collective motion with...

CollectiveMotion withAlignment

AldenAstwood

Collective Motion with Alignment

Alden Astwood

April 19, 2012


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Prior Work

N overdamped Brownian particles coupled by springs in 1D

Simple but exactly solvable

This is a kind of centering interaction


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Another Model: Vicsek

Particles move at constant speed in 2D

Particles rotate to move in the same direction as othersnearby

This is an alignment interaction

Can we do something analytically?


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Single Particle

Start simple and build on it

Begin with a single particle which can point left or right

Flip it randomly at a constant rate F

What is probability to point left/right as a function of t?


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Single Particle

Model with a Master equation:

d

dtP+(t) = F[P−(t) − P+(t)]

d

dtP−(t) = F[P+(t) − P−(t)]

Solution:

P+(t) = P+(0)e−2Ft +

1

2(1 − e−2Ft)

P−(t) = P−(0)e−2Ft +

1

2(1 − e−2Ft)

Long times: P+ = P− = 1/2


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Single Particle, P vs t

0

0.5

1

0 1 2 3 4 5 6

Prob

abili

ty

t/(2F)

Probablity vs time, particle initially pointing right

P+(t)P-(t)


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Single Moving Particle

Now let the particle move at speed c in the direction it ispointing in

Can ask several questions:

What is probability to point left/right as a function of t?On average where is the particle, 〈x〉?What is the MSD, 〈x2〉− 〈x〉2?What is probability to find the particle in the neighborhoodof x, Q(x, t)dx?


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Single Moving Particle – Equations of Motion

Need to formulate equations of motion

Let Q+(x, t)dx be probability to find the particle movingto the right in the neighborhood of x at time t

Similarly define Q−(x, t)

How do they evolve?

∂

∂tQ+(x, t) = ?

∂

∂tQ−(x, t) = ?


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∂

∂tQ+(x, t) =

− c∂

∂xQ+(x, t) + F[Q−(x, t) −Q+(x, t)]

∂

∂tQ−(x, t) =

+ c∂

∂xQ−(x, t) + F[Q+(x, t) −Q−(x, t)]

What are the equations of motion for Q±?

If c = 0, we know the equations of motion

We know solution of ∂∂tf(x, t) = −c ∂∂xf(x, t) is

f(x− ct, 0)

Now we can try to answer some of those questions


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∂

∂tQ+(x, t) =

− c∂

∂xQ+(x, t) +

F[Q−(x, t) −Q+(x, t)]

∂

∂tQ−(x, t) =

+ c∂

∂xQ−(x, t) +

F[Q+(x, t) −Q−(x, t)]




f(x− ct, 0)



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∂

∂tQ+(x, t) = − c

∂

∂xQ+(x, t) + F[Q−(x, t) −Q+(x, t)]

∂

∂tQ−(x, t) = + c

∂

∂xQ−(x, t) + F[Q+(x, t) −Q−(x, t)]




f(x− ct, 0)



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Single Moving Particle – Direction

What is probability to point left/right as a function of t?

Define P±(t) ≡∫∞−∞Q±(x, t)dx

P±(t) obey

d

dtP+(t) = F[P−(t) − P+(t)]

d

dtP−(t) = F[P+(t) − P−(t)]

Flipping happens independently of position


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Single Moving Particle – Average Position

On average, where is the particle?

Define first moment:

〈x(t)〉 =∫∞−∞ x[Q+(x, t) +Q−(x, t)]dx

Evolution is

d

dt〈x(t)〉 = c[P+(t) − P−(t)] = c〈σ(t)〉


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Evolution of average direction is

〈σ(t)〉 = e−2Ft〈σ(0)〉

First moment is then

〈x(t)〉 = 〈x(0)〉+ 1 − e−2Ft

2Fc〈σ(0)〉

〈x(t)〉 ∝ t at short times, constant at long times


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0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6

(<x>

-<x(

0)>

)(2F

/c)

t/(2F)

Single Particle <x> vs t, Initially Pointing Right


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Single Moving Particle – MSD

What is the mean squared displacement 〈x2〉?For Q+(x, 0) = Q−(x, 0) = δ(x)/2, can shew

〈x2(t)〉 = 2c2 2Ft− (1 − e−2Ft)

(2F)2

Ballistic at short times:

〈x2(t)〉 ≈ (ct)2

Diffusive at long times:

