Collective Motion with Alignment
Collective Motion with Alignment Alden Astwood
Collective Motion with Alignment Alden Astwood
April 19, 2012
Prior Work Collective Motion with Alignment Alden Astwood
N overdamped Brownian particles coupled by springs in 1D Simple but exactly solvable This is a kind of centering interaction
Another Model: Vicsek Collective Motion with Alignment Alden Astwood
Particles move at constant speed in 2D Particles rotate to move in the same direction as others nearby This is an alignment interaction Can we do something analytically?
Single Particle Collective Motion with Alignment Alden Astwood
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?
Single Particle Collective Motion with Alignment
Model with a Master equation:
Alden Astwood
d P+ (t) = F[P− (t) − P+ (t)] dt d P− (t) = F[P+ (t) − P− (t)] dt Solution: 1 P+ (t) = P+ (0)e−2Ft + (1 − e−2Ft ) 2 1 P− (t) = P− (0)e−2Ft + (1 − e−2Ft ) 2 Long times: P+ = P− = 1/2
Single Particle, P vs t Collective Motion with Alignment
Probablity vs time, particle initially pointing right 1
Probability
Alden Astwood
P+(t) P-(t)
0.5
0 0
1
2
3 t/(2F)
4
5
6
Single Moving Particle Collective Motion with Alignment Alden Astwood
Now let the particle move at speed c in the direction it is pointing in
-4
-2
0 x(t)
2
4
Can ask several questions: What is probability to point left/right as a function of t? On average where is the particle, hxi? What is the MSD, hx2 i − hxi2 ? What is probability to find the particle in the neighborhood of x, Q(x, t)dx?
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
Need to formulate equations of motion Let Q+ (x, t)dx be probability to find the particle moving to the right in the neighborhood of x at time t Similarly define Q− (x, t) How do they evolve? ∂ Q+ (x, t) = ? ∂t ∂ Q− (x, t) = ? ∂t
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
∂ Q+ (x, t) = ∂t ∂ Q− (x, t) = ∂t What are the equations of motion for Q± ?
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
∂ Q+ (x, t) = ∂t ∂ Q− (x, t) = ∂t What are the equations of motion for Q± ? If c = 0, we know the equations of motion
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
∂ Q+ (x, t) = ∂t ∂ Q− (x, t) = ∂t
F[Q− (x, t) − Q+ (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
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
∂ Q+ (x, t) = ∂t ∂ Q− (x, t) = ∂t
F[Q− (x, t) − Q+ (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 f(x − ct, 0)
∂ ∂t f(x, t)
∂ = −c ∂x f(x, t) is
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
∂ ∂ Q+ (x, t) = − c Q+ (x, t) + F[Q− (x, t) − Q+ (x, t)] ∂t ∂x ∂ ∂ Q− (x, t) = + c Q− (x, t) + F[Q+ (x, t) − Q− (x, t)] ∂t ∂x What are the equations of motion for Q± ? If c = 0, we know the equations of motion We know solution of f(x − ct, 0)
∂ ∂t f(x, t)
∂ = −c ∂x f(x, t) is
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
∂ ∂ Q+ (x, t) = − c Q+ (x, t) + F[Q− (x, t) − Q+ (x, t)] ∂t ∂x ∂ ∂ Q− (x, t) = + c Q− (x, t) + F[Q+ (x, t) − Q− (x, t)] ∂t ∂x What are the equations of motion for Q± ? If c = 0, we know the equations of motion We know solution of f(x − ct, 0)
∂ ∂t f(x, t)
∂ = −c ∂x f(x, t) is
Now we can try to answer some of those questions
Single Moving Particle – Direction Collective Motion with Alignment Alden Astwood
