Phase Transition in Collective Motion

Transcription

1 Phase Transton n Colletve Moton Hefe Hu May 4, 2008 Abstrat There has been a hgh nterest n studyng the olletve behavor of organsms n reent years. When the densty of lvng systems s nreased, a phase transton from a dsordered state nto an ordered state ours:.e. unts, whh move n random dretons below the transton, move together n the approxmately same manner or dreton. Several models and experments are revewed n ths essay.

2 1. Introduton A hgh nterest n studyng the olletve behavor of organsms has been rased n reent years. When the densty of lvng systems s nreased, a phase transton from a dsordered state nto an ordered state ours unts, whh move n random dretons below the transton, move together n the approxmately same manner or dreton (Fg. 1). Fg.1. Fsh shools [Photo: Norbert Wu, 1999] Ths olletve moton emerges at all szes, from ells to whales. Several assumptons based on evolutonary funtons are proposed to explan why organsms tend to behave smlarly, suh as nreasng survvorshp, matng, food fndng, et. [Parrsh et al., 1999]. Besdes evolutonary assumptons, several models based on self-propelled partles have been developed. Also, a few experments were suessfully arred out under laboratory ondtons reently. Models and experments are revewed n the followng setons. 2. Models and smulatons

3 The self-propelled partle models proposed n about last 10 years manly fall nto two ategores: one s assumng the partles have smple short-range nteraton wth some random nose, and the other one employs more omplated nteratons. 2.1 Smple short-range nteraton wth nose added Vsek et al. ntrodued ther model n 1995 wth a smple rule that partles were propelled wth a onstant absolute veloty and the average dreton of moton n a partle s neghborhood was assumed at eah tme step wth some random nose added [vsek et al.1995]. In smulatons, Vsek et al. assumed that N partles were randomly dstrbuted n a square shaped ell at ntal tme (t = 0), and they had the same absolute veloty v and random dstrbuton of veloty dretons θ. The poston of the th partle s x ( t+ Δ t) = x () t + v () t Δt, (1) where the smultaneous veloty v () t step. v ( t+δt) determned by was determned at eah step and Δ t was tme here has an absolute value v, and a dreton gven by θ(t+δt) whh s θ ( t+ Δ t) =< θ( t) > +Δ θ, (2) where <θ(t)> r was the average dreton of partles wthn a rle of radus r (nteraton range) surroundng a partular partle, and Δθ, presentng nose, was a random number hosen n the nterval [-η/2, η/2 ]. Fgure 2 (a - d) shows the veloty feld wth varous nose parameters η and densty ρ= N / L. Fg. 2(d) wth hgh densty and low nose demonstrates the most nterestng result most partles have ordered moton n approxmately the same dreton. The absolute value of the average normalzed veloty was also determned and taken as an order parameter v a r N 1 = v. (3) Nd = 1 When the veloty of partles was randomly dstrbuted ntally v a = 0, and when the moton of all the partles beame ordered v a = 1. Fgure 3 (a) demonstrates v a as a funton of η at fxed densty ρ wth dfferent sample sze N, and fgure 3(b) shows behavor of v a as densty ρ hanges at fxed nose η.

4 Fg. 2 Veloty feld wth dfferent nose parameters η and densty ρ. N = 300 n eah ase. (a) t = 0, L = 7, η =2.0. (b) L = 25, η = 0.1. () L = 7, η = 2.0. (d) L = 5, η = 0.1 Fg. 3 (a) the average normalzed veloty as a funton of nose. (b) the average normalzed veloty as a funton of densty Sne the behavor of v a s smlar to that of the order parameter n equlbrum systems, Vsek et al. assumed that ths knet phase transton was analogous to the phase transton n equlbrum systems, v a β [ η ρ η] and ~ [ ρ ρ ( η) ] ~ ( ), v a δ (4) where η (ρ) and ρ (η) are rtal ponts. β = 0.45±0.07 and δ = 0.35±0.06 were obtaned by lnearly fttng data n plot of dependene of ln v a on ln([ η ( L) η]/ η ( L)) and [ L ] ln( ρ ρ ( ) / ρ ( L)) shown n fgure 4.

5 Fg. 4 (a) dependene of ln v a on ln([ η ( L) η]/ η ( L)) (b) dependene of ln v a on [ L ] ln( ρ ρ ( ) / ρ ( L)) Gregore et al. [Gregore et al., 2003] proposed a mnmal model n 2003 extendng Vsek s model by takng the possble oheson of partles nto aount. 2.2 Nose-ndued transton Erdmann et al. [Erdmann, 2005] proposed a model wth an attratve parabol attratng par potental between self-drven partles. Nose effet was espeally nvestgated on the systems. The moton of N partles was gven by the followng equaton: r = v (5) where F s hosen n the form of N a v = F ( r r ) + ξ ( t), (6) j N j= 1 F v v dependng on the veloty of partles, 2 = (1 ) and ξ s stohast whte fores wth strength D ndependent for dfferent partles: < ξ () t >= 0, < ξ () t ξ ( t ) >= 2 Dδ( t t ) δ. In smulaton, all partles had dental j j postons as well as velotes at t = 0, and nose was ntrodued at tme t = 30. Sne the swarm of partles was not sotrop, mean-square dsperson, parallel and orthogonal to the dreton of ts nstantaneous mean veloty V = (1 / N) v ( t), was montored:

6 N 1 S { } 2 () t = r() () (), 2 t R t V t NV () t = 1 (7) where R N 1 S () { () () ()} 2 t = r, 2 t R t V t NV () t = 1 s the enter of mass of the loud of partles. Fgure 5 demonstrates dependene of the mean veloty of swarm on the nose ntensty D. A sharp transton was found at < D < 0.070, where V dropped dramatally to a small number. Fgure 6 shows the behavor of longtudnal and transverse dspersons as nose nreases. It ndates S () t S () t untl the ntensty of nose D approahes ts rtal pont, whh means the swarm of partles s strongly squeezed n the dreton of mean veloty, and after the transton the longtudnal dsperson approahes the transverse dsperson for a strong nose. Fgure 6 gves the sequental snapshots showng how translatonal moton s transferred to rotatonal mode. (8) Fg. 5 Mean veloty of swarm as a funton of nose ntensty.

