Eur Phys J B (2013) 86 91DOI 101140epjbe2013-30821-1

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

Statistical properties of swarms of self-propelled particleswith repulsions across the order-disorder transition

Maksym Romenskyy and Vladimir Lobaskina

School of Physics Complex and Adaptive Systems Lab University College Dublin Belfield Dublin 4 Ireland

Received 9 September 2012 Received in final form 18 December 2012Published online 11 March 2013 ndash ccopy EDP Sciences Societa Italiana di Fisica Springer-Verlag 2013

Abstract We study dynamic self-organisation and order-disorder transitions in a two-dimensional systemof self-propelled particles Our model is a variation of the Vicsek model where particles align the motionto their neighbours but repel each other at short distances We use computer simulations to measure theorientational order parameter for particle velocities as a function of intensity of internal noise or particledensity We show that in addition to the transition to an ordered state on increasing the particle density asreported previously there exists a transition into a disordered phase at the higher densities which can beattributed to the destructive action of the repulsions We demonstrate that the transition into the orderedphase is accompanied by the onset of algebraic behaviour of the two-point velocity correlation functionand by a non-monotonous variation of the velocity relaxation time The critical exponent for the decay ofthe velocity correlation function in the ordered phase depends on particle concentration at low densitiesbut assumes a universal value in more dense systems

1 Introduction

Self-organisation and non-linear dynamics in systems ofactive objects have been receiving much attention fromthe scientific community during the last years in particu-lar in relation to a number of spectacular biological prob-lems One of the most prominent biological examples isthe swarming behaviour ndash spontaneous onset of organisedmotion of large amount of independent individuals Suchcollective motions are commonly observed [1] in schoolsof fish [2ndash4] bacterial colonies [5] slime moulds [67] andflocks of birds [8] Vortex motions were specifically ob-served long ago in colonies of ants [9] and more recently inbacterial colonies [10] and slime moulds [11] The collectivemotion has been shown to arise from simple rules that arefollowed by individuals and does not involve any centralcoordination external field or constraint Recently enor-mous attention has been attracted also to swarm roboticsa promising new approach to coordinate the behaviour oflarge number of primitive robots in decentralised man-ner through their local interactions A novel area of re-search Swarm Intelligence [12] covers the general featuresof swarming and collective behaviour and expands veryrapidly

Observation of dynamic patterns in ensembles of bi-ological and physical species which can be genericallyreferred to as active species stimulated a significant ef-fort in physics of dynamical systems (for which we referthe reader to comprehensive reviews [13ndash16]) A minimal

a e-mail vladimirlobaskinucdie

model of swarming was suggested by Vicsek et al [17]in 1995 A comparison of subsequent minimal models hasbeen conducted in reference [18] In the Vicsek model theswarming results from the action of a collision-type in-teraction that aligns the velocity of each actively movingparticle in a group to the mean local velocity The sta-tistical properties of the Vicsek model in various imple-mentations the pattern formation and the type of order-disorder transition have been extensively studied by thecommunity [19ndash24] The model was further developed byCouzin et al [25] and later was enriched with biologicallyrelevant features including new types of behavioural in-teraction which allowed one to reproduce several types ofswarming dynamics [2126]

An analogy of the velocity alignment in the Vicsekrsquosto the alignment of magnetic dipoles interacting via lo-cal mean field was explored by Toner and Tu [2728]who predicted an existence of the aligned (ferromagnetic)and isotropic (paramagnetic) steady states in a genericcontinuum model of an active system with aligning in-teractions The transition from the disordered behaviourto aligned collective motion occurs in all these modelson decreasing the noise strength similar to the magnetictransition on decreasing temperature The correspondingphase diagram however is more complex than the typi-cal magnetic one as active particles are not fixed on thelattice and their concentration can vary across the sys-tem The aligning interactions in this case act as a co-hesive factor [14] As a result the ordering occurs atcertain minimum particle number density and the ac-tive particles demonstrate a tendency to form compact

Page 2 of 10 Eur Phys J B (2013) 86 91

ordered flocks The orientational transition in the Vicsekmodel with angular noise was shown to have continuouscharacter [2324]