What is probability to point left/right as a function of t? R∞ Define P± (t) ≡ −∞ Q± (x, t)dx P± (t) obey d P+ (t) = F[P− (t) − P+ (t)] dt d P− (t) = F[P+ (t) − P− (t)] dt Flipping happens independently of position
Single Moving Particle – Average Position Collective Motion with Alignment Alden Astwood
On average, where is the particle? Define first moment: Z∞ hx(t)i = x[Q+ (x, t) + Q− (x, t)]dx −∞
Evolution is d hx(t)i = c[P+ (t) − P− (t)] = chσ(t)i dt
Single Moving Particle – Average Position Collective Motion with Alignment Alden Astwood
Evolution of average direction is hσ(t)i = e−2Ft hσ(0)i First moment is then hx(t)i = hx(0)i +
1 − e−2Ft chσ(0)i 2F
hx(t)i ∝ t at short times, constant at long times
Single Moving Particle – Average Position Collective Motion with Alignment
Single Particle <x> vs t, Initially Pointing Right
Alden Astwood
(<x>-<x(0)>)(2F/c)
1 0.8 0.6 0.4 0.2 0 0
1
2
3 t/(2F)
4
5
6
Single Moving Particle – MSD Collective Motion with Alignment Alden Astwood
What is the mean squared displacement hx2 i? For Q+ (x, 0) = Q− (x, 0) = δ(x)/2, can shew hx2 (t)i = 2c2
2Ft − (1 − e−2Ft ) (2F)2
Ballistic at short times: hx2 (t)i ≈ (ct)2 Diffusive at long times: hx2 (t)i ≈ 2Deff t with Deff =
c2 2F
Single Moving Particle – Average Position Collective Motion with Alignment
Single Particle MSD c=F=1 102
Alden Astwood
101
MSD
100 10-1 10-2 10-3 10-4 10-2
10-1
100 time
101
102
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t)
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t) + Add and subtract equations for ∂Q ∂t and
∂Q− ∂t ,
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t) + Add and subtract equations for ∂Q ∂t and
∂Q− ∂t ,
∂ ∂ [Q+ + Q− ] = −c [Q+ − Q− ] ∂t ∂x ∂ ∂ [Q+ − Q− ] = −c [Q+ + Q− ] − 2F[Q+ − Q− ] ∂t ∂x
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t) + Add and subtract equations for ∂Q ∂t and
∂Q− ∂t ,
∂ ∂ [Q+ + Q− ] = −c [Q+ − Q− ] ∂t ∂x ∂ ∂ [Q+ − Q− ] = −c [Q+ + Q− ] − 2F[Q+ − Q− ] ∂t ∂x Take ∂/∂t of the first and sub the second,
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t) + Add and subtract equations for ∂Q ∂t and
∂Q− ∂t ,
∂ ∂ [Q+ + Q− ] = −c [Q+ − Q− ] ∂t ∂x ∂ ∂ [Q+ − Q− ] = −c [Q+ + Q− ] − 2F[Q+ − Q− ] ∂t ∂x Take ∂/∂t of the first and sub the second, 2 ∂2 ∂ 2 ∂ [Q + Q ] = c [Q+ + Q− ] + 2Fc [Q+ − Q− ] + − 2 2 ∂ t ∂x ∂x
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t) + Add and subtract equations for ∂Q ∂t and
∂Q− ∂t ,
∂ ∂ [Q+ + Q− ] = −c [Q+ − Q− ] ∂t ∂x ∂ ∂ [Q+ − Q− ] = −c [Q+ + Q− ] − 2F[Q+ − Q− ] ∂t ∂x Take ∂/∂t of the first and sub the second, 2 ∂2 ∂ 2 ∂ [Q + Q ] = c [Q+ + Q− ] + 2Fc [Q+ − Q− ] + − 2 2 ∂ t ∂x ∂x
Eliminate
∂ ∂x [Q+
− Q− ]:
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t) + Add and subtract equations for ∂Q ∂t and
∂Q− ∂t ,
∂ ∂ [Q+ + Q− ] = −c [Q+ − Q− ] ∂t ∂x ∂ ∂ [Q+ − Q− ] = −c [Q+ + Q− ] − 2F[Q+ − Q− ] ∂t ∂x Take ∂/∂t of the first and sub the second, 2 ∂2 ∂ 2 ∂ [Q + Q ] = c [Q+ + Q− ] + 2Fc [Q+ − Q− ] + − 2 2 ∂ t ∂x ∂x
Eliminate
∂ ∂x [Q+
− Q− ]:
2 ∂2 ∂ 2 ∂ Q + 2F Q = c Q ∂t2 ∂t ∂x2
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Q(x, t) obeys “telegraph equation” 2 ∂2 ∂ 2 ∂ Q = c Q + 2F Q ∂t2 ∂t ∂x2
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
√ where Λ ≡ (F/c) c2 t2 − x2 .
