Swarming in disordered environments
SSwarming in disordered environments
David Quint and Ajay Gopinathan
Department of Physics, University of California Merced, Merced CA 95343 USA
Abstract
The emergence of collective motion, also known as flocking or swarming, in groups ofmoving individuals who orient themselves using only information from their neighborsis a very general phenomenon that is manifested at multiple spatial and temporal scales.Swarms that occur in natural environments typically have to contend with spatial disordersuch as obstacles that hinder an individual’s motion or communication with neighbors. Westudy swarming particles, with both aligning and repulsive interactions, on percolated net-works where topological disorder is modeled by the random removal of lattice bonds. Wefind that an infinitesimal amount of disorder can completely suppress swarming for parti-cles that utilize only alignment interactions suggesting that alignment alone is insufficient.The addition of repulsive forces between particles produces a critical phase transition froma collectively moving swarm to a disordered gas-like state. This novel phase transition isentirely driven by the amount of topological disorder in the particles environment and dis-plays critical features that are similar to those of 2D percolation, while occurring at a valueof disorder that is far from the percolation critical point.
Introduction
Collective motion of self propelled individuals is a well studied emergent phenomenon [1] thatspans many different length and time scales from biopolymers on a bed of molecular motors [2],swimming bacteria [3, 4], birds and fish [5, 6] to people “moshing” at heavy metal concerts [7].Within the literature that is aimed at studying collective motion in systems of self-propelled par-ticles [8, 9, 10, 11, 12, 13], a main underlying assumption has been that the environment, wherethe particles exists, is continuous, isotopic and ordered. In the natural world there are manyexamples of disordered environments where collective motion can exist. Examples include bats1 a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b hich navigate natural caverns via echolocation, schools of fish that maneuver through densekelp forests, microbial colonies that move about in heterogeneous soil [14], crowds of peo-ple that are evacuating a building [15] and traffic flow in major cities [16]. Given that naturalenvironments can be intrinsically disordered, it is interesting to consider how self propelled in-dividuals maintain an organized state of collective motion without knowledge of a global “roadmap”. What are the necessary physical mechanisms that facilitate the collective movementof particles across a intrinsically disordered network? What is the role of ”thermal” noise in aswarm that moves through a disordered environment? In this manuscript we provide insight intothese questions for the first time, by studying a two dimensional system in which we representthe disordered environment as a percolated lattice that our self-propelled particles inhabit. Model & Simulation
To study swarming behavior in the presence of topological disorder we implement a MonteCarlo lattice gas model [9, 17] that consists of N p interacting mobile particles that occupy a d periodic triangular lattice with L ≡ N lattice sites. The particle density is defined as ρ = N p /N and in this model it can, in general, be larger than the density of lattice sites. Thedynamics of our model [see supplementary information for more details] are such that, at anygiven time step, particle k moves along any of the six lattice bonds (with direction unit vectors u i ) with a velocity v ( k ) whose magnitude is a constant defined to be unity [bond length/timestep] (Fig. 1A). Particle moves are controlled via a standard Monte Carlo procedure with Boltz-mann weights determined by two interaction energies, between nearest neighbor (n.n) particles,in our model. The first is an alignment interaction energy, E ai (Eqn. 1), that makes it favor-able for particles to orient their velocity vectors along the direction of the average velocity oftheir n.n. The magnitude of alignment is controlled by the parameter α , which is set to unityand defines the relevant energy scale. The second is a mutual repulsive interaction energy, E ai n ( i ) − n (0) seen by a particle,where n ( i ) is the number of particles at a n.n. site along a lattice direction u i and n (0) is thenumber of particles at the particle’s current site. The magnitude of the repulsive interaction iscontrolled by the parameter (cid:15) and the local repulsive energy enhances the probability of movingin directions with the greatest drop in density. We introduce topological disorder in our latticesystem by varying p , the probability that a bond exists between two lattice sites as in usual bondpercolation theory [18]. The quantity, − p , therefore represents the environmental disorderfraction. Finally, the parameter T , which enters via the Boltzmann weighting for the MonteCarlo procedure effectively controls the magnitude of thermal noise in our system. We quenchthe disorder in the lattice during each realization of our simulation, allowing us to study theeffect of topological disorder and thermal noise independently and their effect on the fidelity oflocal information that particles use to navigate the disordered environment. E ai = − α u i · n ( j ) (cid:88) k v k (1) E ri = (cid:15) ( n ( i ) − n (0)) (2) Results
In general, the existence of intrinsic noise in a system of mobile interacting particles can drivethe departure from a collectively moving ordered swarm state to a completely disordered gas-like state state [19]. In our system, we consider a different source of disorder, environmentaltopological disorder, which is controlled by the parameter p , in addition to thermal noise. Fo-cusing first on the thermal noise dependence of our model without the introduction of any envi-ronmental disorder ( p = 1 ) (Fig. 1B), we find that there exists a order-disorder phase transition3ear a critical magnitude of thermal noise T c Fig. 1C. This transition is reflected in the sharpdrop of the global alignment order parameter (cid:104) v (cid:105) (Eq. 3) at T c . Here (cid:104) v (cid:105) is the mean velocity ofthe N p particles in our system and can range from unity for a perfectly ordered swarm movingcollectively along a single direction to zero when particle motions are completely uncorrelated. (cid:104) v (cid:105) = 1 N p (cid:12)(cid:12)(cid:12)(cid:12) N p (cid:88) i v i (cid:12)(cid:12)(cid:12)(cid:12) , where | v i | = 1 (3)The phase transition associated with this order parameter