As the transition is associated with orientational sym-metry breaking in all model systems one can expectthat they belong to the same universality class anddemonstrate the same critical properties across the transi-tion [13] A large variety of statistical properties has beenanalysed over the recent years and the validity of the scal-ing relations between critical exponents known from equi-librium phase transitions has been confirmed [232930]The universal properties have been found in a varietyof active systems including those with only pairwise in-teractions [31] Beside the critical behaviour other sta-tistical characteristics can be associated with transitionsuch as cluster size distributions showing a power lawdecay [1820] and giant number fluctuations [1431] Weshould note that some of these features have been alsoobserved in active systems without aligning interactionsbut with just hard-core repulsions so they are not uniqueto the orientational transition [3233] Some other statisti-cal properties like the two-point velocity correlation func-tion however have not been studied in detail Thereforeit would be interesting to test whether they also demon-strate the universal behaviour which is not sensitive tothe details of the particles and their interactions

In this work we use a Vicsek-type model with addedrepulsive interaction (collision avoidance) to study itscomplete phase diagram in terms of particle density andnoise amplitude We analyse the behaviour of the orien-tational order parameter as well as the most importantcorrelation functions across the transition and attempt toidentify the statistical indicators of the orientational or-dering In Section 2 we describe the model and statisticalmethods In Section 3 we describe and discuss the resultsand conclude the discussion in Section 4

2 The model and methods

21 Model and simulation settings

In our two-dimensional model N point particles move witha constant speed v0 inside a square simulation box of sizeLtimesL with periodic boundary conditions so that the parti-cle number density is given by ρ = NL2 The direction ofmotion of each particle is affected by repulsive or aligninginteractions with other particles located at distance equalor smaller than the radius of the zone of repulsion (zor) orzone of alignment (zoa) respectively (see Fig 1) [2534]The first behavioural zone represented in Figure 1 by acircle of smaller radius Rr is responsible for the mainte-nance of a minimum distance between neighbouring par-ticles while the second one with a greater radius Ra isused for aligning the particle velocities The size of therepulsion zone is always set to unity so it defines the unitof length for the simulation We note that Rr can alsobe considered as the body cross-section diameter for theparticle as the penetration into it is penalised

y

x

zor

zoa

Rr Ra

Fig 1 The interaction parameters in the Vicsek-type modelThe particle turns away from the nearest neighbours within thezone of repulsion (the inner circle zor) to avoid collisions andaligns itself with the neighbours within the zone of alignment(the outer circle zoa)

At each time step position of each particle (ri) is com-pared to the location of the nearest neighbours If otherparticles are present within the zone of repulsion the in-dividual attempts to keep its personal space and to avoidcollision by moving away from the neighbours This rulehas been proposed by Couzin et al [2534]

Vi(t + Δt) = R(ξi(t))Vri (t + Δt) (1)

with

Vri (t + Δt) = minusv0

nrsum

j =i

rij(t)∣∣∣∣∣

nrsum

j =i

rij(t)

∣∣∣∣∣

(2)

where rij = rj minus ri nr is a number of particles inside zorand R(ξi(t)) is the rotation matrix

R(ξi(t)) =(

cos ξi(t) minus sin ξi(t)sin ξi(t) cos ξi(t)

)

(3)

ξi(t) is a random variable uniformly distributed around[minusηπ ηπ] where η represents the noise strength

Repulsions have the highest priority in our modeltherefore no alignment rule can be applied if other parti-cles are present in the zor Only if the zone of repulsioncontains no neighbours the particle responds to the nextzone and to align to other particlesrsquo motion

Vi(t + Δt) = R(ξi(t))Vai (t + Δt) (4)

with

Vai (t + Δt) = v0

nasum

j=1

Vj(t)∣∣∣∣∣

nasum

j=1

Vj(t)

∣∣∣∣∣

(5)

where na is a number of particles inside zoaThe above-described behavioural rules are used

throughout the paper however in the last section we also

Eur Phys J B (2013) 86 91 Page 3 of 10

show (where especially noted) some results for modifiedmodel where we replace the uniformly distributed noisewith Gaussian-distributed one [35]

