Revisiting the flocking transition using Active Spins
A. Solon, J. Tailleur
Laboratoire MSC
CNRS - Université Paris Diderot
Models from Statistical Mechanics in Applied Sciences
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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Energy consumption at the microscopic scale
Self-propulsion
Aligning interactions
Collective motion (with long range-order?)
J. Tailleur (CNRS-Univ Paris Diderot)
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The Vicsek model [Vicsek et al. PRL 75, 1226 (1995)]
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The Vicsek model [Vicsek et al. PRL 75, 1226 (1995)]
N self-propelled particles off-lattice
Local alignment rule
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The Vicsek model [Vicsek et al. PRL 75, 1226 (1995)]
N self-propelled particles off-lattice
Local alignment rule
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The Vicsek model [Vicsek et al. PRL 75, 1226 (1995)]
N self-propelled particles off-lattice
Local alignment rule
J. Tailleur (CNRS-Univ Paris Diderot)
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The Vicsek model [Vicsek et al. PRL 75, 1226 (1995)]
N self-propelled particles off-lattice
Local alignment rule
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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Flocking transition
Disordered
Inhomogeneous
noise
Fluctuating
flocking state
or density
Non-equilibrium transition to long-range order in d = 2
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9/9/2013
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A long-standing debate
Simulations are simple but strong finite size effects
Second-order (1995) vs First-order (2004) [Gregoire and Chate,
PRL 2004]
hard to study numerically
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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A long-standing debate
Simulations are simple but strong finite size effects
Second-order (1995) vs First-order (2004) [Gregoire and Chate,
PRL 2004]
hard to study numerically
Analytical descriptions: Boltzmann (Bertin et al.),
phenomenological equations (Toner& Tu, Marchetti et al.)
hard to solve analytically
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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A long-standing debate
Simulations are simple but strong finite size effects
Second-order (1995) vs First-order (2004) [Gregoire and Chate,
PRL 2004]
hard to study numerically
Analytical descriptions: Boltzmann (Bertin et al.),
phenomenological equations (Toner& Tu, Marchetti et al.)
hard to solve analytically
Use a much simpler model: active spins
on lattice
discrete symmetry
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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Active Ising model
1
−
Density ρi = n+
i + ni
J. Tailleur (CNRS-Univ Paris Diderot)
2
3
4
5
...
L
−
Magnetisation mi = n+
i − ni
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Active Ising model
Spin-flip
Wi−
Wi+
1
i
−
Density ρi = n+
i + ni
Local alignment
2
3
4
5
...
L
−
Magnetisation mi = n+
i − ni
n+ −n−
i
i
Wi± = exp(±β n+
)
+n−
i
i
Fully connected Ising models on each site
J. Tailleur (CNRS-Univ Paris Diderot)
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Active Ising model
Spin-flip
Diffusion
Wi−
Wi+
i
−
Density ρi = n+
i + ni
Local alignment
D(1−) D(1+)
D(1+)
1
4
2
3
5
D(1−)
...
L
−
Magnetisation mi = n+
i − ni
n+ −n−
i
i
Wi± = exp(±β n+
)
+n−
i
i
Fully connected Ising models on each site
Self-propulsion
J. Tailleur (CNRS-Univ Paris Diderot)
Diffusion biased by the spins for 6= 0
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Phase diagram in 2d •
Liquid/gas
1.0
8
T
0.8
6
G
G+L
L
0.6
0.4
0
10
ρ`
ρh
4
ρ0
0
m
2
20
0
100
200
300
2.5
8
2
1.5
6
1
4
ρ(x)
0.5
2
m(x)
0
−0.5
0
0
100
200
Gas
J. Tailleur (CNRS-Univ Paris Diderot)