is in good agreement with previousstudies of the classic Vicsek model [8, 9]. To understand the effects of topological disorder onthe formation of a collective swarm we fix the magnitude of thermal noise below the criticalvalue, T < T c . In doing so we hold the system in a regime where swarming would naturallyoccur without environmental disorder. First we consider the case where particles only interactvia a local alignment interaction (Eqn. 1). In Fig. 2A we see the effect of environmental disorderon the formation of a collective swarm as characterized by the order parameter Eqn. 3, averagedover multiple disordered lattice realizations, for a system without repulsive interactions. At lowdensities of particles ( ρ ≤ . ), consistent with earlier studies [9], we find that there existsno collectively moving swarm ( (cid:104) v (cid:105) < ) that involves all particles in our system even for aperfectly ordered lattice. At higher densities, for low values of the disorder fraction, we finda completely ordered swarming state that exhibits a phase transition beyond some fraction ofmissing bonds ( − p ∗ ) in our system. This order-disorder transition is completely determinedby the amount of topological disorder, which is unrelated to the classic Vicsek model transitionthat is induced by thermal noise[8, 9, 19, 20]. To understand if this transition uniquely definesa critical disorder fraction in the thermodynamic sense, we characterized the location of thecritical disorder fraction ( − p ∗ ( L ) ) for finite systems as a function of the system size. Finite sizescaling analysis revealed (Fig. 2B) that, in the large system size limit ( L → ∞ ), the existenceof a critical phase transition that is governed by environmental disorder is fully suppressed, ( − ∗ ( ∞ ) = 0 ). This result suggests that, in the absence of repulsive forces between particles, anyamount of environmental disorder destroys the ability for a system of particles to collectivelyswarm. In contrast to this behavior, we now consider the effect of adding a mutually repulsiveinteraction between neighboring particles (Eqn. 2). In Fig. 2C, we see that the ability of particlesto form a collectively moving swarm is significantly enhanced compared to Fig. 2A even formoderate values of the disorder fraction. At first glance it seems that the effect of addinga repulsive potential between particles has changed the location of where the order-disordertransition ( − p ∗ ( L ) ) takes place along the disorder fraction axis for finite systems. Again,using finite size scaling, we found that there exists a true thermodynamic phase transition froman ordered swarm to a disordered one (Fig. 2B) that occurs at a finite value of the topologicaldisorder fraction. For a specific value of the repulsive interaction, (cid:15) = 10 − , and thermal noise T = 10 − , this turns out to be − p ∗ ( ∞ ) = 0 . . Furthermore, the location of the criticaldisorder fraction moves as the repulsive interaction is changed as shown in Fig. 2D. At a fixedvalue of thermal noise, the location of the critical phase transition is governed by ratio of thetwo interaction energies, (cid:15)/α ( α = 1 ), but the critical scaling behavior near the critical pointmay in fact be universal. The universality class is usually determined by the values of the criticalexponents that govern the scaling of various physical quantities near the critical point. We firstextended our finite size scaling analysis to compute the critical exponents associated with thistransition for a fixed value of (cid:15) = 10 − . The correlation length ξ , which measures the lengthscale over which particle motion is correlated across the disordered lattice, diverges becomingcomparable to the system size ( ξ ∼ L → ( p ∗ ( L ) − p ) ∼ L − /ν ) as we approach the criticaldisorder fraction transition. This allows us to compute the associated exponent ν . From the fitto the scaling law in Fig. 2B, we find that ν = 1 . . We also measured the susceptibilityof the order parameter to the lattice disorder fraction by measuring the peak value of the orderparameter fluctuations as a function of system size [20], χ v = N σ v ( p ∗ ) ∼ L γ/ν . This provided5n estimate for the susceptibility critical exponent, γ = 2 . (Fig. S1A).We also performed an independent measurement of the susceptibility exponent by examininghow the fluctuations in the order parameter, χ = L σ v ( p ) ∼ ( p ∗ ( L ) − p ) − γ , scale with disor-der fraction − p near the critical point for different values of the repulsion magnitude. Wefound that for over three decades of the repulsive interaction magnitude that the estimate for thecritical exponent given by Fig. S1A agreed very well with estimates given by the scaling near p ∗ ( L ) (Fig. S1B), suggesting universal behavior with respect to repulsive strength. Scaling ofthe the order parameter, (cid:104) v (cid:105) ∼ ( p ∗ ( L ) − p ) β with respect to p , revealed the scaling exponent β (Fig. S1C), which was also insensitive to the value of the repulsive interaction. Using the hyperscaling relation β + γ = dν , we were able to infer the scaling exponent ν (Fig. S1D) over thesame repulsive interaction range and found a reasonable comparison with our estimates from fi-nite system size scaling (Fig. 2C). To gain perspective on which universality class best describesswarming in disorder, we compared our exponents to those of two dimensional percolation [21]and the standard Vicsek model [8, 20]. We found that our system follows a universality classthat is closer to that of percolation than the Vicsek type as shown in Fig. S1B-D). It is inter-esting to note that this system is more akin to the percolation type universality class given thatthe critical disorder fraction ( − p ∗ ( ∞ ) = 0 . ) for a repulsive strength of (cid:15) = 10 − isvastly different than that of ordinary connectivity percolation ( − p c ∼ . ) for a triangularlattice [22]. This comparison suggests that swarming phenomena in such systems are extremelysensitive to the effects of ordinary percolation far from the percolation critical point while re-taining the critical behavior of a percolation type system. It is also relevant to note that thisdisorder induced phase transition occurs in a self-propelled collective swarm rather than beinga field driven transition as is common in condensed matter systems.Our results show that a finite amount of repulsion enhances swarming, suggesting that theremight be an optimal degree of repulsion depending on the system parameters. In Fig. 3A we6xamine the effect that repulsion has on the order parameter in the presence of different amountsof disorder. In general, for finite disorder, we found