+
F 2c
I0 (Λ) +
Ft Λ I1 (Λ)
Single Moving Particle – Distribution Function Collective Motion with Alignment
Telegraph Equation Solution, c=F=1, t=0.000000 1
Q(x,t)
Alden Astwood
0.5
0 -4
-2
0 x
2
4
Two Stationary Particles Collective Motion with Alignment Alden Astwood
This simple model is well known (Taylor 1922, Goldstein 1951, Kac 1974, Segel 1978, Othmer et al 1988, Kenkre and 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
Two Stationary Particles – Rates Collective Motion with Alignment
++
Alden Astwood
−+
+−
−−
Two Stationary Particles – Rates Collective Motion with Alignment Alden Astwood
++ F
−+
+−
−−
F: rate to go from unaligned to aligned state
Two Stationary Particles – Rates Collective Motion with Alignment Alden Astwood
++ F f −+
+−
−−
F: rate to go from unaligned to aligned state f: rate to go from aligned to unaligned state
Two Stationary Particles – Rates Collective Motion with Alignment Alden Astwood
++ F
F f
f
−+
+− f
f
F
F −−
F: rate to go from unaligned to aligned state f: rate to go from aligned to unaligned state f < F: aligned states favored
Two Stationary Particles Collective Motion with Alignment Alden Astwood
Master Equation: dP++ dt dP−− dt dP+− dt dP−+ dt
= F[P+− + P−+ ] − 2fP++ = F[P+− + P−+ ] − 2fP−− = f[P++ + P−− ] − 2FP+− = f[P++ + P−− ] − 2FP−+
Can be solved, at long times get P++ = P−− =
1 F 1 f , P+− = P−+ = 2F+f 2F+f
Two Moving Particles Collective Motion with Alignment Alden Astwood
Now suppose they move at speed c Distribution functions become Q++ (x1 , x2 , t) etc Add drift terms: ∂Q++ ∂t ∂Q−− ∂t ∂Q+− ∂t ∂Q−+ ∂t
∂ ∂x1
∂ ∂x2
Q++ + F[Q+− + Q−+ ] − 2fQ++ ∂ ∂ = −c − ∂x − ∂x Q−− + F[Q+− + Q−+ ] − 2fQ−− 1 2 ∂ ∂ = −c ∂x − ∂x Q+− + f[Q++ + Q−− ] − 2FQ+− 1 2 ∂ ∂ = −c − ∂x + ∂x Q−+ + f[Q++ + Q−− ] − 2FQ−+ 1 2 = −c
+
Two Moving Particles Collective Motion with Alignment Alden Astwood
Can solve exactly First two moments fairly easy to find First moment: hx1 i and hx2 i ∝ ct at short times, depending on i.c. hx1 i and hx2 i = const at long times
Two Particles – Second Moment Collective Motion with Alignment Alden Astwood
Second Moment: hx21 i and hx22 i ≈ (ct)2 at short times hx21 i and hx22 i ≈ const + c2
F2 + f2 t at long times Ff(F + f)
Effective diffusion constant Deff = c2
F2 + f2 2Ff(F + f)
Invariant if F and f are interchanged!