300 0
100
200
300
Liquid
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Mean-field and beyond
±
Mean-field equations hf (n±
i )i ' f (hni i)
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Mean-field and beyond
±
Mean-field equations hf (n±
i )i ' f (hni i)
ρ̇ = D̃∆ρ−v∂x m
v∝
ṁ = D̃∆m−v∂x ρ+2m(β − 1) − α
J. Tailleur (CNRS-Univ Paris Diderot)
m3
ρ2
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Mean-field and beyond
±
Mean-field equations hf (n±
i )i ' f (hni i)
ρ̇ = D̃∆ρ−v∂x m
v∝
ṁ = D̃∆m−v∂x ρ+2m(β − 1) − α
m3
ρ2
ρ = ρ0 m = 0 always linearly unstable for T < Tc = 1
No clusters
Continuous transition
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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Mean-field and beyond
±
Mean-field equations hf (n±
i )i ' f (hni i)
ρ̇ = D̃∆ρ−v∂x m
v∝
ṁ = D̃∆m−v∂x ρ+2m(β − 1) − α
m3
ρ2
ρ = ρ0 m = 0 always linearly unstable for T < Tc = 1
No clusters
Continuous transition
MF only valid at ρ = ∞
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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Mean-field and beyond
±
Mean-field equations hf (n±
i )i ' f (hni i)
ρ̇ = D̃∆ρ−v∂x m
v∝
ṁ = D̃∆m−v∂x ρ+2m(β − 1) − α
m3
ρ2
Taylor expansion at finite density βc = 1 + r/ρ
MF only valid at ρ = ∞
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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Mean-field and beyond
±
Mean-field equations hf (n±
i )i ' f (hni i)
ρ̇ = D̃∆ρ−v∂x m
v∝
r
m3
ṁ = D̃∆m−v∂x ρ+2m(β−1 − ) − α 2
ρ
ρ
Taylor expansion at finite density βc = 1 + r/ρ
MF only valid at ρ = ∞
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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Mean-field and beyond
±
Mean-field equations hf (n±
i )i ' f (hni i)
ρ̇ = D̃∆ρ−v∂x m
v∝
r
m3
ṁ = D̃∆m−v∂x ρ+2m(β−1 − ) − α 2
ρ
ρ
Taylor expansion at finite density βc = 1 + r/ρ
MF only valid at ρ = ∞
J. Tailleur (CNRS-Univ Paris Diderot)
Refined-Mean-Field-Model (RMFM)
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Simulations of the RMFM
ρ1
ρ2
ρ`
ρh
T
Gas
0.8
0.7
Liquid
0.6
0
1
2
3
4
2
1
G
J. Tailleur (CNRS-Univ Paris Diderot)
ρ(x)
2
0
100
m(x)
4
2
0
5
6
4
50
Coexistence
ρ0
0.5
0
Spinodals
0
0
50
L+G
100
0
50
100
L
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Simulations of the RMFM
ρ1
ρ2
ρ`
ρh
T
Gas
0.8
0.7
Liquid
0.6
0
1
2
3
4
2
1
G
ρ(x)
2
0
100
m(x)
4
2
0
5
6
4
50
Coexistence
ρ0
0.5
0
Spinodals
0
0
50
L+G
100
0
50
100
L
Same phenomenology as microscopic model
ρ1 (β) < ρ2 (β)
J. Tailleur (CNRS-Univ Paris Diderot)
Always observe phase-separated profiles
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Phase-separated profiles
Propagating shocks between ρ` , m` = 0 and ρh , mh 6= 0
Stationary solutions in comoving frame of velocity c
Dρ00 + cρ0 − vm0 = 0
(1)
m3
r
Dm00 + cm0 − vρ0 + 2(β − 1 − ) − α 2 = 0
ρ
ρ
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
(2)
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Phase-separated profiles
Propagating shocks between ρ` , m` = 0 and ρh , mh 6= 0
Stationary solutions in comoving frame of velocity c
Dρ00 + cρ0 − vm0 = 0
(1)
m3
r
Dm00 + cm0 − vρ0 + 2(β − 1 − ) − α 2 = 0
ρ
ρ
Solvable at large densities ρ1 =
1: Solve (1) to get ρ = ρ` +
v
c
r
β−1
P∞
k=0
(2)
r
−
k
D
c∇ m
2: Expand (2) around ρ1 , inject ρ(m) and truncate
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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Phase-separated profiles
Propagating shocks between ρ` , m` = 0 and ρh , mh 6= 0
Stationary solutions in comoving frame of velocity c
Dρ00 + cρ0 − vm0 = 0
(1)
m3
r
Dm00 + cm0 − vρ0 + 2(β − 1 − ) − α 2 = 0
ρ
ρ
Solvable at large densities ρ1 =
1: Solve (1) to get ρ = ρ` +
v
c
r
β−1
P∞
k=0
(2)
r
−
k
D
c∇ m
2: Expand (2) around ρ1 , inject ρ(m) and truncate
D(1 +
v2
)m00
c2
+ [c −
J. Tailleur (CNRS-Univ Paris Diderot)
v2
c
−
2Dvr
m]m0 − 2r(ρρ12−ρ` ) m
c2 ρ21
1
+