〈x2(t)〉 ≈ 2Defft with Deff =c2

2F


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10-4

10-3

10-2

10-1

100

101

102

10-2 10-1 100 101 102

MSD

time

Single Particle MSD c=F=1


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Single Moving Particle – Distribution Function

Define: Q(x, t) ≡ Q+(x, t) +Q−(x, t)

Add and subtract equations for ∂Q+

∂t and ∂Q−

∂t ,

∂

∂t[Q+ +Q−] = −c

∂

∂x[Q+ −Q−]

∂

∂t[Q+ −Q−] = −c

∂

∂x[Q+ +Q−] − 2F[Q+ −Q−]

Take ∂/∂t of the first and sub the second,

∂2

∂2t[Q+ +Q−] = c

2 ∂2

∂x2[Q+ +Q−] + 2Fc

∂

∂x[Q+ −Q−]

Eliminate ∂∂x [Q+ −Q−]:

∂2

∂t2Q+ 2F

∂

∂tQ = c2 ∂

2

∂x2Q


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Define: Q(x, t) ≡ Q+(x, t) +Q−(x, t)Add and subtract equations for ∂Q+

∂t and ∂Q−

∂t ,

∂

∂t[Q+ +Q−] = −c

∂

∂x[Q+ −Q−]

∂

∂t[Q+ −Q−] = −c

∂

∂x[Q+ +Q−] − 2F[Q+ −Q−]

Take ∂/∂t of the first and sub the second,

∂2

∂2t[Q+ +Q−] = c

2 ∂2

∂x2[Q+ +Q−] + 2Fc

∂

∂x[Q+ −Q−]

Eliminate ∂∂x [Q+ −Q−]:

∂2

∂t2Q+ 2F

∂

∂tQ = c2 ∂

2

∂x2Q


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Q(x, t) obeys “telegraph equation”

∂2

∂t2Q+ 2F

∂

∂tQ = c2 ∂

2

∂x2Q

Remove first term: diffusion equation

Remove second term: wave equation

No propagation faster than c

Solution for Q(x, 0) = δ(x), Q(x, 0) = 0 for |x| < ct:

Q(x, t) = e−Ft{δ(x−ct)+δ(x+ct)

2 + F2c

[I0(Λ) +

FtΛI1(Λ)

]}where Λ ≡ (F/c)

√c2t2 − x2.


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Two Stationary Particles

This simple model is well known (Taylor 1922, Goldstein1951, Kac 1974, Segel 1978, Othmer et al 1988, Kenkreand Sevilla 2007, etc)

Now let’s try to add to it

Go back to stationary particle and give him a buddy

22 = 4 Total Configurations: ++,+−,−+,−−

Keep random flipping, but make aligned states favorable


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Two Stationary Particles – Rates

++

+−−+

−−

Ff

Ff

Ff

Ff

F: rate to go from unaligned to aligned state

f: rate to go from aligned to unaligned state

f < F: aligned states favored


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++

+−−+

−−

F

fF

f

Ff

Ff





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++

+−−+

−−

Ff

Ff

Ff

Ff





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Two Stationary Particles

Master Equation:

dP++

dt= F[P+− + P−+] − 2fP++

dP−−

dt= F[P+− + P−+] − 2fP−−

dP+−

dt= f[P++ + P−−] − 2FP+−

dP−+

dt= f[P++ + P−−] − 2FP−+

Can be solved, at long times get

P++ = P−− =1

2

F

F+ f,P+− = P−+ =

1

2

f

F+ f


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Two Moving Particles

Now suppose they move at speed c

Distribution functions become Q++(x1, x2, t) etc

Add drift terms:

∂Q++

∂t= −c

(∂∂x1

+ ∂∂x2

)Q++ + F[Q+− +Q−+] − 2fQ++

∂Q−−

∂t= −c

(− ∂∂x1

− ∂∂x2

)Q−− + F[Q+− +Q−+] − 2fQ−−

∂Q+−

∂t= −c

(∂∂x1

− ∂∂x2

)Q+− + f[Q++ +Q−−] − 2FQ+−

∂Q−+

∂t= −c

(− ∂∂x1

+ ∂∂x2

)Q−+ + f[Q++ +Q−−] − 2FQ−+


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Two Moving Particles

Can solve exactly

First two moments fairly easy to find

First moment:

〈x1〉 and 〈x2〉 ∝ ct at short times, depending on i.c.