that repulsion has a non-monotonic effecton the ability for particles to form collectively moving swarms. When the magnitude of therepulsion is small, we find that there is no significant enhancement of swarming ability abovewhat is seen for systems without repulsion. As the repulsion is increased, we find that there isa maximal enhancement of the order parameter, represented by the peaks in Fig. 3A. Beyondthis maximum, we find that, for strong values of repulsion, there is a suppression of collec-tive behavior, indicating the existence of an optimal repulsive interaction for a given disorderfraction.To gain insight into this non-monotonic behavior, we look at the instantaneous disorder averagedmobility fraction, which measures the fraction of particles that are not temporarily stuck at adefect, for various values of − p (Fig 3B). Increasing the repulsion magnitude, we find thatthe mobility fraction also increases for a fixed disorder fraction. Visually we can confirm thatparticle motion is becoming less hindered by the existence of lattice defects as shown in thesimulation snapshots Fig. S3B and D when compared to Fig. S3A and C. This result is alsoconsistent with the increase in the critical disorder fraction as (cid:15) is increased at fixed thermalnoise (Fig. 2D). As we approach the maximal repulsive magnitude, we find that the mobilityfraction saturates at fixed disorder fraction (Fig. 3B) and the critical disorder fraction saturatesas well (Fig. 2D). Increasing the repulsive interaction past the optimum value ( (cid:15) ∼ ), wefind that the mobility fraction remains maximal for − p (cid:46) . (Fig. 3B), while the orderparameter in this regime is greatly reduced (Fig. 3A) and there exists no collective state evenfor a disorder fraction equal to zero. It is interesting to point out for a large disorder fraction( − p (cid:38) . ), there is a slight dip in the mobility fraction for repulsion near (cid:15) (cid:38) − . This isdue to the geometric percolation transition for a triangular lattice, which has a critical disorderfraction of − p c = 1 − π/ (cid:39) . [22]. Near this transition, we find that the particles7ecome trapped in local clusters in the lattice and have a slightly higher mobility fraction for − p > − p c . At large values of repulsive strength ( (cid:15) ∼ ) the mobile fraction is greatlyreduced at even small values of the disorder fraction, because particles are forming a gas-likestate that interacts with defects more frequently such that the mobility scales as the number ofmissing bonds in the lattice (cid:104) µ (cid:105) ∼ − p for p (cid:46) as opposed to (cid:104) µ (cid:105) ∼ for − < (cid:15) < .To summarize our findings, we present the phase diagram for fixed thermal noise ( T < T c )defined by the magnitude of the order parameter (Eqn. 3) in Fig. 3C. In general, we find thatfor both large values of the disorder fraction and the repulsion magnitude that swarming issuppressed. Moreover, for a fixed value of the disorder fraction we see that scanning through therepulsion magnitude (Fig. 3C vertical axis) from low to high takes the order parameter througha maximum, as in Fig. 3D. At larger values of thermal noise, we find that the broad enhancedregion is washed away (Fig. S2C). It is interesting to note that there is a weak enhancementeffect from thermal noise, which can be seen in Fig. S2A-C. We find that thermal noise ( T < T c )allows the system to anneal in the presence of topological disorder, which pushes out the order-disorder transition (even in the absence of repulsive interactions(Fig. S3B) to slightly largervalues of the disorder fraction (Fig.2D). Here, due to thermal noise, particles are susceptibleto random fluctuations in their ability to align with their neighbors thus allowing the system tobehave more ergodically. Conclusion
Navigating an intrinsically disordered environment without a priori knowledge of the topologyof the environment possess a significant challenge for individual organisms. The ability to forma collective intelligence in the form of a swarm can greatly reduce the navigational complexityin a disordered environment. A swarm utilizes physical interactions between individuals as ameans of a primitive collective perception in order to overcome environmental obstacles. The8lasmodium of
Phyarum polycephalum when placed in a two dimensional maze uses a col-lective cellular oscillations to find the shortest path between food sources [23, 24]. Microbialcolonies navigate tiny interconnected channels formed by roots of plants [14]. How do thesereal systems deal with the disorder? How do they communicate local information about theirenvironment to their neighbors? Topological disorder is also intrinsic in social networks, theinternet and scientific citation networks [25]. One may ask the question what is the critical localdegree of connectivity that will allow for the entire population to act holistically [26, 27, 25]?In robotics it is becoming more common to utilize robotic drones to explore dangerous envi-ronments in place of humans. What are the necessary interactions that groups of robotic dronesmust possess in order to perform tasks such as search and rescue in an environment where thelocal topology may not be known a priori [28]? Systems of self propelled particles using localnearest neighbor alignment individual particles can form a collectively moving swarm [8, 9].However, in the presence of environmental topological disorder alignment alone is insufficientfor particles to form a swarm. In order for these systems to navigate disorder they must possessinformation not only from local neighbors directions of travel but also the local surrounding en-vironment. Repulsive forces allow particles to communicate local topological features to theirneighbors and restores the ability of the individuals to form a swarm that collectively navigatesthe intrinsically disorder environment. We have shown there exists a new type of dynamicalphase transition driven by environmental disorder and that the ability for agents to collectivelymove in these disordered environments requires the presence of local repulsive forces betweencollectively moving particles. Collective motion in these disordered environments can be opti-mized by tuning the magnitude of the repulsive interaction for a given amount of disorder. Innature there may exist evolutionary pressures that select for better swarming ability within agroup of individuals. It is interesting to speculate whether organisms that routinely deal withdisordered environments have evolved mechanisms that effectively mediate repulsive interac-9ions similar to the one we have studied here.