Two Particles – Correlation Collective Motion with Alignment Alden Astwood
Are the positions of the two birds correlated? Look at hx1 x2 i hx1 x2 i ≈ 0 at short times, depending on i.c. hx1 x2 i ≈ const + c2
F2 − f2 t Ff(F + f)
So the positions of the two birds are positively correlated if F>f
Two Particles – Reduced Quantities Collective Motion with Alignment Alden Astwood
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, ∂ ∂t Q(x, t)
=−
R R Rt 0
A(x1 − x10 , x2 − x20 , t − t 0 )Q(x10 , x20 , t 0 )dt 0 dx10 dx20
Generalization to N Particles Collective Motion with Alignment Alden Astwood
Distribution functions become Qσ 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): X X ∂ ∂ [wσ Qσ Qσ σn ~ + ~ ,~ σ 0 Qσ ~ 0 − wσ ~ 0 ,~ σ Qσ ~] ~ = −c ∂t ∂xn 0 n σ ~
N Particles – Rates Collective Motion with Alignment Alden Astwood
How to construct rates wσ ~ 0 ,~ σ? Borrow from an existing model: Ising model Ising model itself has no dynamics There is a way to add dynamics (Glauber 1963)
N Particles – Glauber Dynamics Collective Motion with Alignment Alden Astwood
Start with Ising model Hamiltonian X H(~σ) = − Jmn σm σn m,n −βH(~ σ) /Z In steady state, demand Pσ ~ =e
Only allow one spin flip at a time Detailed balance: In steady state, each term in: X d [wσ Pσ ~ 0 − wσ ~ 0 ,~ σ Pσ ~] ~ ,~ σ 0 Pσ ~ =0= dt 0 σ ~
is also zero
N Particles – What can we do Collective Motion with Alignment Alden Astwood
Can’t solve for probabilities Pσ ~ (t) in Glauber dynamics Little hope for finding Qσ x, t) ~ (~ What can we get? Average spin hσm i and pair correlations hσm σn i These are related to hxm i and hxm xn i
Moments – Hard Way Collective Motion with Alignment Alden Astwood
How to calculate moments? Fourier transform eqns of motion for Qσ 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
Moments – Easy way Collective Motion with Alignment Alden Astwood
Instead, write “Langevin equations”: x˙ m (t) = cσm (t)
Moments – Easy way Collective Motion with Alignment
Instead, write “Langevin equations”:
Alden Astwood
x˙ m (t) = cσm (t) Integrate: Zt xm (t) − xm (0) = c 0
dt 0 σm (t 0 )
Moments – Easy way Collective Motion with Alignment
Instead, write “Langevin equations”:
Alden Astwood
x˙ m (t) = cσm (t) Integrate: Zt xm (t) − xm (0) = c
dt 0 σm (t 0 )
0
Ensemble average (taking xm (0) = 0): Zt
dt 0 hσm (t 0 )i 0 Zt Zt 2 0 hxm (t)xn (t)i = c dt dt 00 hσm (t 0 )σn (t 00 )i hxm (t)i = c
0
0
N Particles – 1st Moment Collective Motion with Alignment Alden Astwood
We have:
Zt hxm (t)i = c
dt 0 hσm (t 0 )i
0
How does hσm i evolve? In “mean field” approximation, hσm i = hσi and τ
d hσi = −hσi + tanh(βJzhσi) dt
Fixed points satisfy hσi∗ = tanh(βJzhσi∗ )
hσi Fixed Points Collective Motion with Alignment
Fixed Points vs Temperature
Alden Astwood
1
Stable Unstable
*
0.5
0
-0.5
-1 0
0.5
1 1/(βJz)
1.5
2
N Particles – 1st Moment Collective Motion with Alignment Alden Astwood
Above critical point, only fixed point is hσi = 0, so hxi const at long times
N Particles – 1st Moment Collective Motion with Alignment Alden Astwood
Above critical point, only fixed point is hσi = 0, so hxi const at long times Nonzero solutions for hσi below critical point, so hxi ∝ ct at long times
N Particles – 1st Moment Collective Motion with Alignment Alden Astwood
Above critical point, only fixed point is hσi = 0, so hxi const at long times Nonzero solutions for hσi below critical point, so hxi ∝ ct at long times At the critical point, hσi = 0 in steady state, but √ √ the dynamics are very slow, hσi ∝ 1/ t, so hxi ∝ t
N Particles – 2nd Moment Collective Motion with Alignment Alden Astwood
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?