P (ξi(t)) =1

radic2ξi(t)η

eminusξ2i (t)2η2

(6)

where η is the noise strength In addition in this ver-sion of the model particles with no neighbours are notsubjected to angular noise The repulsive interactions arealso noise-free so the term R(ξi(t)) in equation (1) dis-appears This kind of individual and collective behaviourwhich implies some persistence in motion is well suitedfor modelling biological systems and has been extensivelyused in references [253436ndash39]

In both variations of the model the particle positionsare updated by streaming along to new direction accord-ing to

ri(t + Δt) = ri(t) + Vi(t + Δt)Δt (7)

We use the standard Vicsek updating scheme (the forwardupdating rule as defined in Refs [1824])

In our calculations we assume a constant propulsionspeed v0 = 1 and time step Δt = 1 We start the sim-ulation by randomly placing the particles inside the boxand assigning random velocity directions Total number oftime steps is at least 2times107 for all simulations To set therequired density we keep the box size constant (L = 500)and vary the number of particles in the interval from 250to 150 000 The statistics is collected in the steady state(after 50 000 time steps) and each characteristic of mo-tion is calculated by averaging over 5 independent runsAs we mentioned above the size of the zone of repulsionis set to unity in our simulations Rr = 1 Moreover weused the same radius of the alignment zone Ra = 5 in allruns

22 Motion statistics

To analyse the collective behaviour of the model weperform a series of simulations changing the densityρ or η the noise strength We characterise the collec-tive motion with three different correlationdistributionfunctions The velocity autocorrelation function is calcu-lated as

C(t) =1N

langNsum

i=1

Vi(0) middotVi(t)|Vi(0)| middot |Vi(t)|

rang

(8)

The radial distribution function is defined by

g(r) =L2

N(N minus 1)

langNsum

i=1

Nsum

j =i

δ(r minus |rij |)rang

(9)

where δ is the Dirac delta function 〈middot〉 stands for the en-semble average The two-point velocity correlation func-tion is calculated as

C(r) =1

N(N minus 1)

langNsum

i=1

Nsum

j =i

Vi(t) middotVj(t)|Vi(t)| middot |Vj(t)|

rang

(10)

where i and j label particles separated by distancer = |rij | With this definition two particles with parallel(antiparallel) velocities give a correlation of +1 (minus1) Theangular brackets denote the ensemble average To char-acterise the swarming behaviour of the particles we alsoperform a cluster analysis Cluster in our model is definedas a group of particles with a distance between neighbourssmaller or equal to the radius of alignment zone Ra ieparticles interacting directly or via neighbouring agentsare included into one cluster This definition is similar toone proposed in reference [18] We calculate the numberof clusters and mean cluster size

We characterise the orientational ordering by the po-lar order parameter which quantifies the alignment ofthe particle motion to the average instantaneous velocityvector

ϕ(t) =1

Nv0

∣∣∣∣∣

Nsum

i=1

Vi(t)

∣∣∣∣∣ (11)

This order parameter has been extensively used to de-scribe the orientational ordering in various systems ofself-propelled particles [17212440] It turns zero in theisotropic phase and assumes finite positive values in the or-dered phase which makes it easy to detect the transitionHowever at low densities it may be more difficult to detectthe transition due to relatively small number of particlesconstituting the system and large density fluctuations Tolocate transition points precisely we also calculated theBinder cumulant [41]

GL = 1 minus 〈ϕ4L〉t

3〈ϕ2L〉2t

(12)

where 〈middot〉t stands for the time average and index L denotesthe value calculated in a system of size L The most im-portant property of the Binder cumulant is a very weakdependence on the system size so GL takes a universalvalue at the critical point which can be found as the in-tersection of all the curves GL obtained at different systemsizes [22] at fixed density To detect the transition pointsin qminusρ plane precisely we plot three curves for different Lat constant density and find the point where they crosseach other Then we use those points to construct thephase diagram

3 Results and discussion

31 Collective motion

We performed simulations of a system of active particlesto fully map the region of the ordered collective behaviouron the ρminusη plane To illustrate the role of the both param-eters we describe below the variation of the order alongthe iso-density and iso-η paths

