l

0.8, 0.5

0.5, 1.0

2.0, 0.1a r

a r

C C

l l

Example 1: Example 2:

A few examples:

Why are they qualitatevely different?

What if we add more particles?

Naive parameters:

0.8, 0.5

0.5, 1.0

2.0, 0.1a r

a r

C C

l l

N=100 N=200 N=300

The density is increasing!

Why is the system not extensive?

Example 2

Another example:

3, 0.5

0.5, 1.0

2.0, 0.1a r

a r

C C

l l

Example 3

Persisting double spiral Higher self propulsion

Double spirals

What is the role of the potential?

D. Ruelle,

Statistical Mechanics, Rigorous results

if not: CATASTROPHIC COLLAPSE!

From Statistical Mechanics:

Given a many-body microscopic system

Is a `real’ macroscopic description possible?i.e. thermodynamics

Interactions must obey ‘H-stability’ constraints

H-stability:

N

ijji iBNxxU allfor |)(|

A system of N >> 1 interacting agents

is H-stable if a non-negative

constant B exists such that:

where the l.h.s. is the total potential felt by the i-th agent

Pairwise interactions:

H-stable constraints on the two-body potential

( ) 0nU r d r

Catastrophic !

Pair-wise potential:

d

STABLE CATASTROPHE

2( ) 0U r d r 2( ) 0U r d r

On lattice

Pair-wise potential

Qualitatively similar

Soft-core, exponentially decaying, minimum exists

TWO particles will find a minimum, optimal distance

in BOTH cases

An H-Stable condition:

( ) 0nU r d r

Catastrophic !

Pair-wise potential:

d

STABLE CATASTROPHE

2( ) 0U r d r 2( ) 0U r d r

Infinite number of particles on lattice

Pair-wise potential

An H-Stable condition:

H-stability: Guiding interaction criteria

Morse Potential and H-stability:2

i

i

v

v

Catastrophic:

particles collapse as

Stable:

volume occupied as

N D’Orsogna et al 2005

1l C

l C

/r al l l

/r aC C C

in rotation

N

Morse Potential and H-stability:2

i

i

v

v

Catastrophic:

particles collapse as

Stable:

volume occupied as

N D’Orsogna et al 2005

1l C

l C

/r al l l

/r aC C C

in rotation

N

1

Catastrophic features and patterns:

U(r)

r

Negative area

C

1l C

l C

l

No intrinsic separation

Self-propelling speed

N=60

N=100

N=40 N=150

Catastrophic features and patterns:

Minimum at r=0 borderline

C

1l C

l C

l

U(r)

r

Negative area

Intrinsic length-scale=0

Self-propelling speed

N=60

N=60 N=100 N=200 N=300

Ring Formation:

/ 22 sin(2 / ) / 2 sin(2 / ) /

1

sin2

a r

Nr N l r N la r

n a r

C C ne e

r l l N

Implicit formula:

Excellent agreement!Number of particles

Rin

g r

adiu

s

Catastrophic Features:

C

1l C

l C

l

U(r)

r

Negative area

Finite intrinsic separation

Self-propelling speed

N=40 N=100N=150

Potential features and patterns:

r

1r

1C C

Potential Features:

Optimal spacing

`Crystalline’ at large N

Small values

r

1r

1C C

U(r)

rSTABLE

Pair-wise

Rigid disk Flock

Different

initial conditions

Example 1

Potential Features:

Optimal spacing

Collapse at large N

Larger values

U(r)

r

CATASTROPHIC

r

1r

1C C

Pair-wise

Example 2

Core free vortices, Catastrophic

N

Flock

Catastrophic Vortices:

Fly apart increases with N:

Centrifugal force mv2/r vs. interactions

mr) force vs. N-dense system

~ r max

Area decreases with N!

fixed, catastrophic vortex regime

Clockwise - Counterclockwise

coexistence

Other potentials?

Lennard-Jones Hard disks

Always stable

2( )

( )i j i j

a r

ii i i i i j

j

x x x x

l li j a r

vm v v U x x

t

U x x C e C e

p

2( )

( )i j i j

a r

ii i i i i j

j

x x x x

l li j a r

vm v v U x x

t

U x x C e C e

Power law divergences

p >= 2

Separatrix l C(2p+1)/2 =1

Always stable

p < 2

Stable vs. catastrophic

Ideas:

Add hard core

Increase N:

decrease optimal distance

until hard core barrier is hit

stability

2 particle optimal distance Increase N:

decrease optimal distance

until optimal distance =0

catastrophe

Convergence without crashing?

How to go from discrete to continuum?

