Analytic Models of Collective Motion
Analytic Models of Collective Motion Alden Astwood
June 8, 2012
Collective Motion
• Start with a model of individual behavior • Try to find out what happens to the group • One way: computer simulation • If parameters change, must rerun simulation
‘Hybrid’ Approach
• Postulate equations of motion for macroscopic quantities • Example: diffusion equation with memory
∂ρ(x, t) = ∂t
Zt M(t − t 0 ) 0
∂2 ρ(x, t 0 ) 0 dt ∂x2
• Estimate unknown parameters using simulation • Can memory be found analytically for simple models?
Part 1: A Simple Model with Centering
Description of the Model
• A set of (Brownian) random walkers in 1D • Interact via spring forces (centering) • Selective bias: only affects ‘informed’ individuals • Simple but exactly solvable
Centering Model – Solution Procedure
1. Write equations of motion for bird positions X x˙ m = −γ (xm − xn ) + vm + Γm (t) n
2. 3. 4. 5.
Write Fokker-Planck equation for probability distribution Solve F-P equation with standard techniques Integrate full solution to get coarse grained quantities Work backwards to get eqns of motion/memory
Centering Model – Dynamics
• If all birds start together in a clump: • Informed birds pull ahead • Uninformed birds get dragged along • Spreading due to noise: • MSD ∼ 2D0 t at short times • MSD ∼ 2D0 t/N at long times
Centering Model – Density Total Density of Birds, t=0.013333 5
Density
4 3 2 1 0 -1
0
1
2 x
3
4
Memory
• What eqn of motion does the density obey? • ρ(x, t) is a sum of two Gaussians • Can write an evolution equation like:
∂ρ(x, t) = ∂t
Zt
Z∞ dx −∞
dt 0 M(x − x 0 , t − t 0 )
0 0
∂2 ρ(x 0 , t 0 ) ∂x 02
• Memory must be nonlocal in time AND space
Part 2: A Model with Alignment Interaction
Motivation • Lots of models have ‘alignment’ interaction (e.g. Vicsek): • Birds move at constant speed • Birds change direction to align with neighbors • Can we calculate memory for a model with alignment?
Single Flipping Particle • To start with: a 1D single particle model: • Single particle moving at constant speed • Can point left or right • Flips randomly at a constant rate
-4
-2
0 x(t)
2
4
Single Moving Particle – Memory
• What is the probability density? • Exponential memory:
∂P(x, t) = c2 ∂t
Zt
0
dt 0 e−2F(t−t ) 0
∂2 P(x, t 0 ) ∂x2
• Equivalent to telegraph equation • MSD goes like t2 at short times and t at long times
Single Moving Particle – Distribution Function Telegraph Equation Solution, c=F=1, t=0.000000
P(x,t)
1
0.5
0 -4
-2
0 x
2
4
Adding Another Particle
• This simple 1 particle model is well known • Now we add a second particle • 22 = 4 Total Configurations: ++, +−, −+, −− • Construct flipping rates so that aligned states are favored
Two Particles – Rates ++
−+
+−
−−
Two Particles – Rates ++ F −+
+−
−−
• F: rate to go from unaligned to aligned state
Two Particles – Rates ++ F f −+
+−
−−
• F: rate to go from unaligned to aligned state • f: rate to go from aligned to unaligned state
Two Particles – Rates ++ 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
Two Moving Particles – What Happens?
• Can solve for all probabilities exactly • MSD again goes like t2 at short times, t at long times
Deff =
F2 + f2 2Ff(F + f)
• Positions of the two particles are correlated i.e.
hx1 x2 i − hx1 ihx2 i = 6 0
Two Moving Particles – Memory
• Again, memory is nonlocal in both time AND space:
∂P(x, t) = ∂t
Zt
Z∞ dx −∞
dt 0 M(x − x 0 , t − t 0 )
0 0
∂2 P(x 0 , t 0 ) ∂x 02
• For CM distribution, memory is
˜ M(k, ) =
2c2 4f(F + f) + 2(F + 2f) + 2c2 k2 + 2
Generalization to N Particles
• How to construct flipping rates? • Infinite range interaction: flipping dynamics completely
independent of motion • Treat birds like Ising spins with all to all coupling • Construct flipping rates as in Glauber dynamics
Generalization to N Particles
• MSD related to two time spin-spin correlation function • Phase transition as interaction strength is changed • hx2 i ∼ 2c2 τt at long times above critical point • Don’t know yet what happens at/below critical point • Very difficult to get exact memory
Lessons Learned
• Analytic calculations possible for two simple models • We find memories which are nonlocal in time and space • Lots of further work to be done: • N particle near/below critical point • Finite range interactions • Many other generalizations
June 8, 2012
Collective Motion
• Start with a model of individual behavior • Try to find out what happens to the group • One way: computer simulation • If parameters change, must rerun simulation
‘Hybrid’ Approach
• Postulate equations of motion for macroscopic quantities • Example: diffusion equation with memory
∂ρ(x, t) = ∂t
Zt M(t − t 0 ) 0
∂2 ρ(x, t 0 ) 0 dt ∂x2
• Estimate unknown parameters using simulation • Can memory be found analytically for simple models?