2rv 2
m−
cρ21
3
αm
=0
ρ2
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1
10 / 24
Symmetric front solutions β ' 1
mh
(tanh(q ± (x − ct)) + 1)
2
r
4r
√
c= v
q± = ±
mh =
3α
3ρ1 αD
m± (x) =
ρ−ρl
1.0 ρh −ρl
0.8
0.6 q=0.051
0.4
0.2
0.0
0.2
0
100
J. Tailleur (CNRS-Univ Paris Diderot)
T =0.83
q=-0.051
micro
fit tanh
x
200
300
400
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Asymmetric front solutions β > 1
mh
(tanh(q ± (x − ct)) + 1)
2
2Dr2
r
r2
±
√
c = v+
q
=
±
−
3vαρ21
3ρ1 αD 6αvρ21
m± (x) =
ρ−ρl
1.0 ρh −ρl
0.8
0.6 q=0.078
0.4
0.2
0.0
0.2
0
100
J. Tailleur (CNRS-Univ Paris Diderot)
mh =
4r
8Dr3
− 2 2 2
3α 9v α ρ1
T =0.5
q=-0.14
micro
fit tanh
x
200
300
400
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The flock fly faster than the birds
1.25
1.20
7
c
v
micro
mean-field
1.15
5
3
1
1.10
1.05
1.00
T
0.95
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
J. Tailleur (CNRS-Univ Paris Diderot)
0
200
400
0
200
400
15
13
11
9
7
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The flock fly faster than the birds
1.25
1.20
7
c
v
micro
mean-field
1.15
5
3
1
1.10
1.05
1.00
T
0.95
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
200
400
0
200
400
15
13
11
9
7
2
2Dr
c = v+ 3vαρ
2
1
v
2Dr2
3vαρ21
microscopic velocities
FKPP-like contribution
J. Tailleur (CNRS-Univ Paris Diderot)
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A liquid-gas transition in the canonical ensemble
1.0
ρ1
ρ2
ρ`
ρh
T
Gas
0.8
Liquid
0.6
Spinodals
Coexistence
ρ0
0
5
10
15
2
1
2
0
0
50
100
0
0
50
100
No length selection mechanism
J. Tailleur (CNRS-Univ Paris Diderot)
ρ(x)
4
2
0
m(x)
6
4
0
50
100
Phase-separation
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A liquid-gas transition in the canonical ensemble
1.0
ρ1
ρ2
ρ`
ρh
T
Gas
0.8
Liquid
0.6
Spinodals
Coexistence
ρ0
0
5
10
15
2
1
2
0
0
50
100
ρ(x)
4
2
0
m(x)
6
4
0
0
50
100
No length selection mechanism
0
50
100
Phase-separation
Gas and liquid have different symmetries
First-order liquid-gas transition with ρc = ∞
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Hysteresis loops
1
φ
ρ
0.8
RMFM
0.6
ρ0 = 3
ρ0 = 2.6
ρ0 = 3.4
4
up
down
0.4
ρ0 = 2.3
3
0.2
ρ0
0
2
1
3
2
4
x
0
φ
ρ
0.8
micro 2d
0.6
up
down
0.4
0.2
20
40
ρ0 = 2
ρ0 = 3
ρ0 = 3.5
6
60
80
100
ρ0 = 4
ρ0 = 4.5
4
2
ρ0
0
1
J. Tailleur (CNRS-Univ Paris Diderot)
2
3
4
5
6
x
0
0
50
100 150 200 250
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A difficult Finite-Size Scaling
First-order but in the temperature-density ensemble
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A difficult Finite-Size Scaling
First-order but in the temperature-density ensemble
Magnetization ∝ liquid fraction
J. Tailleur (CNRS-Univ Paris Diderot)
Continuous when L = ∞
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A difficult Finite-Size Scaling
First-order but in the temperature-density ensemble
Magnetization ∝ liquid fraction
Finite size of interfaces
Continuous when L = ∞
Discontinuous jump of m
Width and mean of the peaks go to 0 as L → ∞
10
p(m)
10
ρ=2.19
ρ=2.22
ρ=2.26
5
p(m)
ρ=2
ρ=2.011
ρ=2.03
5
m
0
−0.8 −0.4
0
L = 80
J. Tailleur (CNRS-Univ Paris Diderot)
0.4
0.8
m
0
−0.8 −0.4
0
0.4
0.8
L = 150
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Flocking in 1d: Role of fluctuations
In 1D, single-site fluctuations can flip a domain
80
t=0
40
40
ρh
1
J. Tailleur (CNRS-Univ Paris Diderot)
1500
50
0
3000
t = 200
100
0
20
v2
0
150
m
60
v1
20
t = 10
Reversals •
0
1
1500
3000
1
1500
3000
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Flocking in 1d: Role of fluctuations
In 1D, single-site fluctuations can flip a domain
80
t=0
40
40
ρh
1
1500
50
0
3000
t = 200
100
0
20
v2
0
150
m
60
v1
20
t = 10
Reversals •
0
1
1500
3000
1
1500
3000
In 2D: fluctuations of a complete line of length ∝ L
Vanishing probability as L → ∞
J. Tailleur (CNRS-Univ Paris Diderot)
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Flocking in 1d: Role of fluctuations