First we look at the behaviour of the system at con-stant density ρ = 004 while varying the intensity of angu-lar noise η We expect the onset of ordered behaviour atlow noise levels and a disordered phase in the strong noise

Page 4 of 10 Eur Phys J B (2013) 86 91

(a)

(b)

Fig 2 Simulation snapshots of a swarm in the Vicsek-typemodel (ρ = 004) (a) At zero noise (b) At η = 06

regime Figure 2 shows the alignment of particles veloc-ities at different noise strengths At zero noise (Fig 2a)interacting particles are aligned almost perfectly They canform massive clusters and orientation of each individualcompletely coincides with the direction of movement of thegroup With an increase of the noise amplitude the size ofthe correlated domain shrinks and at high levels of noiseparticles cannot form any stable clusters (Fig 2b) Wenote that this kind of behaviour is an inalienable featureof models with metric interactions Recent studies [4243]on metric-free models showed that topological interactionscause significantly higher cohesion of the particles leadingto formation of one or several large clusters with higherresistance to noise

Analysis of the velocity autocorrelation function in ourmodel (Fig 3a) shows that at η = 0 motion of particles ismost persistent The change of the direction of motion isrealised only through collisions between different clustersHowever we see a fast decorrelation of the velocity inpresence of strong angular noise The qualitative changefrom the slow decaying regime to a fast decay happens atη ge 03 (Fig 3a) Plots of the radial distribution function(Fig 3b) and mean cluster size (Fig 4) show that theparticles tend to form clusters with the density of up to3-4 times above the average The cluster size can be aslarge as several hundred particle sizes (as set by Rr) and

(a)

1 10 100t

001

01

1

C(t

)

η=0η=02η=03η=06η=08

(b)

0 25 50 75r

0

1

2

3

4

g(r)

η=0η=02η=03η=06η=08

(c)

1 10 100r

0001

001

01

C||(r

)

η=0η=02η=03η=06η=08

Fig 3 Statistical properties of the Vicsek-type model at vari-ous noise magnitude (N = 10 000 ρ = 004) (a) Semi-log plotof velocity autocorrelation function C(t) over time t (b) Ra-dial distribution function g(r) (c) Spatial velocity correlationfunction C(r)

can include hundreds of individuals The spatial range ofthe alignment extends as far as 10 particle sizes at highnoise (η = 06 08 in Fig 3c) but becomes much larger atthe lower noise levels On changing the noise the decayalso changes qualitatively the spatial velocity correlationdecays exponentially at high noise and with a power lawat the lower noise levels

Eur Phys J B (2013) 86 91 Page 5 of 10

1 10 100 1000 10000m

1e-05

00001

0001

001

p(m

)

η=0η=02η=03η=06η=08

0 02 04 06 08η

0

10

20

30

40

ltmgt

Fig 4 Cluster statistics for the Vicsek-type model at constantdensity (ρ = 004) The exponent p(m) prop mminusζ for the straightsegment η = 01 ndash ζ asymp 077 η = 02 ndash ζ asymp 10 η = 04 ndashζ asymp 065 Inset average cluster size vs the noise strength

Figure 4 presents the cluster statistics for the samedensity ρ = 004 The main plot shows the cluster sizedistribution on a log-log scale We observe two qualita-tively different distributions at low noise η = 0 to 03the curves have a straight segment at large numbers whichindicates the power law decay of the distribution The al-gebraic decay can be associated with a scale-free systemwhere a formation of arbitrarily large clusters is possibleAt η gt 03 only exponential decay is observed and theformation of large clusters is suppressed The plot in theinset shows the mean cluster size as a function of the noiseamplitude Quite expectedly the mean cluster size is de-creasing with the noise amplification

