· Irene Giardina ! Dept. of Physics, University of Rome La Sapienza & CNR-ISC ! CNR ! Dept. of Physics, Rome ! University La Sapienza ! Institute ! for Complex Systems !

Irene Giardina!

Dept. of Physics, University of Rome La Sapienza & CNR-ISC !

CNR!

Dept. of Physics, Rome!University La Sapienza!

Institute !for Complex Systems! Accademia Nazionale dei Lincei, Rome 2017

Dynamic  scaling  in  natural  swarms    

Collec3ve  behaviour  in  biological  systems  

insects  

mammals    fish   birds  

bacteria       cells  

self-­‐organized  collec5ve  behaviour  

0

0.5

1

d eDeseigne  et  al,  PRL  2010  Narayan  et  al,  Science  2007   Bricard  et  al,  Nature  2013  

Analogy  with  sta3s3cal  physics  and  ac3ve  non-­‐living  ma;er    

Hope:                  details  do  not  ma/er  –  few  features  determine  the  universality  class  

                                       simple  models  capture  the  large  scale  behavior  of  ac;ve  ma/er  

v

i(t +1) =

v

i(t) + J

v

k(t) +

ξ

i

k∈i

∑ri(t +1) =

ri(t)+vi(t +1)

vi= v

0

self-­‐propelled  par5cles  

Vicsek  model  

Are  these  hopes/assump3ons  jus3fied  for  animal  groups  

 and  complex  biological  systems?  

Collec3ve  behaviour  in  animal  groups  

movie by C. Carere - Starflag!

movie by S. Melillo, SWARM!

Flocks   Swarms  

Global  order  

Scale  free  correla;ons  -­‐    Collec;ve  turns  

Pnas  105  (2008),  Pnas  107  (2010),    Pnas  109  (2012)  

 Nature  Phys  10  (2014),  Jstat  2015,  Nature  Phys  12  (2016)  

PRL  118  (2017)  

Plos  Comp.  Biol  10  (2014)    ,  PRL  113  (2014)  

Nature  Phys    13  (2017)  

No  global  order  

Correla;ons  –  quasi  cri;cal  behavior  

Can  we  define  classes  of  behavior  ?  

scaling  in  cri;cal  phenomena  

sta;c  scaling  

sta3c  scaling  hypothesis:  

anomalous    dimension  

correla;on  func;on  in  momentum  space:  control  parameters  

microscopic  scale  

correla;on  length:  

dynamic  scaling  

dynamic  scaling  hypothesis:  

dynamical  renormaliza;on  group  idea:  

dynamic  cri;cal  exponent  

consequences  of  dynamic  scaling  

collapse  

0 60

t

0

1

C

^ (k,t

)

0 60

kzt

0

1C

^ (k

,t)

just  one  func;on  

cri;cal  slowing  down  

systems  strongly  correlated  in  space  are  also  strongly  correlated  in  ;me  

scaling  

renormaliza;on  group  

universality  

universality  in  biological  systems?  

start  from  scaling  

experiments  

this  is  our  `spin’  

!i(t

0)

!j(t

0+t)

rij

c)

velocity  correla;on  func;on  in  real  space  and  ;me  

0.5t

0

1

C

^(k

,t)

k = 4.53k = 7.77k = 12.9

k = 17.5k = 23.9

dynamic  correla;ons  in  k-­‐space  

natural  swarms  

dynamic  scaling  hypothesis  

0 0.4t

0

1

C

^ (k,t

)

ξ = 12.7

ξ = 10.6

ξ = 8.98

ξ = 7.34

ξ = 6.06

0 8k

zt

0

1

C

^ (k,t

)

2 3logk

-2

0

log

τ k0 60

t

0

1

C

^ (k,t

)

ξ = 1.56

ξ = 1.26

ξ = 1.04

ξ = 0.85

ξ = 0.71

0 60

kzt

0

1

C

^ (k,t

)

-0.5 0 0.5

logk

2

3

4

log

τ k

z ~ 1

z ~ 2

d) e) f)

Natural Swarms

Vicsek Swarms

dynamic  scaling  holds  and  a  new  exponent  emerges  

•   natural  swarms:  

•   Vicsek    swarms:  

how  anomalous  is  the  exponent  z  =  1  ?  

•   Heisenberg/Ising  model  (Model  A):  z  =  2    

•   Non-­‐dissipa;ve  an;ferromagnet  (linear  spin-­‐wave)  (Model  G):  z  =  1.5  

•   Quantum  ferromagnet  (Model  J):  z  =  2.5    

•   Vicsek  model,  ordered  phase  (flocks):  z  =  1.6    

•   Vicsek  model,  disordered  phase  (swarms):  z  =  2  

equilibrium  models:  

ac;ve  ma/er  models:  

such  a  small  value  of  z    seems  to  require  a  non-­‐trivial  renormaliza;on  structure  

scaling  

renormaliza;on  group  

universality  

non-­‐dissipa;ve  relaxa;on  

0.20 t/k

-0.2

0

log(C

(k,t))

^

P(h

)

15

a)

b)

c)

k

15.00

x

0

0.5

1

h(x

)

Vicsek SwarmsNatural Swarms

b)

exponen;al  relaxa;on  -­‐  dissipa;ve    

non-­‐exponen;al    relaxa;on  –  non-­‐dissipa;ve    

t

C

^ (t)

t

C

^ (t)

t

C

^ (t)

0 0.25 0.5 0.75 10.1

x

0

1

h(x

)

deeply underdamped

underdamped

ligthly underdamped

critically damped

lightly overdamped

overdamped

deeply overdamped

effectively exponential

a) b) c)

d)

underdamped  

cri;cally  damped   overdamped  underdamped  

overdamped  

stochas;c  harmonic  oscillator  

swarms  

Vicsek SwarmsNatural Swarms

second-­‐order  iner;al  dynamics  ?  

the  system  has  no  long-­‐range  order  

why  don’t  we  observe  a  purely  dissipa;ve  regime?  

ξ ξL

hydrodynamic  regime   cri;cal  regime  

hydrodynamics  breaks  down  

ξ ~ L

1 2 3

0.1

0.3

ξ

b)

L

hydrodynamic  vs  cri;cal  regime  

natural  swarms  

swarms  are  always  in  the    

cri;cal  regime  

near-­‐cri;cal  censorship  of  hydrodynamics  

conclusions  

dynamic  scaling  holds  in  natural  swarms  

anomalous  cri;cal  exponent  and  non-­‐dissipa;ve  relaxa;on  

near-­‐cri;cal  censorship  of  hydrodynamics  

arXiv:1611.08201  

Nature  Physics,  06/2017  

Andrea Cavagna

Alessandro  AVanasi  

Lorenzo  Del  Castello  

Agnese  d’Orazio  

Asja  Jelic  

Stefania  Melillo  

Leonardo  Parisi  

Oliver  Pohl  

Edward  Shen  

Edmondo  Silvestri  

Massimiliano  Viale  

Tomas Grigera

COBBS  Group  –  Collec;ve  Behaviour  in  Biological  Systems  

Daniele  Con5