In 1D, single-site fluctuations can flip a domain
80
t=0
40
40
ρh
1
1500
50
0
3000
t = 200
100
0
20
v2
0
150
m
60
v1
20
t = 10
Reversals •
0
1
1500
3000
1
1500
3000
In 2D: fluctuations of a complete line of length ∝ L
Vanishing probability as L → ∞
Similar to other 1D flocking models [Czirók et al.1999, O’loan&
Evans 1999]
J. Tailleur (CNRS-Univ Paris Diderot)
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No ordered phase in the thermodynamic limit
The ordered liquid is metastable in finite systems •
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No ordered phase in the thermodynamic limit
The ordered liquid is metastable in finite systems •
Mean time until reversal τR ∝
1
L
Small but finite probability of flipping portions of ` sites
Proba that no such portions flipped until time t : P ∝ exp(−cLτ /`)
100000
10000
τR
1000
100
100
1000
10000
100000
L
J. Tailleur (CNRS-Univ Paris Diderot)
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No ordered phase in the thermodynamic limit
The ordered liquid is metastable in finite systems •
Mean time until reversal τR ∝
1
L
Small but finite probability of flipping portions of ` sites
Proba that no such portions flipped until time t : P ∝ exp(−cLτ /`)
100000
10000
τR
1000
100
100
1000
10000
100000
L
Spreading time ∝ L
J. Tailleur (CNRS-Univ Paris Diderot)
Ordered liquid never seen for large L
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No ordered phase in the thermodynamic limit
The ordered liquid is metastable in finite systems •
Mean time until reversal τR ∝
1
L
Small but finite probability of flipping portions of ` sites
Proba that no such portions flipped until time t : P ∝ exp(−cLτ /`)
100000
10000
τR
1000
100
100
1000
10000
100000
L
Spreading time ∝ L
Ordered liquid never seen for large L
Only two phases in the thermodynamic limit
J. Tailleur (CNRS-Univ Paris Diderot)
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Dynamics in the low temperature phase •
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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Dynamics in the low temperature phase •
80
t=0
40
v1
40
20
ρh
1
J. Tailleur (CNRS-Univ Paris Diderot)
150
1500
100
0
50
0
3000
t = 200
m
20
v2
0
t = 10
60
0
1
1500
3000
1
1500
3000
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Dynamics in the low temperature phase •
80
t=0
40
v1
40
20
ρh
1
150
1500
100
0
50
0
3000
t = 200
m
20
v2
0
t = 10
60
0
1
1500
3000
1
1500
3000
Duration of reversal ∝ Length of cluster
Length of cluster ∝ Mean time between reversal
h Time during reversals i ∝ h Time between reversal i
J. Tailleur (CNRS-Univ Paris Diderot)
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Dynamics in the low temperature phase •
80
t=0
40
v1
40
20
ρh
0
1500
100
0
50
0
3000
t = 200
m
20
v2
1
150
t = 10
60
0
1
1500
3000
1
1500
3000
Duration of reversal ∝ Length of cluster
Length of cluster ∝ Mean time between reversal
h Time during reversals i ∝ h Time between reversal i
10
p(|m|)
1
10−1
No ergodicity breaking
L=1000
L=2000
L=4000
L=8000
10−2
10−3
10−4
0.1
J. Tailleur (CNRS-Univ Paris Diderot)
0.5
1
|m|
2
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A Nightmarish FSS
ρ < ρ` : disordered gas
ρ > ρ1 : alternating clusters
ρ` < ρ < ρ1 : phase coexistence
Binder cumulant G(β) = 1 −
J. Tailleur (CNRS-Univ Paris Diderot)
hm4 i
3hm2 i2
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A Nightmarish FSS
ρ < ρ` : disordered gas
ρ > ρ1 : alternating clusters
ρ` < ρ < ρ1 : phase coexistence
Binder cumulant G(β) = 1 −
β =1.54
β =1.55
β =1.56
L=1024
L=2048
hm4 i
3hm2 i2
β =2.96
β =3.04
β =3.1
L=4096
L=8192
L=1000
L=2000
L=4000
L=8000
6
15
p(m)
0.6
G
5
10
p(m)
0.5
0
3
0.2
5
m
0
−0.2
0
2
1.4
1.6
1.8
D = 1; ρ = 3
−0.5
1
β
0.0
0.2
G
4
0.4
m
0
2.0
−0.4
0
0.4
−1
β
2.5
3.0
3.5
D = 10; ρ = 0.5
Very difficult to analyze!