We analyse now how the collective motion changesupon variation of the density at constant noise η = 04(Figs 5 and 6) At low density (ρ = 0004) the size of thecorrelated domain is minimal particles form small clus-ters (Fig 5) that move in random directions The clustersgrow with the increase of the density and motion becomesmore persistent Behaviour of the velocity autocorrelationfunction (Fig 6a) at these conditions follows two distinctpatterns we see either fast decay within 10 to 30 timeunits (ρ = 0004 or ρ = 05) or a very slow decay (alldensities in between) such that there is no visible decayin the correlation function over hundred time steps Theformer pattern is typical for stochastic disordered systemswhile the latter one corresponds to a very persistent andordered motion Plot of the radial distribution function inFigure 6b shows that maximum local density of particlesis observed within the zone of alignment but the increaseddensity region spans to very large distances in the systemsshowing the persistent motion We see a similar qualita-tive difference in the spatial velocity correlations for thesesystems as shown in Figure 6c two of them ρ = 0004and ρ = 05 decay exponentially while the other fourshow an algebraic decay with C(r) prop rminus spanning overinterval of more than hundred interparticle distances

In Figure 7 we present the clustering statistics at vari-able density Similar to the noise level series we observetwo types of cluster size distributions at low densitiesρ = 0004 and 0016 we see exponential distributions

(a)

(b)

(c)

Fig 5 Typical snapshots for the Vicsek-type model at variousdensities (η = 04) (a) ρ = 0004 (b) ρ = 01 (c) ρ = 055

while the four other curves at ρ gt 0016 contain a straightsegment which corresponds to a power law decay More-over at the three highest densities we also observe a peakat large particle numbers at m 10 000 which is closeto the total number of particles in the system We canconclude that particles travel in a single flock most ofthe time The inset plot shows the mean cluster size asa function of the density We note that at ρ lt 0024 noclusters are formed while the cluster size starts growingrapidly at ρ gt 01 ie there is a sharp transition into theldquocondensedrdquo phase on increasing density We note that the

Page 6 of 10 Eur Phys J B (2013) 86 91

(a)

1 10 100t

001

01

1

C(t

)

ρ=0004ρ=015ρ=025ρ=03ρ=04ρ=05

(b)

0 2 4 6 8 10r

05

1

15

2

25

g(r)

ρ=0004ρ=015ρ=025ρ=03ρ=04ρ=05

(c)

1 10 100r

00001

0001

001

01

1

C||(r

)

ρ=0004ρ=015ρ=025ρ=03ρ=04ρ=05

Fig 6 Statistical properties of the Vicsek-type model in pres-ence of constant angular noise (η = 04) (a) Semi-log plot ofvelocity autocorrelation function C(t) over time t (b) Radialdistribution function g(r) (c) Spatial velocity correlation func-tion C(r)

cluster size distribution does not show the same exponentp(m) prop mminusζ as myxobacterial systems or hard rods [31]where an exponent ζ of about 09 has been found Al-though the bulk of our measurements shows an exponentζ = 09 divide 11 there are some deviations at low densitieswhere the exponent becomes smaller (ζ asymp 066) and athigh noise strength (ζ gt 15) This can be also compared

1 10 100 1000 10000m

1e-05

00001

0001

001

01

p(m)

0 02 04ρ

0

1e+04

2e+04

ltmgt

ρ=0004ρ=0016ρ=01ρ=015ρ=02ρ=025

Fig 7 Cluster statistics for the Vicsek-type model in presenceof constant angular noise (η = 04) The exponent p(m) prop mminusζ

for the straight segment ρ = 004 ndash ζ asymp 066 ρ = 01 ndashζ asymp 099 ρ = 015 ndash ζ asymp 11 ρ = 02 and ρ = 025 ndash ζ asymp 12Inset average cluster size vs the density ρ

with the cluster size distributions obtained for the origi-nal Vicsek model [1820] where authors found exponentsranging from ζ = 05 to ζ = 15 depending on the level ofnoise and the density Although our simulation settingsthe particle speed and the size of the alignment zone dif-fer from those in papers by Huepe and Aldana [18] ourvalues of ζ lie in the same range The differences in clus-ter size distributions between the Vicsek-type models andsystems of rod-like particles [31] might be related eitherto isotropic character of interaction or more long-rangealigning interactions thus leading to stronger cohesion inthe former systems