Further Work Collective Motion with Alignment Alden Astwood
Try to get (approximate) memory for reduced quantities (density etc) Lots of things could be added: Add bias Higher Dimensions – XY model/Heisenberg model? Finite range interactions
Collective Motion with Alignment Alden Astwood
April 19, 2012
Prior Work Collective Motion with Alignment Alden Astwood
N overdamped Brownian particles coupled by springs in 1D Simple but exactly solvable This is a kind of centering interaction
Another Model: Vicsek Collective Motion with Alignment Alden Astwood
Particles move at constant speed in 2D Particles rotate to move in the same direction as others nearby This is an alignment interaction Can we do something analytically?
Single Particle Collective Motion with Alignment Alden Astwood
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?
Single Particle Collective Motion with Alignment
Model with a Master equation:
Alden Astwood
d P+ (t) = F[P− (t) − P+ (t)] dt d P− (t) = F[P+ (t) − P− (t)] dt Solution: 1 P+ (t) = P+ (0)e−2Ft + (1 − e−2Ft ) 2 1 P− (t) = P− (0)e−2Ft + (1 − e−2Ft ) 2 Long times: P+ = P− = 1/2
Single Particle, P vs t Collective Motion with Alignment
Probablity vs time, particle initially pointing right 1
Probability
Alden Astwood
P+(t) P-(t)
0.5
0 0
1
2
3 t/(2F)
4
5
6
Single Moving Particle Collective Motion with Alignment Alden Astwood
Now let the particle move at speed c in the direction it is pointing in
-4
-2
0 x(t)
2
4
Can ask several questions: What is probability to point left/right as a function of t? On average where is the particle, hxi? What is the MSD, hx2 i − hxi2 ? What is probability to find the particle in the neighborhood of x, Q(x, t)dx?
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
Need to formulate equations of motion Let Q+ (x, t)dx be probability to find the particle moving to the right in the neighborhood of x at time t Similarly define Q− (x, t) How do they evolve? ∂ Q+ (x, t) = ? ∂t ∂ Q− (x, t) = ? ∂t
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
∂ Q+ (x, t) = ∂t ∂ Q− (x, t) = ∂t What are the equations of motion for Q± ?
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
∂ Q+ (x, t) = ∂t ∂ Q− (x, t) = ∂t What are the equations of motion for Q± ? If c = 0, we know the equations of motion
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
∂ Q+ (x, t) = ∂t ∂ Q− (x, t) = ∂t
F[Q− (x, t) − Q+ (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
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
∂ Q+ (x, t) = ∂t ∂ Q− (x, t) = ∂t
F[Q− (x, t) − Q+ (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 f(x − ct, 0)
∂ ∂t f(x, t)
∂ = −c ∂x f(x, t) is
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
∂ ∂ Q+ (x, t) = − c Q+ (x, t) + F[Q− (x, t) − Q+ (x, t)] ∂t ∂x ∂ ∂ Q− (x, t) = + c Q− (x, t) + F[Q+ (x, t) − Q− (x, t)] ∂t ∂x What are the equations of motion for Q± ? If c = 0, we know the equations of motion We know solution of f(x − ct, 0)
∂ ∂t f(x, t)
∂ = −c ∂x f(x, t) is
Single Moving Particle – Equations of Motion Collective Motion with Alignment Alden Astwood
∂ ∂ Q+ (x, t) = − c Q+ (x, t) + F[Q− (x, t) − Q+ (x, t)] ∂t ∂x ∂ ∂ Q− (x, t) = + c Q− (x, t) + F[Q+ (x, t) − Q− (x, t)] ∂t ∂x What are the equations of motion for Q± ? If c = 0, we know the equations of motion We know solution of f(x − ct, 0)
∂ ∂t f(x, t)
∂ = −c ∂x f(x, t) is
Now we can try to answer some of those questions