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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G(β) = 1 −
hm4 i
3hm2 i2
across a transition
Simplest case of phase coexistence:
P (m, β) ∝ α(β) e−
J. Tailleur (CNRS-Univ Paris Diderot)
(m+m0 )2
2σ 2
m2
+ (1 − 2α(β)) e− 2σ2 + α(β) e−
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(m−m0 )2
2σ 2
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G(β) = 1 −
hm4 i
3hm2 i2
across a transition
Simplest case of phase coexistence:
P (m, β) ∝ α(β) e−
minα (G) =
(m+m0 )2
2σ 2
m2
+ (1 − 2α(β)) e− 2σ2 + α(β) e−
(m−m0 )2
2σ 2
−1
(σ/m0 )2 (12+36(σ/m0 )2 )
J. Tailleur (CNRS-Univ Paris Diderot)
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G(β) = 1 −
hm4 i
3hm2 i2
across a transition
Simplest case of phase coexistence:
P (m, β) ∝ α(β) e−
minα (G) =
1
P (m)
(m−m0 )2
2σ 2
G
0.4
0
0.2
m
0.0
0.6
m0 = 1
σ = 0.45
m2
+ (1 − 2α(β)) e− 2σ2 + α(β) e−
−1
(σ/m0 )2 (12+36(σ/m0 )2 )
0.6
m0 = 1
σ = 0.25
(m+m0 )2
2σ 2
−2
−1
0
1
−1
2
P (m)
α
0.0
1
0.2
0.4
0.6
0.2
0.4
0.6
0.8
G
0.4
0
0.2
m
0.0
−2
J. Tailleur (CNRS-Univ Paris Diderot)
−1
0
1
2
−1
α
0.0
0.8
9/9/2013
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G(β) = 1 −
hm4 i
3hm2 i2
across a transition
Simplest case of phase coexistence:
P (m, β) ∝ α(β) e−
minα (G) =
1
P (m)
(m−m0 )2
2σ 2
G
0.4
0
0.2
m
0.0
0.6
m0 = 1
σ = 0.45
m2
+ (1 − 2α(β)) e− 2σ2 + α(β) e−
−1
(σ/m0 )2 (12+36(σ/m0 )2 )
0.6
m0 = 1
σ = 0.25
(m+m0 )2
2σ 2
−2
−1
0
1
−1
2
P (m)
α
0.0
1
0.2
0.4
0.6
0.2
0.4
0.6
0.8
G
0.4
0
0.2
m
0.0
−2
No negative peak
J. Tailleur (CNRS-Univ Paris Diderot)
−1
0
1
2
−1
α
0.0
0.8
2nd order transition
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Merging of the peaks as L increases (D=10, eps=0.9, beta=1.7)
8
6
4
p(m)
0.6
L =4000
L =8000
L =16000
0.4
0.2
2
0
G(m)
L =2000
L =4000
L =8000
L =16000
L =32000
L =64000
0.0
m
0.2
0.1
0.0
J. Tailleur (CNRS-Univ Paris Diderot)
0.1
0.2
0.2
1.1
ρ0
1.2
1.3
1.4
1.5
9/9/2013
1.6
22 / 24
Merging of the peaks as L increases (D=10, eps=0.9, beta=1.7)
8
p(m)
6
0.6
L =4000
L =8000
L =16000
4
0.4
0.2
2
0
G(m)
L =2000
L =4000
L =8000
L =16000
L =32000
L =64000
0.0
m
0.2
0.1
0.0
0.1
0.2
L < Lc critical nucleus
L & Lc
discontinuous
L Lc
continuous
J. Tailleur (CNRS-Univ Paris Diderot)
0.2
1.1
ρ0
1.2
1.3
1.4
1.5
1.6
continuous
9/9/2013
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Merging of the peaks as L increases (D=10, eps=0.9, beta=1.7)
8
p(m)
6
0.6
L =4000
L =8000
L =16000
4
0.4
0.2
2
0
G(m)
L =2000
L =4000
L =8000
L =16000
L =32000
L =64000
0.0
m
0.2
0.1
0.0
0.1
0.2
L < Lc critical nucleus
L & Lc
discontinuous
L Lc
continuous
0.2
1.1
ρ0
1.2
1.3
1.4
1.5
1.6
continuous
Grand canonical ensemble would be much simpler!