From the results presented in this subsection we canconclude that motion in our model becomes persistent ina limited range of noise level and densities Outside thisregion the motion is predominantly random with short-range temporal and spatial correlations The two types ofcollective dynamics are characterised by qualitatively dif-ferent statistical properties exponentially decaying cor-relations in the disordered state and the algebraic decayand long-range correlations and scale-free behaviour in theordered state It is clearly seen that the persistence is acollective effect which macroscopically corresponds to biggroups of particles moving in the same direction and sta-bilising each otherrsquos direction of motion We also see thatin addition to orientational ordering a density separationsimilar to a vapour condensation into a liquid phase is tak-ing place The aligning interaction between particles actsin this case as an effective attraction which suppresses therelative motion of the particles thus leading to particleaggregation

32 Orientational ordering

As one can see from the simulation snapshots the velocityalignment in our model leads to global symmetry break-ing when most of the particles end up moving steadily inthe same direction (Figs 2a and 5b) In this subsectionwe will analyse the transition in some more detail

Eur Phys J B (2013) 86 91 Page 7 of 10

0 20000 40000 60000 80000 1e+05t

0

02

04

06

08

1

ϕ

η=00

η=01

η=02

η=03

η=04η=06

Fig 8 Evolution the orientational order parameter in theVicsek-type model at constant density (ρ = 004) The initialconfiguration at t = 0 is disordered

The evolution of the orientational order parameter isillustrated by the time series in Figure 8 for various levelsof noise At weak noise η = 0minus02 the model demon-strates highly ordered motion with the order parameterfluctuating about the finite mean value At the higher levelof noise η = 03 04 we see a borderline behaviour withlarge fluctuations which are characteristic for the criticalregion where the system fluctuates between the orderedand disordered states Finally at η = 06 the order pa-rameter is zero within the statistical uncertainty Figure 9shows the behaviour of the order parameter for variousvalues of the swarm density Here we see a wildly fluctu-ating order parameter at ρ = 0004 and ρ = 004 then thefinite ϕ values with some fluctuations at ρ = 01 02 03and the absence of order at ρ = 055 In contrast withthe dependence on the noise strength where the order-ing was gradually decreasing upon increase of the noisethe concentration behaviour is non-monotonous The or-dering is weak at small and high densities and high inbetween The highest values of ϕ are reached at ρ = 01This finding is in agreement with the observation madeearlier (see Fig 5c) the spatial velocity correlation spansover smaller distances on increasing the particle numberdensity We can relate this to a larger concentration oforientational fluctuations at the higher densities as com-pared to the small system The orientational fluctuationsin the motion of the individuals lead to particle overlapsand the temporary loss of alignment due to the repulsionswhich leads to globally less ordered motion

It is clearly seen from the Figures 8 and 9 that symme-try breaking into an orientationally ordered state is possi-ble only after reaching certain levels of the noise strengthand particle number density We combined a large set ofthe order parameter data for various values of ρ and ηin Figure 10c The plot confirms the existence of the opti-mum intervals for both density and the magnitude of noisewithin which the global ordering (non-zero average ϕ) ismaximal The minimum density required for the orderingis quite small In dense systems the range of the orderedbehaviour is widest at η = 0 and is getting narrower uponincreasing η We also note that the density at which the

0 20000 40000 60000 80000 1e+05t

0

01

02

03

04

05

06

ϕ

ρ=01

ρ=02

ρ=03

ρ=004

ρ=0004ρ=055

Fig 9 Evolution of the orientational order parameter in theVicsek-type model at constant noise (η = 03) The initialconfiguration at t = 0 is disordered

maximum ordering is observed is increasing from ρ asymp 01at small η = 0 to ρ asymp 018 at η = 046 The station-ary value of the order parameter in each configuration isdetermined by the competition between the disorderingnoise and the aligning interactions Therefore the rangeof densities at which our model exhibits a transition tothe ordered state depends on the level of noise

In Figures 10a and 10b ϕ is plotted for various valuesof ρ and η At zero noise and low densities the order pa-rameter is close to 1 which corresponds to a completelyaligned system As the density and the noise increase theorder parameter eventually decays to zero We find that ϕvaries continuously across the transition on varying theparticle number density at the fixed noise level (see Fig 6and the map on Fig 11a) Similarly the transformationthat happens on increasing the noise level when the den-sity is fixed is continuous with the order parameter decay-ing to zero (see Fig 3 and the map on Fig 11a) in agree-ment with previous observation for the model with thesame alignment and noise type [162340] This conclusionis also confirmed by the behaviour of Binder cumulantsacross the transition