Single Moving Particle – Direction Collective Motion with Alignment Alden Astwood
What is probability to point left/right as a function of t? R∞ Define P± (t) ≡ −∞ Q± (x, t)dx P± (t) obey d P+ (t) = F[P− (t) − P+ (t)] dt d P− (t) = F[P+ (t) − P− (t)] dt Flipping happens independently of position
Single Moving Particle – Average Position Collective Motion with Alignment Alden Astwood
On average, where is the particle? Define first moment: Z∞ hx(t)i = x[Q+ (x, t) + Q− (x, t)]dx −∞
Evolution is d hx(t)i = c[P+ (t) − P− (t)] = chσ(t)i dt
Single Moving Particle – Average Position Collective Motion with Alignment Alden Astwood
Evolution of average direction is hσ(t)i = e−2Ft hσ(0)i First moment is then hx(t)i = hx(0)i +
1 − e−2Ft chσ(0)i 2F
hx(t)i ∝ t at short times, constant at long times
Single Moving Particle – Average Position Collective Motion with Alignment
Single Particle <x> vs t, Initially Pointing Right
Alden Astwood
(<x>-<x(0)>)(2F/c)
1 0.8 0.6 0.4 0.2 0 0
1
2
3 t/(2F)
4
5
6
Single Moving Particle – MSD Collective Motion with Alignment Alden Astwood
What is the mean squared displacement hx2 i? For Q+ (x, 0) = Q− (x, 0) = δ(x)/2, can shew hx2 (t)i = 2c2
2Ft − (1 − e−2Ft ) (2F)2
Ballistic at short times: hx2 (t)i ≈ (ct)2 Diffusive at long times: hx2 (t)i ≈ 2Deff t with Deff =
c2 2F
Single Moving Particle – Average Position Collective Motion with Alignment
Single Particle MSD c=F=1 102
Alden Astwood
101
MSD
100 10-1 10-2 10-3 10-4 10-2
10-1
100 time
101
102
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t)
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t) + Add and subtract equations for ∂Q ∂t and
∂Q− ∂t ,
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t) + Add and subtract equations for ∂Q ∂t and
∂Q− ∂t ,
∂ ∂ [Q+ + Q− ] = −c [Q+ − Q− ] ∂t ∂x ∂ ∂ [Q+ − Q− ] = −c [Q+ + Q− ] − 2F[Q+ − Q− ] ∂t ∂x
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t) + Add and subtract equations for ∂Q ∂t and
∂Q− ∂t ,
∂ ∂ [Q+ + Q− ] = −c [Q+ − Q− ] ∂t ∂x ∂ ∂ [Q+ − Q− ] = −c [Q+ + Q− ] − 2F[Q+ − Q− ] ∂t ∂x Take ∂/∂t of the first and sub the second,
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t) + Add and subtract equations for ∂Q ∂t and
∂Q− ∂t ,
∂ ∂ [Q+ + Q− ] = −c [Q+ − Q− ] ∂t ∂x ∂ ∂ [Q+ − Q− ] = −c [Q+ + Q− ] − 2F[Q+ − Q− ] ∂t ∂x Take ∂/∂t of the first and sub the second, 2 ∂2 ∂ 2 ∂ [Q + Q ] = c [Q+ + Q− ] + 2Fc [Q+ − Q− ] + − 2 2 ∂ t ∂x ∂x
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t) + Add and subtract equations for ∂Q ∂t and
∂Q− ∂t ,
∂ ∂ [Q+ + Q− ] = −c [Q+ − Q− ] ∂t ∂x ∂ ∂ [Q+ − Q− ] = −c [Q+ + Q− ] − 2F[Q+ − Q− ] ∂t ∂x Take ∂/∂t of the first and sub the second, 2 ∂2 ∂ 2 ∂ [Q + Q ] = c [Q+ + Q− ] + 2Fc [Q+ − Q− ] + − 2 2 ∂ t ∂x ∂x
Eliminate
∂ ∂x [Q+
− Q− ]:
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Define: Q(x, t) ≡ Q+ (x, t) + Q− (x, t) + Add and subtract equations for ∂Q ∂t and
∂Q− ∂t ,
∂ ∂ [Q+ + Q− ] = −c [Q+ − Q− ] ∂t ∂x ∂ ∂ [Q+ − Q− ] = −c [Q+ + Q− ] − 2F[Q+ − Q− ] ∂t ∂x Take ∂/∂t of the first and sub the second, 2 ∂2 ∂ 2 ∂ [Q + Q ] = c [Q+ + Q− ] + 2Fc [Q+ − Q− ] + − 2 2 ∂ t ∂x ∂x
Eliminate
∂ ∂x [Q+
− Q− ]:
2 ∂2 ∂ 2 ∂ Q + 2F Q = c Q ∂t2 ∂t ∂x2
Single Moving Particle – Distribution Function Collective Motion with Alignment Alden Astwood
Q(x, t) obeys “telegraph equation” 2 ∂2 ∂ 2 ∂ Q = c Q + 2F Q ∂t2 ∂t ∂x2
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
√ where Λ ≡ (F/c) c2 t2 − x2 .