〈x1〉 and 〈x2〉 = const at long times


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Two Particles – Second Moment

Second Moment:

〈x21〉 and 〈x2

2〉 ≈ (ct)2 at short times

〈x21〉 and 〈x2

2〉 ≈ const + c2 F2 + f2

Ff(F+ f)t at long times

Effective diffusion constant

Deff = c2 F2 + f2

2Ff(F+ f)

Invariant if F and f are interchanged!


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Two Particles – Correlation

Are the positions of the two birds correlated?

Look at 〈x1x2〉

〈x1x2〉 ≈ 0 at short times, depending on i.c.

〈x1x2〉 ≈ const + c2 F2 − f2

Ff(F+ f)t

So the positions of the two birds are positively correlated ifF > f


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Two Particles – Reduced Quantities

What is equation of motion for

Q ≡ Q++ +Q−− +Q+− +Q−+ ?

Unfortunately not so simple as telegraph equation

Can write with nonlocal memory in time AND space,

∂∂tQ(x, t) = −

∫ ∫ ∫t0 A(x1 − x

′1, x2 − x

′2, t− t ′)Q(x ′1, x ′2, t ′)dt ′dx ′1dx

′2


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Generalization to N Particles

Distribution functions become Q~σ(~x, t)

~σ an N element “vector” which represents directions

2N possible configurations

~x an N dimensional vector which represents positions

Equations of motion (general form):

∂

∂tQ~σ = −c

∑n

σn∂

∂xnQ~σ +

∑~σ ′

[w~σ,~σ ′Q~σ ′ −w~σ ′,~σQ~σ]


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N Particles – Rates

How to construct rates w~σ ′,~σ?

Borrow from an existing model: Ising model

Ising model itself has no dynamics

There is a way to add dynamics (Glauber 1963)


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N Particles – Glauber Dynamics

Start with Ising model Hamiltonian

H(~σ) = −∑m,n

Jmnσmσn

In steady state, demand P~σ = e−βH(~σ)/Z

Only allow one spin flip at a time

Detailed balance: In steady state, each term in:

d

dtP~σ = 0 =

∑~σ ′

[w~σ,~σ ′P~σ ′ −w~σ ′,~σP~σ]

is also zero


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N Particles – What can we do

Can’t solve for probabilities P~σ(t) in Glauber dynamics

Little hope for finding Q~σ(~x, t)

What can we get?

Average spin 〈σm〉 and pair correlations 〈σmσn〉These are related to 〈xm〉 and 〈xmxn〉


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Moments – Hard Way

How to calculate moments?

Fourier transform eqns of motion for Q~σ(~x, t)

Rewrite moments in terms of k-space derivatives

Hope you can solve resulting eqns of motion

Easy to get first moment this way, hard to get second


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Moments – Easy way

Instead, write “Langevin equations”:

xm(t) = cσm(t)

Integrate:

xm(t) − xm(0) = c

∫t0dt ′σm(t ′)

Ensemble average (taking xm(0) = 0):

〈xm(t)〉 = c∫t

0dt ′〈σm(t ′)〉

〈xm(t)xn(t)〉 = c2

∫t0dt ′∫t

0dt ′′〈σm(t ′)σn(t

′′)〉


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N Particles – 1st Moment

We have:

〈xm(t)〉 = c∫t

0dt ′〈σm(t ′)〉

How does 〈σm〉 evolve?

In “mean field” approximation, 〈σm〉 = 〈σ〉 and

τd

dt〈σ〉 = −〈σ〉+ tanh(βJz〈σ〉)

Fixed points satisfy

〈σ〉∗ = tanh(βJz〈σ〉∗)


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〈σ〉 Fixed Points

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2

<σ>*

1/(βJz)

Fixed Points vs Temperature

StableUnstable


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N Particles – 1st Moment

Above critical point, only fixed point is 〈σ〉 = 0, so 〈x〉const at long times

Nonzero solutions for 〈σ〉 below critical point, so 〈x〉 ∝ ctat long times

At the critical point, 〈σ〉 = 0 in steady state, but thedynamics are very slow, 〈σ〉 ∝ 1/

√t, so 〈x〉 ∝

√t


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N Particles – 2nd Moment

Won’t say much for now

Expressions for two time pair correlation exist in literature(Suzuki and Kubo 1968)

Expect diffusive motion above critical point, what is Deff?

What happens at and below critical point?


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Further Work

Try to get (approximate) memory for reduced quantities(density etc)

Lots of things could be added:

Add biasHigher Dimensions – XY model/Heisenberg model?Finite range interactions

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