From phase to micro-phase separation in flocking models · 2017-06-06 · From phase to micro-phase...

From phase to micro-phase separation in flockingmodels

A. Solon, H. Chaté, J. Tailleur

Laboratoire MSCCNRS - Université Paris Diderot

Advances in Nonequilibrium Statistical Mechanics

J. Tailleur (CNRS-Univ Paris Diderot) GGI-28/05/2014 1 / 16

Energy consumption at the microscopic scale Self-propulsion

Aligning interactions

Collective motion (with long range-order?)


The Vicsek model [Vicsek et al. PRL 75, 1226 (1995)]

N self-propelled particles off-lattice

Local alignment rule


















Flocking transition [Grégoire, Chaté, PRL (2004)]

Disordered Inhomogeneous Fluctuatingflocking state

noise or density

Non-equilibrium transition to long-range order in d = 2


A long-standing debate

Simulations are simple but strong finite size effects2nd-order (1995) vs 1st-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 Ising spinson latticediscrete symmetry
















Active Ising model

1 2 3 4 5 . . . L

D(1−ε) D(1+ε) D(1+ε) D(1−ε)

Biased diffusionSpin-flip

i

W−iW+

i

Density ρi = n+i + n−i Magnetisation mi = n+i − n−i

Local alignment W±i = exp(±βmiρi

)

Fully connected Ising models on each site

Self-propulsion Diffusion biased by the spins for ε 6= 0


Active Ising model

1 2 3 4 5 . . . L

D(1−ε) D(1+ε) D(1+ε) D(1−ε)

Biased diffusion

Spin-flip

i

W−iW+

i



)




Active Ising model

1 2 3 4 5 . . . L

D(1−ε) D(1+ε) D(1+ε) D(1−ε)

Biased diffusionSpin-flip

i

W−iW+

i



)




Phase diagram in 2d

m

Quench ordered •Quench disordered •

0 10 20

0.4

0.6

0.8

1.0T

ρ0

GG+L

Lρ`

ρh

0 100 200 300−0.5

0

0.5

1

1.5

2

2.5

0 100 200 300

0

2

4

6

8

0 100 200 300

0

2

4

6

8

ρ(x)

m(x)

Liquid/gas

Gas Liquid


Movies/coarsening_1000x1000_CIord.avi

Movies/coarsening_400x400_CIdesord.avi

Mean-field and beyond

Mean-field equations 〈f(n±i )〉 ' f(〈n±i 〉)

ρ = D∆ρ−v∂xm v ∝ ε

m = D∆m−v∂xρ+2m(β − 1)− αm3

ρ2

ρ = ρ0 m = 0 always linearly unstable for T < Tc = 1

No clustersContinuous transition

MF only valid at ρ =∞ Refined-Mean-Field-Model (RMFM)





m = D∆m−v∂xρ+2m(β − 1)− αm3

ρ2








m = D∆m−v∂xρ+2m(β − 1)− αm3

ρ2








m = D∆m−v∂xρ+2m(β − 1)− αm3

ρ2


No clustersContinuous transitionMF only valid at ρ =∞

Refined-Mean-Field-Model (RMFM)





m = D∆m−v∂xρ+2m(β − 1)− αm3

ρ2

Finite density: fluctuations βc = 1 + r/ρ

Continuous transition

MF only valid at ρ =∞






m = D∆m−v∂xρ+2m(β−1− r

ρ)− αm

3

ρ2



MF only valid at ρ =∞






m = D∆m−v∂xρ+2m(β−1− r

ρ)− αm

3

ρ2





Simulations of the RMFM

0 5 10 15

0.6

0.8

1.0T

ρ0

Gas

Liquid

ρ1

ρ2

ρ`

ρh

0 50 100

0

1

2

0 50 100

0

2

4

0 50 100

0

2

4

6m(x)

ρ(x)