7 Fg. 6 Tme snapshots of a swarm wth nose ntensty D = 0.070

8 3. Experments Although there are a huge number of examples of olletve moton observed n nature, only a few experments were arred out n the laboratory ondtons. Beo et al. [Beo, 2006] presented suh an experment on fsh shools by trakng the moton of every young tlapa fsh. A thn ontaner (40m 30m 2m) wth water was used, thus the moton of fsh ould be onsdered as n two dmensons. The ontaner was llustrated by a homogeneous lght soure and the moton of fsh was reorded by a CCD amera below the ontaner. Fgure 7 shows two typal trajetores of all fsh for dfferent fsh denstes (fsh per m 2 ). Fg. 7 Left: the trajetores of 20 fsh wth fsh densty 350 f/m 2. Rght: the trajetores of 20 fsh wth fsh densty 905 f/m 2 The nearest neghbor dstane D was omputed and found to be dstrbuted obeyng a lognormal law: 1 ( log( D) μ ) 2 PD ( ) = exp, (9) 2 DS 2π 2S where μ s a sale parameter and S s the shape parameter. Then the poston of the maxmum n the dstrbuton D 1 was alulated. The dependene of D 1 on the fsh densty ρ s llustrated n fgure 8, showng a sharp transton at a rtal densty ρ. The behavor of D 1 (ρ) was ftted wth a empral law: α a1( ρ ρ) + D1, when ρ < ρ D1 =, (10) D1, when ρ ρ where a 1 s a fttng parameter. The fttng gves ρ =527±126 f / m 2, D 1, = 1.21±0.17 m and exponent α = 0.7±0.3. Correlatons between dfferent fsh velotes were also measured. The orrelaton between speeds of neghborng fsh was found to be very

9 weak, but the orrelaton between relatve orentatons of neghborng fsh was strong. Fgure 9 shows the dstrbuton of the relatve orentaton θ nn defned as the relatve angle between velotes of neghborng fsh. An exponental ft was used to fnd the wdth σ of the dstrbuton. Then the ooperatveness was measured by the nverse of the wdth σ -1 and plotted n fgure 10 as a funton of fsh densty. Another empral law was appled to ft the urve: b2 when ρ ρ 1 < σ ( ρ) = β, (11) a3 ( ρ ρ) when ρ ρ where a 3 and b 2 are fttng parameters. ρ = 472±38 f/m 2 and β = 0.7±0.3 were obtaned by fttng. Fg. 8 The most probable nterdstane D 1 as a funton of fsh densty. Fg. 9 Dependene of ooperatveness on fsh densty. B. Szabo et al. [B. Szabo et al.,2006] presented a better ontrolled experment n 2006 usng tssue ells. A omputer-ontrolled tme-lapse mrosope was used to montor the moton of ells, whh s presented n fgure 10 for three dfferent denstes. In hgh densty ase, ells show ordered moton. The order parameter was hosen to be tme average of the sum of the normalzed velotes dvded by the number N of ells measured: N 1 v( tk) V =, (12) N v ( t ) = 1 k tk where t k s the tme elapsed. Fgure 11 shows the order parameter as a funton of normalzed ell densty ndatng a sharp phase transton ours as the normalzed ell densty nreases. Ths experment was also nterpreted by the model proposed by

10 Vsek et al.. Fg. 10 Moton of ells for three dfferent denstes. (a) 1.8 ells/ μm 2. (b) 5.3 ells/ μm 2. () 14.7 ells/ μm 2. Fg. 11 Dependene of order parameter on normalzed ell densty.

11 4. Conluson Several models were establshed for olletve moton of organsms. Computatonal smulatons n these models were also performed, whh gave the theoretal evdene of olletve behavor, and were used to nterpret expermental observatons. However, the lak of experments under laboratory ondtons lmts a better understandng of ths vtal phenomenon. How to perform better experments whh an be analyzed quanttatvely s a key pont.

12 Referenes Ch. Beo, N. Vandewalle, J. Delourt and P. Ponn, Expermental evdenes of a strutural and dynamal transton n fsh shool, Physa A, 367 (2006) U. Erdmann and W. ebelng, Nose-ndued transton from translatonal to rotatonal moton of swarms, Phys. Rev. E 71, (2005) G. Gregore, H. Chate and Y. Tu, Movng and stayng together wthout a leader, Physa D, 181 (2003) J. K. Parrsh and L. Edelsten-Keshet, Complexty, Pattern, and Evolutonary Trade-offs n Anmal Aggregaton, Sene, 284, 99 (1999) B. Szabo, G.J. Szollos, B. Gon, Zs. Jurany, D. Selmez, and T. Vsek, Phase transton n the olletve mgraton of tssue ell: Experment and model, Phys. Rev. E 74, (2006) T. Vsek, A. Czrok, E. Ben-Jaob, I. Cohen and O. Shohet, Novel Type of Phase Transton n a System of Self-Drven Partles, Phys. Rev. Lett., 75, 1226 (1995)