Supplementary Information
Model-Monte Carlo
In this section we detail our simulation methods. Lattice configurations were generated ran-domly for a given value of the disorder fraction − p and system size N = L . During anyparticular realization of the simulation, the bond topology was kept fixed. Particle updates werecarried out at each time step by selecting a particle at random. The particle’s direction of mo-tion along one of the six lattice directions ( i = [1 , ) was updated by computing the probabilityto move P i (Eqn.S 1) in a given direction, u i using the local alignment field (Eqn. 1) and re-pulsive particle-particle number density (Eqn. 2). In general each of lattice sites has a number n ( j ) ≥ particles which maybe be overlapped and can point in any of the six lattice directions.Topological disorder is coded into the update probability, P i by the condition that when a bondexists between two neighboring lattice sites then η i = 1 and conversely when a bond is missing η i = 0 . Trial moves were selected using a standard METROPOLIS Monte Carlo method whereparticle moves were accepted/rejected by comparing a randomly generated number ≤ r ≤ to a mapping of the move probabilities (Eqn. 1) to the interval [0,1]. Lattice directions that havea missing bond are assigned a move probability equal to zero. If a particle encounters a brokenbond it will wait there until the next move update. The total number of particle updates to reachsteady state are system size dependent and vary from for N sw = 16 to for N sw = 128 .All averaged quantities are disorder averaged over different realizations of the lattice networkat fixed values of − p . P i ≡ P ( u i ) = 1 Z η i (cid:89) j =1 exp (cid:20) − β (cid:18) − αη j u i · f ( j ) (cid:19)(cid:21) × exp( − βE ri ) , (1)10here Z = (cid:80) n =1 P n . Model-Discussion
To understanding how topological disorder affects the ability for particles to form a swarm wecouple the disorder directly to the dynamics of the particles local information. This is achievedby allowing particles to only communicate with their n.n when a bond exists between them,determined by the conditional η i . Treating disorder this way, missing bonds are representedas infinite barriers that forbid local information from influencing particle motion as well asinhibiting motion along that bond. When − p > the available amount of local informationis limited as there is an average coordination number ( ¯ z = pz , z = 6 for a triangular lattice)available to each particle. In Fig. 1A we see an example of how a missing bond affects thenext possible move that a particle can make. Consider the center particle in Fig. 1A and forthe moment we will set (cid:15) to zero. If all lattice bonds were present (i.e all bonds are black) theparticle would determine that the most probable direction would be to continue on along itscurrent direction given by the grey horizontal arrow (along the u direction). In the case whenthe one bond is missing (grey) the particle has limited information about the local alignmentfield and will most likely choose to move along the white arrow, which is along the u direction.Furthermore, particles can passively run into a dead end where a bond is missing from thelattice. Once this happens motion ceases for that particular particle until a new direction iscomputed using Eqn. 1 that takes the particle away from the broken bonds. For this reasonbroken bond directions are not allowed to be chosen by the Monte Carlo to avoid particlesbecoming permanently stuck at a broken bond.11 inite Size Scaling To provide estimates for critical exponents we measured the system order parameter, (cid:104) v (cid:105) andthe fluctuations of the order parameter, σ v , for various system sizes at a fixed density, ρ = 1 .Using the finite size scaling functions below we extracted both the critical disorder fractionsthreshold and critical exponents (Fig. 2B). To locate the critical disorder fraction in Fig. 2Bwe use the fact that the correlation length will scale as the system size near the critical disorderfraction, ξ L , p ∗ ( L ) − p ∗ ( ∞ ) ∼ L − λ (2)where λ = 1 /ν .The susceptibility is related to the fluctuations of the order parameter near the critical disorderfraction, χ v = Nk B T (cid:2) (cid:104) v (cid:105) − (cid:104) v (cid:105) (cid:3) = L σ v . (3)The peak of the susceptibility also scales as a function of system size (Fig. S1A) , χ v ( p ∗ ( L ) , L ) ∼ L γ/ν . (4)To check the universality of these exponents over the range of repulsion magnitudes we fit boththe order parameter and the susceptibility near the critical disorder fraction, p ∗ ( L ) (Fig. S1Band C). (cid:104) v (cid:105) ∼ ( p − p ∗ ( L )) β (5)and χ v ∼ ( p − p ∗ ( L )) − γ ∼ L σ v . (6)Using these estimates we used the hyper scaling relation, β + γ = dν (7)to compute ν when varying the repulsive interaction(Fig. S1D).12 eferences [1] T. Vicsek and A. Zaferis, Physics Reports , 71 (2012).