Spinodals

Coexistence

G L+G L

Same phenomenology as microscopic model


Hysteresis loops

1 2 3 4 5 60

0.2

0.4

0.6

0.8

1 φ

ρ0interface effect

nucleation

spinodaldecomposition

updown

0 200 400 600 800

123456

ρ

x

ρ0 =1.2

ρ0 =2

ρ0 =3

ρ0 =4

1.5 2.0 2.5 3.0 3.5 4.00

0.2

0.4

0.6

0.8

1 φ

ρ0

updown

0 200 400 600 800

2

3

4 ρ

x

ρ0 =1.7

ρ0 =2

ρ0 =2.4

ρ0 =2.8

Micro 2d

RMFM


Active spins – summary

New flocking model with discrete symmetry using active spins

Flocking trans. Liquid-gas transition in canonical ensemble

Symmetry of the liquid phase ρc =∞

T

ρ0

G L+G L

Tc, ρcEquilibirum

Liquid-gas

Tc, ρc =∞

Active

Liquid-gas


Back to the Vicsek model (with H. Chaté)

Phase diagram: liquid-gas picture seems ok

But phase separation micro-phase separationρ1 ≤ ρ2

AIM

VM

Movies/movie_band_break_800x100_eta04_rho225.avi

Back to the Vicsek model (with H. Chaté)

Phase diagram: liquid-gas picture seems ok

But phase separation micro-phase separationρ1 ≤ ρ2

AIM

VM

Movies/movie_band_break_800x100_eta04_rho225.avi

Spinodals Quenches shows different regimes • •

Finite-size scaling of order parameter

Hysteresis:


Movies/movie_CIdesord_nucleation_eta0.4_rho0.9_vector.avi

Movies/movie_CIdesord_spinodal_eta0.4_rho2_vector.avi

Spinodals Quenches shows different regimes • •Finite-size scaling of order parameter

Hysteresis:




Spinodals Quenches shows different regimes • •Finite-size scaling of order parameter

Hysteresis:




Hydrodynamics: phase vs micro-phase separationBoth type of propagative solutions exist for generichydrodynamic descriptions [Caussin et al., PRL 2014]

Scalar order parameter (Ising)

∂tρ = −v∂xm

∂tm+ ξm∂xm = D∇2m− λ∂xρ+

[(ρ− ρc)−

m2

P 20 ρ

]m

Vectorial order parameter (Vicsek)

∂tρ = −∇.~m

∂t ~m+ ξ(~m∇).~m = D∇2 ~m− λ∇ρ+

[(ρ− ρc)−

|~m|2

P 20 ρ

]~m

0.8

1.2

1.6

ρ

PDE Ising PDE Vicsek

Hydrodynamics: phase vs micro-phase separationBoth type of propagative solutions exist for generichydrodynamic descriptions [Caussin et al., PRL 2014]Scalar order parameter (Ising)

∂tρ = −v∂xm


[(ρ− ρc)−

m2

P 20 ρ

]m


∂tρ = −∇.~m

∂t ~m+ ξ(~m∇).~m = D∇2 ~m− λ∇ρ+

[(ρ− ρc)−

|~m|2

P 20 ρ

]~m

0.8

1.2

1.6

ρ


Hydrodynamics: phase vs micro-phase separationBoth type of propagative solutions exist for generichydrodynamic descriptions [Caussin et al., PRL 2014]Scalar order parameter (Ising)

∂tρ = −v∂xm


[(ρ− ρc)−

m2

P 20 ρ

]m


∂tρ = −∇.~m

∂t ~m+ ξ(~m∇).~m = D∇2 ~m− λ∇ρ+

[(ρ− ρc)−

|~m|2

P 20 ρ

]~m

0.8

1.2

1.6

ρ


Fluctuations play a crucial role

PDEs + noises do a good job: • •

∂tm = [...] + η ∂t ~m = [...] + ~η

t=400 t=106

scalarm

vectorial~m

0

1

2

3ρ

103 104 105 106 107

101

102

103

104 ∆n

nn0.5

n0.8

sSDEvSDEAIMVM

The nature of the phase-separated states stems from theinterplay between fluctuations and symmetry of the orderparameter


Movies/2000x100_A4_Lb1000_scalaire.avi

Movies/2000x100_A4_Lb1000.avi

Fluctuations play a crucial role

PDEs + noises do a good job: • •

∂tm = [...] + η ∂t ~m = [...] + ~η

t=400 t=106

scalarm

vectorial~m

0

1

2

3ρ

103 104 105 106 107

101

102

103

104 ∆n

nn0.5

n0.8

sSDEvSDEAIMVM

The nature of the phase-separated states stems from theinterplay between fluctuations and symmetry of the orderparameter


Movies/2000x100_A4_Lb1000_scalaire.avi

Movies/2000x100_A4_Lb1000.avi

Conclusion

Flocking trans. Liquid-gas transition in canonical ensemble

Symmetry of the liquid phase ρc =∞

Different universality classes:

Ising phase separationVicsek micro-phase separation

Active Ising Model [A. Solon, J.T., PRL 111 078101, (2013)]

Study of Hydrodynamic equations [JB. Caussin, A. Solon, A. Peshkov, H.