We should note that observed behaviour of the or-der parameter and correlation functions closely resem-bles those for Heisenberg ferromagnet as has been dis-cussed previously [13142329] An extensive discussion ofthe finite size scaling and critical exponents in the Vicsekmodel can be found in references [2329] The change inthe behaviour of C(r) as seen in Figures 3c and 6ccan be associated with a symmetry breaking in the sys-tem and a transition into the aligned state In the two-dimensional XY spin model the decay of the two-pointcorrelation function is exponential in the isotropic phasewhile at the critical point the decay becomes algebraicC(r) prop rminus [4445] Note that generally the correlationfunction of the order parameter scales at the critical pointas G(r) prop rminusd+2minus in d-dimensional systems [46] In twodimensions the simple spin-wave theory leads to a tem-perature dependent exponent = kBT4πJ where kB

and T are the Boltzmann constant and absolute temper-ature respectively and J is the spin coupling constant

Page 8 of 10 Eur Phys J B (2013) 86 91

(a)

0 01 02 03 04 05 06ρ

0

02

04

06

08

1

ϕ

η=0η=01η=02η=03η=04η=046η=05

(b)

0 01 02 03 04 05η

0

02

04

06

08

1

ϕ

ρ=004ρ=01ρ=02ρ=03

(c)

0 01

02 03

04 05

0 01 02 03 04 05 06

0 02 04 06 08

1

η

ρ

ϕ 0 02 04 06 08 1

Fig 10 Orientational order parameter for the Vicsek-typemodel (a) The iso-η curves (b) The iso-density curves (c)The surface plot

while an inclusion of interaction between spin vorticesyields an exponent = 14 [4547] In two-dimensionalnematics an algebraic decay of order parameter corre-lations have been previously found across the whole ne-matic region although with a temperature-dependent ex-ponent [48]

In our system the correlation function indeed decaysexponentially in the symmetric (disordered) phase In thenon-symmetric (ordered) phase it varies according to apower law C(r) prop rminusd+2minus where the exponent takesvalues from = 05 to asymp 097 At the transition point

(a)

0 01 02 03 04 05 06ρ

0

02

04

06

08

1

η

UniformGaussianDisordered state

Ordered state

Figs 3 4

Figs 6 7

(b)

0001 001 01 1ρ

01

1

η

UniformGaussian

~ρ045

Fig 11 Phase diagram for the Vicsek-type model (a) Com-plete diagram in ηminusρ plane The red and blue dots show thesettings corresponding to series shown in Figures 3 4 6 and 7respectively (b) Log-log plot showing transition region at lowdensities

however the exponent is expected to satisfy the Fisherrsquosscaling law γν = 2minus [46] where γ and ν are the criticalexponents for isothermal succeptibility and the fluctuationcorrelation radius respectively Indeed in the limit of lowconcentrations where the repulsions are not importantwe have = 05 thus 2 minus = 15 which is in agreementwith the result for γν = 147 obtained previously for thestandard 2D Vicsek model [23] The other values of inour system are obtained inside the ordered region far fromthe critical line so the scaling relations are not expectedto hold

In Figure 12 we plotted the values of the exponent for two versions of the model (a) with uniformly dis-tributed and (b) Gaussian noise (we consider it necessaryto remind a reader that in this variation of the model par-ticles with no neighbours as well as non-interacting agentsare not subjected to the action of noise) [35] In contrastto the XY model in the Vicsek-type system the valueof seems to be insensitive to the ldquotemperaturerdquo (thenoise strength) and depends only on the density We notethat the lower values of the exponent are observed onlyat low densities where the orientational phase transitionis accompanied by the gas-liquid-type transition We can

Eur Phys J B (2013) 86 91 Page 9 of 10

(a)

0 01 02 03 04 05 06ρ

05

06

07

08

09

1

η

η=0η=01η=02η=03η=04

(b)