+
F 2c
I0 (Λ) +
Ft Λ I1 (Λ)
Single Moving Particle – Distribution Function Collective Motion with Alignment
Telegraph Equation Solution, c=F=1, t=0.000000 1
Q(x,t)
Alden Astwood
0.5
0 -4
-2
0 x
2
4
Two Stationary Particles Collective Motion with Alignment Alden Astwood
This simple model is well known (Taylor 1922, Goldstein 1951, Kac 1974, Segel 1978, Othmer et al 1988, Kenkre and 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
Two Stationary Particles – Rates Collective Motion with Alignment
++
Alden Astwood
−+
+−
−−
Two Stationary Particles – Rates Collective Motion with Alignment Alden Astwood
++ F
−+
+−
−−
F: rate to go from unaligned to aligned state
Two Stationary Particles – Rates Collective Motion with Alignment Alden Astwood
++ F f −+
+−
−−
F: rate to go from unaligned to aligned state f: rate to go from aligned to unaligned state
Two Stationary Particles – Rates Collective Motion with Alignment Alden Astwood
++ F
F f
f
−+
+− f
f
F
F −−
F: rate to go from unaligned to aligned state f: rate to go from aligned to unaligned state f < F: aligned states favored
Two Stationary Particles Collective Motion with Alignment Alden Astwood
Master Equation: dP++ dt dP−− dt dP+− dt dP−+ dt
= F[P+− + P−+ ] − 2fP++ = F[P+− + P−+ ] − 2fP−− = f[P++ + P−− ] − 2FP+− = f[P++ + P−− ] − 2FP−+
Can be solved, at long times get P++ = P−− =
1 F 1 f , P+− = P−+ = 2F+f 2F+f
Two Moving Particles Collective Motion with Alignment Alden Astwood
Now suppose they move at speed c Distribution functions become Q++ (x1 , x2 , t) etc Add drift terms: ∂Q++ ∂t ∂Q−− ∂t ∂Q+− ∂t ∂Q−+ ∂t
∂ ∂x1
∂ ∂x2
Q++ + F[Q+− + Q−+ ] − 2fQ++ ∂ ∂ = −c − ∂x − ∂x Q−− + F[Q+− + Q−+ ] − 2fQ−− 1 2 ∂ ∂ = −c ∂x − ∂x Q+− + f[Q++ + Q−− ] − 2FQ+− 1 2 ∂ ∂ = −c − ∂x + ∂x Q−+ + f[Q++ + Q−− ] − 2FQ−+ 1 2 = −c
+
Two Moving Particles Collective Motion with Alignment Alden Astwood
Can solve exactly First two moments fairly easy to find First moment: hx1 i and hx2 i ∝ ct at short times, depending on i.c. hx1 i and hx2 i = const at long times
Two Particles – Second Moment Collective Motion with Alignment Alden Astwood
Second Moment: hx21 i and hx22 i ≈ (ct)2 at short times hx21 i and hx22 i ≈ const + c2
F2 + f2 t at long times Ff(F + f)
Effective diffusion constant Deff = c2
F2 + f2 2Ff(F + f)
Invariant if F and f are interchanged!