Chaté , T. Dauxois, J.T., V. Vitelli, D. Bartolo et al., PRL 112 148102, (2014)]

AIM (follow-up) and Vicsek: hopefully next week on the arxiv !


Phase-separated profiles

Propagating shocks between ρ`,m` = 0 and ρh,mh 6= 0

Stationary solutions in comoving frame of velocity c

Dρ′′ + cρ′ − vm′ = 0 (1)

Dm′′ + cm′ − vρ′ + 2m(β − 1− r

ρ)− αm

3

ρ2= 0 (2)

Solvable at large densities ρ1 = rβ−1 � r

1: Solve (1) to get ρ = ρ` + vc

∑∞k=0

(− D

c ∇)km

2: Expand (2) around ρ1, inject ρ(m) and truncate

D(1 + v2

c2)m′′ + [c− v2

c −2Dvrc2ρ21

m]m′−2r(ρ1−ρ`)ρ21

m+ 2rvcρ21m2− αm3

ρ21= 0





Dρ′′ + cρ′ − vm′ = 0 (1)

Dm′′ + cm′ − vρ′ + 2m(β − 1− r

ρ)− αm

3

ρ2= 0 (2)



∑∞k=0

(− D

c ∇)km


D(1 + v2

c2)m′′ + [c− v2

c −2Dvrc2ρ21

m]m′−2r(ρ1−ρ`)ρ21


ρ21= 0





Dρ′′ + cρ′ − vm′ = 0 (1)

Dm′′ + cm′ − vρ′ + 2m(β − 1− r

ρ)− αm

3

ρ2= 0 (2)



∑∞k=0

(− D

c ∇)km


D(1 + v2

c2)m′′ + [c− v2

c −2Dvrc2ρ21

m]m′−2r(ρ1−ρ`)ρ21


ρ21= 0


Symmetric front solutions β ' 1

m±(x) =mh

2(tanh(q±(x− ct)) + 1)

c= v mh =4r

3αq± = ± β − 1

3√αD

' ±0.0518

0 100 200 300 4000.2

0.0

0.2

0.4

0.6

0.8

1.0

x

ρ−ρlρh−ρl

q=0.051 q=-0.051

T=0.83

microfit tanh


Symmetric front solutions β ' 1

m±(x) =mh

2(tanh(q±(x− ct)) + 1)

c= v mh =4r

3αq± = ± β − 1

3√αD' ±0.0518

0 100 200 300 4000.2

0.0

0.2

0.4

0.6

0.8

1.0

x

ρ−ρlρh−ρl

q=0.051 q=-0.051

T=0.83

microfit tanh


Asymmetric front solutions β > 1

m±(x) =mh

2(tanh(q±(x− ct)) + 1)

c = v+2Dr2

3vαρ21q± = ± r

3ρ1√αD− r2

6αvρ21mh =

4r

3α− 8Dr3

9v2α2ρ21

0 100 200 300 4000.2

0.0

0.2

0.4

0.6

0.8

1.0

x

ρ−ρlρh−ρl

q=0.078 q=-0.14

T=0.5

microfit tanh


The flock fly faster than the birds

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.95

1.00

1.05

1.10

1.15

1.20

1.25 cv

T

micromean-field

0 200 4001

3

5

7

0 200 4007

9

11

13

15

c = v+ 2Dr2

3vαρ21

v microscopic velocities

2Dr2

3vαρ21FKPP-like contribution


The flock fly faster than the birds

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.95

1.00

1.05

1.10

1.15

1.20

1.25 cv

T

micromean-field

0 200 4001

3

5

7

0 200 4007

9

11

13

15

c = v+ 2Dr2

3vαρ21

v microscopic velocities

2Dr2

3vαρ21FKPP-like contribution


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From phase to micro-phase separation in flocking models · 2017-06-06 · From phase to micro-phase...

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