[2] V. Schaller et al. , Nature Letters , 73 (2010).[3] H. Zhang, A. Be’er, E. Florin, and H. L. Swinney, PNAS , 13626 (2010).[4] S. Mishra, A. Baskaran, and M. C. Marchetti, PRE , 061916 (2010).[5] Y. Katz et al. , PNAS , 18720 (2011).[6] C. C. Ioannou and I. D. Couzin, Science , 1212 (2012).[7] J. L. Silverberg, M. Bierbaum, J. P. Sethna, and I. Cohen, ArXiv arXiv:1302.1886 , 4(2013).[8] T. Vicsek et al. , PRL. , 1226 (1995).[9] Z. Csah´ok and T. Vicsek, PRE. , 5297 (1995).[10] A. Czir´ok, A.-L. Barab´asi, and T. Vicsek, PRL , 209 (1999).[11] A. Czir´ok, M. Vicsek, and T. Vicsek, Physica A , 299 (1999).[12] M. R. D’Orsogna, Y. L. Chuang, A. L. Bertozzi, and L. S. Chayes, PRL , 104302 (2006).[13] C. A. Yates et al. , PNAS , 5464 (2009).[14] F. Per´ez-Reche, W. O. S.N. Taraskin, L. d. F. C. M.P. Viana, and C. Gilligan, PRL. ,098102 (2012).[15] D. Helbing, I. Farkas, and T. Vicsek, Nature Letters , 487 (2000).[16] D. Helbing, Review of Modern Physics , 1068 (2001).[17] C. A. Weber, V. Schaller, A. R. Bausch, and E. Frey, PRE , 030901 (2012).[18] D. ben Avraham and S. Havlin, Diffusion and reactions in fractals and disordered systems ,1st ed ed. (Cambridge University Press, ADDRESS, 2000).[19] M. Aldana et al. , PRL , 095702 (2006).[20] G. Baglietto and E. V. Albano, PRE , 021125 (2008).[21] H. Kesten, Communications in Mathematical Physics , 109 (1987).[22] M. F. Sykes and J. W. Essam, Journal of Mathematical Physics , 1117 (1964).1323] T. Nakagaki, H. Yamada, and A. T ´oth, Nature: Brief Communications , 470 (2000).[24] T. Nakagaki, H. Yamada, and A. T ´oth, Biophysical Chemistry , 47 (2001).[25] R. Albert and A.-L. Baraba´ai, Reviews of modern physics , 988 (2002).[26] D. J. Watts and S. H. Strogatz, Nature Letters , 440 (1998).[27] D. J. Watts, Small Worlds: The Dynamics of Networks between Order and Randomness ,1st ed ed. (Princeton University Press, ADDRESS, 1999).[28] A. Jadbabaie, J. Lin, and A. S. Morse, IEEE Transactions on Automatic Control , 988(2003). 14 i v k A BC ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p h v i E r i = ✏ ( n ( i ) n ( )) . ( ) P i ⌘ P ( u i ) = Y j = e x p ✓ ↵ u i · f ( j ) ◆ ⇥ e x p ( E r i ) , ( ) E i = X j = ✓ ↵ u i · f ( j ) ◆ E r i , ( ) A li g n m e n t ( ) ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p h v i log( T ) T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . Noise( ) − − − C D
Figure 1: ( A ) Diagram of a lattice unit cell with particles (red) occupying lattice sites. Solidblack bars are occupied lattice bonds and grey bars are deleted bonds. The blue triangles rep-resent current particle velocity directions. White/Grey arrows indicate some potential latticedirections along which the center particle can move at the next time step. In the absence ofany repulsion, if the deleted bond (grey bond) were still present, the updated direction of thecenter particle would most probably be along the grey arrow, since that is the local averagevelocity direction. However, since the bond is not present, the most probable direction is alongthe white arrow. Note that the absence of the bond not only disallows motion in that directionbut also inhibits transfer of information. (see supplementary information)( B ) A snapshot of atypical simulation showing finite sized groups of particles which are collectively moving in adisorder free ( − p = 0 ) lattice. ( C ) Order-disorder transition driven by the thermal noise ina system of particles ( ρ = 1 . and − p = 0 ) without repulsive interactions for a system sizeof N = 1024 . The blue dash line indicates the location of the critical transition temperature, T c (cid:39) . , which was determined numerically. ( D ) A snapshot of a simulation of a system withdisorder ( − p = 0 . ), where missing bonds are not drawn. Black filled circles are particlesthat are temporarily stuck at a lattice bond defect.15 − − L log( L ) p ⇤ ( L ) p ⇤ ( ) ⇠ L = 0 . > = 0 . ⌫ = 1 . ⌫ = 1 . ✏T c ' . T )log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T ) ✏ = 0 . ✏ = 10 L log( L ) p ⇤ ( L ) p ⇤ ( ) ⇠ L = 0 . > = 0 . ⌫ = 1 . ⌫ = 1 . ✏T c ' . T )log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T ) ✏ = 0 . ✏ = 10 ✏ = 10 ✏ = 2 . · ✏ = 7 . · ✏ = 10 ✏ = 10 ✏ = 10 h µ i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 1 . ⇢ = 2 . ⇢ = 4 . E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)3 ✏ = 10 ✏ = 2 . · ✏ = 7 . · ✏ = 10 ✏ = 10 ✏ = 10 h µ i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 1 . ⇢ = 2 . ⇢ = 4 . E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)3 L log( L ) p ⇤ ( L ) p ⇤ ( ) ⇠ L = 0 . > = 0 . ⌫ = 1 . ⌫ = 1 . ✏T c ' . T )log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T ) ✏ = 0 . ✏ = 10 ✏ = 10 ✏ = 2 . · ✏ = 7 . · ✏ = 10 ✏ = 10 ✏ = 10 h µ i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 1 . ⇢ = 2 . ⇢ = 4 . E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)3 ✏ = 10 ✏ = 2 . ⇥ ✏ = 