Two Particles – Correlation Collective Motion with Alignment Alden Astwood
Are the positions of the two birds correlated? Look at hx1 x2 i hx1 x2 i ≈ 0 at short times, depending on i.c. hx1 x2 i ≈ const + c2
F2 − f2 t Ff(F + f)
So the positions of the two birds are positively correlated if F>f
Two Particles – Reduced Quantities Collective Motion with Alignment Alden Astwood
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, ∂ ∂t Q(x, t)
=−
R R Rt 0
A(x1 − x10 , x2 − x20 , t − t 0 )Q(x10 , x20 , t 0 )dt 0 dx10 dx20
Generalization to N Particles Collective Motion with Alignment Alden Astwood
Distribution functions become Qσ 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): X X ∂ ∂ [wσ Qσ Qσ σn ~ + ~ ,~ σ 0 Qσ ~ 0 − wσ ~ 0 ,~ σ Qσ ~] ~ = −c ∂t ∂xn 0 n σ ~
N Particles – Rates Collective Motion with Alignment Alden Astwood
How to construct rates wσ ~ 0 ,~ σ? Borrow from an existing model: Ising model Ising model itself has no dynamics There is a way to add dynamics (Glauber 1963)
N Particles – Glauber Dynamics Collective Motion with Alignment Alden Astwood
Start with Ising model Hamiltonian X H(~σ) = − Jmn σm σn m,n −βH(~ σ) /Z In steady state, demand Pσ ~ =e
Only allow one spin flip at a time Detailed balance: In steady state, each term in: X d [wσ Pσ ~ 0 − wσ ~ 0 ,~ σ Pσ ~] ~ ,~ σ 0 Pσ ~ =0= dt 0 σ ~
is also zero
N Particles – What can we do Collective Motion with Alignment Alden Astwood
Can’t solve for probabilities Pσ ~ (t) in Glauber dynamics Little hope for finding Qσ x, t) ~ (~ What can we get? Average spin hσm i and pair correlations hσm σn i These are related to hxm i and hxm xn i
Moments – Hard Way Collective Motion with Alignment Alden Astwood
How to calculate moments? Fourier transform eqns of motion for Qσ 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
Moments – Easy way Collective Motion with Alignment Alden Astwood
Instead, write “Langevin equations”: x˙ m (t) = cσm (t)
Moments – Easy way Collective Motion with Alignment
Instead, write “Langevin equations”:
Alden Astwood
x˙ m (t) = cσm (t) Integrate: Zt xm (t) − xm (0) = c 0
dt 0 σm (t 0 )
Moments – Easy way Collective Motion with Alignment
Instead, write “Langevin equations”:
Alden Astwood
x˙ m (t) = cσm (t) Integrate: Zt xm (t) − xm (0) = c
dt 0 σm (t 0 )
0
Ensemble average (taking xm (0) = 0): Zt
dt 0 hσm (t 0 )i 0 Zt Zt 2 0 hxm (t)xn (t)i = c dt dt 00 hσm (t 0 )σn (t 00 )i hxm (t)i = c
0
0
N Particles – 1st Moment Collective Motion with Alignment Alden Astwood
We have:
Zt hxm (t)i = c
dt 0 hσm (t 0 )i
0
How does hσm i evolve? In “mean field” approximation, hσm i = hσi and τ
d hσi = −hσi + tanh(βJzhσi) dt
Fixed points satisfy hσi∗ = tanh(βJzhσi∗ )
hσi Fixed Points Collective Motion with Alignment
Fixed Points vs Temperature
Alden Astwood
1
Stable Unstable
*
0.5
0
-0.5
-1 0
0.5
1 1/(βJz)
1.5
2
N Particles – 1st Moment Collective Motion with Alignment Alden Astwood
Above critical point, only fixed point is hσi = 0, so hxi const at long times
N Particles – 1st Moment Collective Motion with Alignment Alden Astwood
Above critical point, only fixed point is hσi = 0, so hxi const at long times Nonzero solutions for hσi below critical point, so hxi ∝ ct at long times
N Particles – 1st Moment Collective Motion with Alignment Alden Astwood
Above critical point, only fixed point is hσi = 0, so hxi const at long times Nonzero solutions for hσi below critical point, so hxi ∝ ct at long times At the critical point, hσi = 0 in steady state, but √ √ the dynamics are very slow, hσi ∝ 1/ t, so hxi ∝ t
N Particles – 2nd Moment Collective Motion with Alignment Alden Astwood
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?
Further Work Collective Motion with Alignment Alden Astwood
Try to get (approximate) memory for reduced quantities (density etc) Lots of things could be added: Add bias Higher Dimensions – XY model/Heisenberg model? Finite range interactions