Part 1: A Simple Model with Centering
Description of the Model
• A set of (Brownian) random walkers in 1D • Interact via spring forces (centering) • Selective bias: only affects ‘informed’ individuals • Simple but exactly solvable
Centering Model – Solution Procedure
1. Write equations of motion for bird positions X x˙ m = −γ (xm − xn ) + vm + Γm (t) n
2. 3. 4. 5.
Write Fokker-Planck equation for probability distribution Solve F-P equation with standard techniques Integrate full solution to get coarse grained quantities Work backwards to get eqns of motion/memory
Centering Model – Dynamics
• If all birds start together in a clump: • Informed birds pull ahead • Uninformed birds get dragged along • Spreading due to noise: • MSD ∼ 2D0 t at short times • MSD ∼ 2D0 t/N at long times
Centering Model – Density Total Density of Birds, t=0.013333 5
Density
4 3 2 1 0 -1
0
1
2 x
3
4
Memory
• What eqn of motion does the density obey? • ρ(x, t) is a sum of two Gaussians • Can write an evolution equation like:
∂ρ(x, t) = ∂t
Zt
Z∞ dx −∞
dt 0 M(x − x 0 , t − t 0 )
0 0
∂2 ρ(x 0 , t 0 ) ∂x 02
• Memory must be nonlocal in time AND space
Part 2: A Model with Alignment Interaction
Motivation • Lots of models have ‘alignment’ interaction (e.g. Vicsek): • Birds move at constant speed • Birds change direction to align with neighbors • Can we calculate memory for a model with alignment?
Single Flipping Particle • To start with: a 1D single particle model: • Single particle moving at constant speed • Can point left or right • Flips randomly at a constant rate
-4
-2
0 x(t)
2
4
Single Moving Particle – Memory
• What is the probability density? • Exponential memory:
∂P(x, t) = c2 ∂t
Zt
0
dt 0 e−2F(t−t ) 0
∂2 P(x, t 0 ) ∂x2
• Equivalent to telegraph equation • MSD goes like t2 at short times and t at long times
Single Moving Particle – Distribution Function Telegraph Equation Solution, c=F=1, t=0.000000
P(x,t)
1
0.5
0 -4
-2
0 x
2
4
Adding Another Particle
• This simple 1 particle model is well known • Now we add a second particle • 22 = 4 Total Configurations: ++, +−, −+, −− • Construct flipping rates so that aligned states are favored
Two Particles – Rates ++
−+
+−
−−
Two Particles – Rates ++ F −+
+−
−−
• F: rate to go from unaligned to aligned state
Two Particles – Rates ++ F f −+
+−
−−
• F: rate to go from unaligned to aligned state • f: rate to go from aligned to unaligned state
Two Particles – Rates ++ 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
Two Moving Particles – What Happens?
• Can solve for all probabilities exactly • MSD again goes like t2 at short times, t at long times
Deff =
F2 + f2 2Ff(F + f)
• Positions of the two particles are correlated i.e.
hx1 x2 i − hx1 ihx2 i = 6 0
Two Moving Particles – Memory
• Again, memory is nonlocal in both time AND space:
∂P(x, t) = ∂t
Zt
Z∞ dx −∞
dt 0 M(x − x 0 , t − t 0 )
0 0
∂2 P(x 0 , t 0 ) ∂x 02
• For CM distribution, memory is
˜ M(k, ) =
2c2 4f(F + f) + 2(F + 2f) + 2c2 k2 + 2
Generalization to N Particles
• How to construct flipping rates? • Infinite range interaction: flipping dynamics completely
independent of motion • Treat birds like Ising spins with all to all coupling • Construct flipping rates as in Glauber dynamics
Generalization to N Particles
• MSD related to two time spin-spin correlation function • Phase transition as interaction strength is changed • hx2 i ∼ 2c2 τt at long times above critical point • Don’t know yet what happens at/below critical point • Very difficult to get exact memory
Lessons Learned
• Analytic calculations possible for two simple models • We find memories which are nonlocal in time and space • Lots of further work to be done: • N particle near/below critical point • Finite range interactions • Many other generalizations