7 . ⇥ ✏ = 10 ✏ = 10 ✏ = 10 h µ i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 1 . ⇢ = 2 . ⇢ = 4 . E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)3 ✏ = 10 ✏ = 2 . · ✏ = 7 . · ✏ = 10 ✏ = 10 ✏ = 10 h µ i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 1 . ⇢ = 2 . ⇢ = 4 . E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)3 ✏ = 10 ✏ = 2 . · ✏ = 7 . · ✏ = 10 ✏ = 10 ✏ = 10 h µ i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 1 . ⇢ = 2 . ⇢ = 4 . E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)3 ✏ = 10 ✏ = 2 . ⇥ ✏ = 7 . ⇥ ✏ = 10 ✏ = 10 ✏ = 10 h µ i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 1 . ⇢ = 2 . ⇢ = 4 . E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)3 ✏ = 10 ✏ = 2 . · ✏ = 7 . · ✏ = 10 ✏ = 10 ✏ = 10 h µ i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 1 . ⇢ = 2 . ⇢ = 4 . E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)3 L l og ( L ) p ⇤ ( L ) p ⇤ ( ) ⇠ L = . ( ) > = . ( ) ⌫ = . ( ) ⌫ = . ✏ T c ' . l og ( T ) l og ( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i l og ( T ) ✏ = . ✏ = ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p h v i E r i = ✏ ( n ( i ) n ( )) . ( ) P i ⌘ P ( u i ) = Y j = e x p ✓ ↵ u i · f ( j ) ◆ ⇥ e x p ( E r i ) , ( ) E i = X j = ✓ ↵ u i · f ( j ) ◆ E r i , ( ) A li g n m e n t ( ) ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p h v i E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)1 Disorder Fraction ( ) ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p ⇤ ( L ) p ⇤ ( ) p ⇤ > ( ) ✏ T c ' . l og ( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p ⇤ ( L )1 p ⇤ ( ) = 01 p ⇤ > ( ) = 0 . p ⇤ ( L ) p ⇤ ( ) ⇠ L > = 0 . = 0 . ✏T c ' . ✏ )1 A ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p h v i E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)1 Disorder Fraction ( ) ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p h v i E r i = ✏ ( n ( i ) n ( )) . ( ) P i ⌘ P ( u i ) = Y j = e x p ✓ ↵ u i · f ( j ) ◆ ⇥ e x p ( E r i ) , ( ) E i = X j = ✓ ↵ u i · f ( j ) ◆ E r i , ( ) A li g n m e n t ( ) BC D
Figure 2: ( A ) Global alignment, in the absence of repulsion, as measured by Eqn. 3 as a func-tion of the disorder fraction in the lattice for different densities [ ρ = 0 . (black), ρ = 0 . (red), ρ = 1 . (blue), ρ = 2 . (green) ]. ( B ) Finite system size critical disorder fraction ( − p ∗ ( L ) ) forsystems without repulsion [ (cid:15) = 0 (green filled circles)] and with repulsive interactions [ (cid:15) > (blue filled circles)]. Fitting to finite size scaling, ( p ∗ ( L ) − p ∗ ( ∞ )) ∼ L − /ν , predicted a criticaldisorder fraction for systems without repulsive interactions (black dashed line) that is zero andfor systems with repulsion (red dashed line) is found to be − p ∗ ( ∞ ) = 0 . (black solidline). The critical scaling exponent ν in the case for repulsive interactions is . . ( C ) Globalalignment for a system of particles that interact with both a local alignment and repulsive fieldsfor ρ = 1 . and (cid:15) = [5 . ∗ − (black), − (red), . ∗ − (blue), − (green)]. The lo-cation of the order-disorder transition is pushed out allowing particles to swarm in the presenceof moderate environmental disorder. ( D ) The location of the critical disorder as a function oftemperature at various values of the repulsive interaction strength (see inset). For low valuesof repulsion the critical disorder changes very little over temperature. At intermediate repul-sion values we see there is an enhancement effect with increasing thermal noise. At repulsivestrengths near maximal (Fig. 1D) we find little change of the critical disorder fractions overentire range in T . All data shown in (A), (C) and (D) is for a system size N = 1024 . All data in(A), (B) and (C) presented here was obtained at a thermal noise of T = 10 − e pu l s i o n ( ) ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p ✏ l og ( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i l og ( T ) R e pu l s i o n ( ) ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p ✏ l og ( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i l og ( T ) R e pu l s i o n ( ) ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p ✏ l og ( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i l og ( T ) R e pu l s i o n ( ) ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p ✏ l og ( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i l og ( T ) R e pu l s i o n ( ) ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p ✏ l og ( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i l og ( T ) R e pu l s i o n ( ) ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p ✏ l og ( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i l og ( T ) R e pu l s i o n ( ) ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p ✏ l og ( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i l og ( T ) C Disorder ( ) ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p✏ log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T )1 R e pu l s i o n ( ) ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p ✏ l og ( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i l og ( T ) h µ i T = T = T = T = ⇢ = . ⇢ = . ⇢ = . ⇢ = . ⇢ = . ⇢ = . ⇢ = . E r i = ✏ ( n ( i ) n ( )) . ( ) P i ⌘ P ( u i ) = Y j = e x p ✓ ↵ u i · f ( j ) ◆ ⇥ e x p ( E r i ) , ( ) E i = X j = ✓ ↵ u i · f ( j ) ◆ E r i , ( ) M o b ili t y ( ) Disorder ( ) ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p✏ log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T )1 B L log( L ) p ⇤ ( L ) p ⇤ ( ) ⇠ L = 0 . > = 0 . ⌫ = 1 . ⌫ = 1 . ✏T c ' . T )log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T ) ✏ = 0 . ✏ = 10 L log( L ) p ⇤ ( L ) p ⇤ ( ) ⇠ L = 0 . > = 0 . ⌫ = 1 . ⌫ = 1 . ✏T c ' . T )log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T ) ✏ = 0 . ✏ = 10 ✏ = 10 ✏ = 2 . · ✏ = 7 . · ✏ = 10 ✏ = 10 ✏ = 10 h µ i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 1 . ⇢ = 2 . ⇢ = 4 . E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)3 ✏ = 10 ✏ = 2 . · ✏ = 7 . · ✏ = 10 ✏ = 10 ✏ = 10 h µ i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 1 . ⇢ = 2 . ⇢ = 4 . E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)3 ✏ = 10 ✏ = 2 . · ✏ = 7 . · ✏ = 10 ✏ = 10 ✏ = 10 h µ i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 1 . ⇢ = 2 . ⇢ = 4 . E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)3 ✏ = 10 ✏ = 2 . · ✏ = 7 . · ✏ = 10 ✏ = 10 ✏ = 10 h µ i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ⇢ = 1 . ⇢ = 2 . ⇢ = 4 . E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)3 − − − ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p h v i E r i = ✏ ( n ( i ) n ( )) . ( ) P i ⌘ P ( u i ) = Y j = e x p ✓ ↵ u i · f ( j ) ◆ ⇥ e x p ( E r i ) , ( ) E i = X j = ✓ ↵ u i · f ( j ) ◆ E r i , ( ) A li g n m e n t ( ) Repulsion ( ) ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p✏ log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T )1 A Figure 3: ( A ) Behavior of the order parameter for particular values of the disorder fraction[ p = 1 . (red), . (blue), . (green)] clearly shows a non-monotonic dependence on themagnitude of the order parameter (log horizontal axis) with a central maximum for curveswith p < . ( B )The ensemble averaged instantaneous mobility of particles as function ofenvironmental disorder. For smaller values of the disorder fraction, the effect of increasingthe repulsive energy significantly increases particle mobility allowing particles to swarm. Asthe disorder fraction is increased, the magnitude of the mobility generally decreases as expected.Interestingly, for values of the disorder fraction near the connectivity bond percolation threshold( − p c (cid:39) . ) we find a significant dip in the mobility of particles at large values of therepulsive energy parameter ( (cid:15) (cid:38) ). This signifies that particles are becoming trapped inlocal disconnected clusters in the lattice. ( C ) Phase diagram defined by the magnitude (colorbar) of the order parameter (Eqn. 3) for the two parameters (cid:15) and − p at T = 10 − . Thelarge peak centered near log( (cid:15) ) ∼ − shows the effect of repulsion has on the persistence ofswarming the presence of disorder. 17 ⇤ ( L ) p ⇤ ( ) ⇠ L > = 0 . = 0 . ⌫ > = 1 . ⌫ = 1 . ✏T c ' . ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T ) ✏ = 0 . ✏ = 10 ✏ = 10 ✏ = 10 ✏ = 10 ✏ = 10 L log( L ) p ⇤ ( L ) p ⇤ ( ) ⇠ L = 0 . > = 0 . ⌫ = 1 . ⌫ = 1 . ✏T c ' . T )log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T ) ✏ = 0 . ✏ = 10 DA ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p ⇤ ( L ) p ⇤ ( ) = p ⇤ > ( ) = . ( ) l og ( v ( p ⇤ ( L ) , L )) l og ( v ) ⌫ + = . = . ( ) > ⌫ + = . = . ( ) p ⇤ ( L ) p ⇤ ( ) ⇠ L > = 0 . = 0 . ⌫ > = 1 . ⌫ = 1 . ✏T c ' . ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T ) ✏ = 0 . ✏ = 10 ✏ = 10 ✏ = 10 ✏ = 10 ✏ = 10 B ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p ⇤ ( L )1 p ⇤ ( ) = 01 p ⇤ > ( ) = 0 . v ( p ⇤ ( L ) , L ))log( v ) ⌫ + 1 = 2 . = 3 . > ⌫ + 1 = 2 . = 2 . p ⇤ ( L ) p ⇤ ( ) ⇠ L > = 0 . = 0 . ⌫ > = 1 . ⌫ = 1 . ✏T c ' . ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T ) ✏ = 0 . ✏ = 10 ✏ = 10 ✏ = 10 ✏ = 10 ✏ = 10 C ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p ⇤ ( L )1 p ⇤ ( ) = 01 p ⇤ > ( ) = 0 . v ( p ⇤ ( L ) , L ))log( v ) ⌫ + 1 = 2 . = 3 . > ⌫ + 1 = 2 . = 2 . − − − − − γ ex γ avg γ perc γ VM − − − − − β ex β avg β perc β VM − − − − − ν ex ν avg ν perc ν VM log( L ) p ⇤ ( L ) p ⇤ ( ) ⇠ L = 0 . > = 0 . ⌫ = 1 . ⌫ = 1 . ✏T c ' . T )log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T ) ✏ = 0 . ✏ = 10 ✏ = 10 ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p ⇤ ( L )1 p ⇤ ( ) = 01 p ⇤ > ( ) = 0 . v ( p ⇤ ( L ) , L ))log( v ) ⌫ + 1 = 2 . = 3 . > ⌫ + 1 = 2 . = 2 . Figure S1: ( A ) Finite size scaling fit (red dashed line) of the susceptibility (blue filled circles)at finite repulsion ( (cid:15) = 