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
22 / 24
Active spins – summary
New flocking model using active Ising spins
Flocking trans.
Liquid-gas transition in canonical ensemble
Symmetry of the liquid phase
ρc = ∞
No spontaneous symmetry breaking in 1d
FSS very difficult in the canonical ensemble
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
23 / 24
Active spins – summary
New flocking model using active Ising spins
Flocking trans.
Liquid-gas transition in canonical ensemble
Symmetry of the liquid phase
ρc = ∞
No spontaneous symmetry breaking in 1d
FSS very difficult in the canonical ensemble
Perspectives:
Is all this generic?
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
23 / 24
More general flocking models
Different symmetries for liquid and gas
Critical point at ρc = ∞
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
24 / 24
More general flocking models
Different symmetries for liquid and gas
Critical point at ρc = ∞
Symmetry of order parameter + coupling to self-propulsion
Form of the liquid phase
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
24 / 24
More general flocking models
Different symmetries for liquid and gas
Critical point at ρc = ∞
Symmetry of order parameter + coupling to self-propulsion
Form of the liquid phase
Ising interaction
ρ1
J. Tailleur (CNRS-Univ Paris Diderot)
<
ρ2
9/9/2013
24 / 24
More general flocking models
Different symmetries for liquid and gas
Critical point at ρc = ∞
Symmetry of order parameter + coupling to self-propulsion
Form of the liquid phase
Off-lattice Ising interaction
ρ1
J. Tailleur (CNRS-Univ Paris Diderot)
<
ρ2
9/9/2013
24 / 24
More general flocking models
Different symmetries for liquid and gas
Critical point at ρc = ∞
Symmetry of order parameter + coupling to self-propulsion
Form of the liquid phase
4-color Potts interaction
ρ1
J. Tailleur (CNRS-Univ Paris Diderot)
<
ρ2
9/9/2013
24 / 24
More general flocking models
Different symmetries for liquid and gas
Critical point at ρc = ∞
Symmetry of order parameter + coupling to self-propulsion
Form of the liquid phase
XY interaction
ρ1
J. Tailleur (CNRS-Univ Paris Diderot)
<
ρ2
9/9/2013
24 / 24
More general flocking models
Different symmetries for liquid and gas
Critical point at ρc = ∞
Symmetry of order parameter + coupling to self-propulsion
Form of the liquid phase
Vicsek model
ρ1
J. Tailleur (CNRS-Univ Paris Diderot)
<
ρ2
9/9/2013
24 / 24
More general flocking models
Different symmetries for liquid and gas
Critical point at ρc = ∞
Symmetry of order parameter + coupling to self-propulsion
Form of the liquid phase
Nematic Vicsek model
ρ1
J. Tailleur (CNRS-Univ Paris Diderot)
<
ρ2
9/9/2013
24 / 24
More general flocking models
Different symmetries for liquid and gas
Critical point at ρc = ∞
Symmetry of order parameter + coupling to self-propulsion
Form of the liquid phase
Nematic Vicsek model
ρ1
<
ρ2
A.Solon, J. Tailleur, Phys. Rev. Lett. 111, 078101 (2013)
J. Tailleur (CNRS-Univ Paris Diderot)
9/9/2013
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