Irving Kirkwood:

1 1

1 1

1

( ,..., , ,..., , )

d ... ... 1

k k k

N N

N N

Nk

R R pk k

f R R p p t

f R dR dp dp

pff U f

t m

f: Probability distribution function in phase space

Hamiltonian equations of motion, U potential

Liouville equation

1 1

1 1

1

( ,..., , ,..., , )

d ... ... 1

k k k

N N

N N

Nk

R R pk k

f R R p p t

f R dR dp dp

pff U f

t m

f: Probability distribution function in phase space

Hamiltonian equations of motion, U potential

Liouville equation

1 1

1 1

( ,..., , ,..., )

, ... ...

N N

N N

a R R p p

a f a f dR dR dp dp

a: Dynamic variable

expectation value = Macroscopic value of a

Use Liouville equation to find dynamics of variable <a,f> Hydrodynamics equations, JCP 1950

Irving Kirkwood:

Irving Kirkwood 2:

1

( , ) ( ),N

k kk

r t m R r f

MACROSCOPIC DENSITY

1

( , ) ( , ) ( ),N

k kk

r t u r t p R r f

MEAN FIELD VELOCITY

2

1

( , ) ( ),2

Nk

k kk k

pE r t R r f

m

KINETIC ENERGY DENSITY

Continuity equation, momentum transport, energy transport

Non-Hamiltonian systems?

But: These Liouville equations are valid for conserved systems!

CAN PROVE existence of Liouville’s equation for NON Hamiltonian systems

CAN generalize Irving Kirkwood continuum limit!

Our simple model becomes:

1

( ) ( )N

i ii

r m r r

average in phase space

Continuum:

Euler

Irving-Kirkwood

2( ) ( ) ( ) ...

( ) 0

r

vv v v v r U r r dr

t

vt

Continuum swarmsset rotational velocities

0

( ) ( ) lnR U r R dR D r

Density implicitly defined

( sin ,cos )v

(r)

radius

Constant speed

H-unstable

DiscreteContinuum

Continuum equations:

Linear stability analysis around

0

ˆv v

Uniform density, velocity

TRANSLATIONAL MOTION:

, , , r a a rC C l l Predict instabilities, most unstable wavelengths

2( ) ( ) ( )

( ) 0

r

vv v v v r U r r dr

t

vt

Unstable at short wavelengths

Unstable at long wavelengths

Stable

Un

stab

le

H-unstable 1

1.0, 0.5

0.5, 7.9

2.0, 0.5a r

a r

C C

l l

H-unstable 2

1.0, 0.5

0.5, 1.0

2.0, 0.5a r

a r

C C

l l

Noise?

2( )

( )i j i j

a r

ii i i i i j

j

x x x x

l li j a r

vm v v U x x

t

U x x C e C e

t

tt’t-t’

white noise

tTransition from flock to vortex

For large noise values

Noise?

tcrit

ical

COM is fixed

t

COM is moving

noise (regular time increases)

x o

f ce

nte

r of

mass

FLOCK, VCOM is constant

SWARM, VCOM is 0

noise

Variable masses:

Variable massesVortex

Segregation

mi ri= Interactions

Same segregation behavior for variable i-

s

Site avoidance:

Split Patterns

Site convergence:

T=0 Flock

Medium attraction to target

Wait a little bit

T=Tfinal Swarm

Site convergence:

T=0 Flock

Medium attraction to target

When center of mass is close to target

Turn on noise

Randomize

Turn off noise

T=Tfinal Swarm

Biologically relevant?

Under starving conditions the bacteria will aggregate

2D double spirals collapse into 3D aggregates

Direct interactions

Myxococcus xanthus

Stigmatella aurantiaca

t

Maybe!

Conclusions:

Potential determines stability of structures in large agent limit

H-stability

statistical mechanics – biology – device control

Can apply to other potentials in literature

For stability: Just Add Hard Core repulsion!

Can tune cross-over from stable-dispersive

to catastrophic-site convergent

( ) 0nU r d r

Catastrophic !

d spacing

d

the

U(r)

r

Why this condition?

U at r ~ infinity 0

U on lattice at d

If then it is more favorable to be collapsed

rdrUdrrrUddNeighborsdU 2)( 2)(~0~)()(

0)( 2 rdrU

1 1 1 1( ,..., , ,..., , ) ( ,..., , ,..., , )NN N s s sf R R p p t f R R p p t

f + Liouville

BBGKY hierarchy equations

An application, BBGKY

1( , , )f R p t

1 1( ,..., , ,..., , )Ns s sf R R p p t

Single particle distribution function

Probability of finding s<N particles with specified positions, momenta

Framework:

discrete:

individual based,

Lagrange

i=1, …, N particles

ODE-s

(Vicsek 1995)

continuum:

Fluid-like fields, Euler

PDEs

(Toner 1998)

( , )x t

( , )v x t

( ),ix t

( )iv t

Follow i-th motion

Fields at x,t

x