10 − ) as a function of system size. The critical exponent, γ , associ-ated with the scaling of the susceptibility is shown. ( B ) Extracted critical exponents with errorbars for γ (blue circles and black bars) from scaling analysis near the critical disorder fractiontransition ( − p ∗ ( L ) ) plotted against the log of the repulsion magnitude. The grey dashed lineis the average of the these values while the red line is the predicted exponent for percolationand the green line is the predicted exponent for the standard Vicsek model. ( C ) Filled circlesare the extracted exponents, β for the scaling of the order parameter near the critical disorderfraction [see supplementary material]. The dashed lines have the same color arrangement as theprevious figure in the series. ( D ) Filled circles are the estimates for the critical exponent, ν , as-sociated with the correlation length, plotted over the magnitude of the repulsive interaction [seesupplementary material]. Dashed lines indicate the averaged exponent and expected exponentsfrom percolation and the standard Vicsek model. The color arrangement is the same as in theprevious figures in this series. 18 − − ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p h v i E r i = ✏ ( n ( i ) n ( )) . ( ) P i ⌘ P ( u i ) = Y j = e x p ✓ ↵ u i · f ( j ) ◆ ⇥ e x p ( E r i ) , ( ) E i = X j = ✓ ↵ u i · f ( j ) ◆ E r i , ( ) A li g n m e n t ( ) Repulsion ( ) ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p✏ log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T )1 ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p✏ log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T )1 ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p✏ log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T )1 ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p✏ log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T )1 ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p h v i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p h v i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p h v i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p h v i T = 10 T = 10 T = 10 T = 10 ⇢ = 0 . ⇢ = 0 . ⇢ = 0 . ✏ b i = C ⇤ [ ( k c o s ( ¯ i , i + i ) k i ) + ( k s i n ( ¯ i , i + i ) k i ) ] ( ) ✏ t i = K ⇤ [ ( i , i ± s i n ( ✓ i , i + ) R ! ) ] ( ) ✏ b i n d i = V s i n ( i )( ) C ' [ . . ] p N ⇤ n m d e g ! ' . d e g n m k i ' [ . . ] n m K i s a f r ee p a r a m e t e r R ' [ ] n m p ⇤ ⇠ . = k B T ¯ z z = p p h v i E r i = ✏ ( n ( i ) n ( )) . ( ) P i ⌘ P ( u i ) = Y j = e x p ✓ ↵ u i · f ( j ) ◆ ⇥ e x p ( E r i ) , ( ) E i = X j = ✓ ↵ u i · f ( j ) ◆ E r i , ( ) A li g n m e n t ( ) ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p h v i E ri = ✏ ( n ( i ) n (0)) . (4) P i ⌘ P ( u i ) = Y j =1 exp ✓ ↵ u i · f ( j ) ◆ ⇥ exp( E ri ) , (5) E i = X j =1 ✓ ↵ u i · f ( j ) ◆ E ri , (6)1 Disorder Fraction ( )
A BC
Disorder ( ) ✏ bi = C ⇤ [( k cos( ¯ i,i +1 i ) k i ) + ( k sin( ¯ i,i +1 i ) k i ) ] (1) ✏ ti = K ⇤ [( i,i ± sin(2 ¯ ✓ i,i +1 )2 R ! ) ] (2) ✏ bindi = V sin ( i ) (3) C ' [0 . . pN ⇤ nm deg ! ' . degnmk i ' [0 . . nm K is a free parameter R ' [10 ] nmp ⇤ ⇠ . = k B T ¯ zz = p p✏ log( ✏ ) p ⇤ p > p ⇤ p ⇠ p ⇤ p < p ⇤ h v i log( T )1 Figure S2: ( A ) Increasing thermal noise, for system without repulsion, but below the critical T c value there is an annealing effect that pushes the order-disorder transition to slight higher valuesof disorder fraction. ( B )The combination of both the repulsive energy and the temperatures near T c ( T = 10 − ) both help the ability for swarming behavior near the critical bond occupationprobability p ∗ for (cid:15)/α (cid:28) − as compared to Fig. 3A. ( C ) A phase diagram for larger thermalnoise magnitude ( T = 10 − ) near the critical value T c (compare to Fig. 1C). The prominentpeak near log( (cid:15) ) ∼ − has diminished significantly, but there also is a slight thermal noiseeffect which restores swarming for weak values of the repulsion magnitude ( log( (cid:15) ) ∼ − ) andsmall values of the disorder fraction. 19igure S3: Snapshots of a simulations with and without repulsive energy. ( A ) In early simula-tion times without repulsion, particles scatter off locations where bonds are missing and beginto form compact single file lines. ( B ) With repulsion ( (cid:15) = 10 − ) in early times, particles formextended groups that move together around defect bonds and fill in gaps caused by defects. ( C )At late times without repulsion, particles form an extended single line that finds the a defect freelattice direction. Black particles are stuck at a broken bond indefinitely in this case. ( D ) In latetimes with repulsion, we find an collectively moving ordered swarm which can avoid defectsand remain ordered by moving around particles which are temporarily stuck at a broken bond.All snapshots are for disorder fractions of − p = 0 . at a thermal noise of T = 10 −2