0 01 02 03 04 05 06ρ

05

06

07

08

09

1

η

η=0η=01η=02η=03η=04η=05η=06η=07

Fig 12 Behaviour of the scaling exponent of the velocity cor-relation function C(r) prop rminus for the Vicsek-type model with(a) uniformly distributed and (b) Gaussian noise for variousnoise amplitudes η within the ordered phase

speculate that the crossover is reflecting the competitionbetween the cluster size and the range of alignment Thecorrelations are fully developed at densities ρ gt 02 so theexponent remains constant at the higher densities Thehigher value of the exponent in the Vicsek-type model ascompared to the 2D XY is a consequence of the isotropyof the interaction which represents the local mean fieldand is not bound to the lattice directions Another impor-tant difference between the two models is that the activeswarm with aligning interactions must be able to developa true long-range order ie the ordered cluster can growindefinitely in a large system as it follows from the nonlin-ear continuum theory [13] The transition lines in the ηminusρplane are best seen in Figure 11b In the low-density re-gion we observe the same power law ηc prop ρκ with κ = 045for both types of noise This finding is in agreement withrecent theoretical and computational results for the orig-inal Vicsek model [212229] The phase diagram we ob-serve in Figures 11a and 11b possesses one new importantfeature as compared to previously reported ones We seethe breakdown of the global order at high densities Inour system this breakdown is related to the misalignmentcaused by more frequent particle collisions as the systembecomes more crowded To confirm this hypothesis wehave calculated the collision frequency as a function ofthe density at various noise levels Indeed we found thatthe relative frequency of collisions continuously grows withthe density and reaches 08 at ρ = 06 (ie 80 of time theparticles are just bouncing back from each other) We alsofound that a removal of this mechanism or replacement ofit by a more traditional excluded volume interaction re-moves the re-entrant transition at the higher densities InFigure 13 we present results of simulations performed forvarious radii of the repulsion zone for the variation of ourmodel with uniformly distributed noise It is clearly seenthat upon increase of Rr the transition point moves to thelower values of the noise amplitude (Fig 13a) At the sametime in the density scan (Fig 13b) the re-entrant tran-sition into disordered phase happens earlier for the biggerradii of the repulsion zone and disappears at Rr = 0 Weshould note that the range of the ordered behaviour onthe phase diagram depends also on the propulsion speedand the time step and decrease of either of these variables

(a)

0 01 02 03 04 050

02

04

06

08

1

ϕ

Rr=0Rr=1Rr=15

η

(b)

0 01 02 03 04 05ρ

0

02

04

06

08

ϕ

Rr=0

Rr=1

Rr=15

Fig 13 Orientational order parameter for the Vicsek modelwith different radii of the repulsion zone Rr at various levels ofthe noise (a) at ρ = 004 and (b) at variable density at η = 04

leads to a shift of the transition points to higher densi-ties Therefore this destruction of order seems to be aspecific feature of the collision avoidance rule used in theCouzinrsquos version of the model [253449] We can imaginethat in real life systems this kind of behaviour would becharacteristic for swarms of agents with step-like motionsuch as high-density human crowds

4 Conclusions

We have studied the statistical properties of a system ofself-propelled particles with aligning interactions and col-lision avoidance representing a variation of the Vicsekmodel across the dynamical order-disorder transition Weobserved that in addition to the orientational transitionon increasing the particle number density reported ear-lier there exists a re-entrant transition into the disorderedphase at high concentrations which is related to the ac-tion of short-range repulsions destroying the local orderThe transition in our system is continuous along both den-sity and noise strength axes in agreement with previousobservations of models with the same alignment and up-dating rules We have demonstrated that orientationallyordered phase is characterised by algebraic behaviour ofthe two-point velocity correlation function and cluster sizedistribution The critical exponent for the decay of thetwo-point velocity correlation function exhibits a crossoverfrom = 05 to a density-independent value asymp 1 onincreasing density

We are grateful to Prof L Schimansky-Geier Prof M Barand Prof H-B Braun for insightful discussions Financialsupport from the Irish Research Council for Science Engi-neering and Technology (IRCSET) is gratefully acknowledgedThe computing resources were provided by UCD and IrelandrsquosHigh-Performance Computing Centre

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