Bond disorder enhances the information transfer in polar flock
BBond disorder enhances the information transfer in polar flock
Jay Prakash Singh , ∗ Sameer Kumar , † and Shradha Mishra ‡ , Department of Physics, Indian Institute of Technology (BHU), Varanasi, India 221005 (Dated: February 9, 2021)Collection of self-propelled particles (SPPs) exhibit coherent motion and show true long-rangeorder in two-dimensions. Inhomogeneity, in general destroys the usual long-range order of the polarSPPs. We model a system of polar self-propelled particles with inhomogeneous interaction strengthor bond disorder . The system is studied near the order-to-disorder transition for different strengthsof the disorder. The nature of phase transition changes from discontinuous to continuous typeby tuning the strength of the disorder. The bond disorder also enhances the ordering near thetransition due to the formation of a homogeneous flock state for the large disorder. It leads to fasterinformation transfer in the system and enhances the system’s information entropy. Our study givesa new understanding of the effect of intrinsic inhomogeneity in the self-propelled particle system.
I. INTRODUCTION
Collective behavior of a large number of self-propelledparticles (SPPs) or “flocking” is ubiquitous. Examplesof such systems range from a few micrometers, e.g., actinand tubulin filaments, molecular motors [1, 2], unicellu-lar organisms such as amoebae and bacteria [3], to sev-eral meters, e.g., bird flock [4], fish school [5] and humancrowd [6] etc . Interestingly, these systems show a collec-tive motion on a scale much larger than each individual,and hence long-range ordering (LRO) is observed in two-dimensional.A minimal model for understanding the basic features ofthe collective behavior of polar self-propelled particlesor “polar flock” was introduced in 1995 by T. Vicsek etal. [7]. In the last three decades, many variants of theVicsek model are studied to understand various featuresof different model systems [9–12].In these studies, authors mainly consider a collection ofSPPs in a homogeneous system or medium. Recently,there is a growing interest to understand the effects andadvantages of different kinds of inhomogeneities, whichare omnipresent in nature. Many studies show thatthe inhomogeneity can destroy the LRO present in aclean system [13–21] whereas a few studies discuss spe-cial kinds of inhomogeneities that enhance the orderingin the system [22, 23]. Therefore, the inhomogeneitycan be useful for many practical applications, e.g., crowdcontrol and faster evacuation etc.[24, 27–32].In the Vicsek model, each individual interacts througha short-range alignment interaction and the strength ofthe interaction is the same for all the particles. But innatural systems, each particle can have a different abil-ity to influence its neighbors based on their individualintelligence or physical strength, etc. However, scien-tists have not paid much attention to understand theeffects of different interaction strengths in a polar flock.In a recent study, William et al. show that the varying ∗ [email protected] † [email protected] ‡ [email protected] interaction strength of the SPPs results in maximum en-tropy. Hence, more information transfer among the par-ticles [33][43]. Surprisingly, in this work, we note thatthe presence of inhomogeneity in the form of the parti-cles’ different interaction ability, the system approachesa more homogeneous state near to the point of order-disorder transition. More importantly, the flock’s re-sponse is faster for higher disorder in the interactionstrength among the SPPs because each SPP neighbor isupdated more frequently, which leads to faster informa-tion transfer within the flock. We also calculate the in-formation entropy [44–46] for different disorder and findthat the larger the disorder more is the information en-tropy of the system.We also characterised the effect of bond disorder on thenature disorder-to-order phase transition in the system.Which is a matter of great interest in many previousstudies [10, 11]. We find that the nature of disorder-to-order phase transition changes from discontinuous typeto continuous type by tuning the strength of bond disor-der. Also, the system shows the enhanced ordering nearthe transition point for the larger disorder.The rest of the paper is organized as follows. In Sec.II,we discuss the model and simulation details. In Sec.III,the results from the numerical simulations are discussed.In Sec.IV, we conclude the paper with a summary anddiscussion of the results. II. MODEL
We consider a collection of N polar self-propelled par-ticles (SPPs) moving on a two-dimensional substrate.SPPs interact through a short-range alignment inter-action within a small interaction radius R I [7, 9, 10].Moreover, the strength of interaction of each SPP withits neighbors is different , unlike the Vicsek model [7] ofuniform interaction strength. Each SPP is defined by itsposition r i ( t ) and orientation θ i ( t ), and it moves alongits direction vector n i ( t ) = (cos( θ i ( t )) , sin( θ i ( t ))) with afixed speed v . The two update equations for the posi-tion r i ( t ) and the direction vector n i ( t ) are given by, r i ( t + ∆ t ) = r i ( t ) + v n i ( t )∆ t (1) a r X i v : . [ c ond - m a t . s o f t ] F e b n i ( t + ∆ t ) = (cid:80) j ∈ R I J j n j ( t ) + ηN i ( t ) ξ i ( t ) w i ( t ) (2)The first equation represents the particle’s motion due toits self-propelled nature, along the direction vector n i ( t )with fixed speed v . ∆ t = 1 . i th particle with its neighbors withinthe interaction radius ( R I ), and J j is the interactionstrength of the j th neighbor. The probability distribu-tion of the interaction strength J , P ( J ), is obtained froma uniform distribution of range [1 − (cid:15) : 1 + (cid:15) ] [35], where (cid:15) measures the degree of disorder. (cid:15) = 0 corresponds tothe uniform interaction strength ( J i = 1 for all the parti-cles) like the Vicsek model [7] whereas (cid:15) = 2 correspondsto the maximum disorder in the system.Furthermore, the second term in the Eq.2 denotes thevector noise, which measures the particle’s error whilefollowing its neighbors. ξ i ( t ) is a random unit vector,where N i ( t ) denotes the number of neighbors within theinteraction radius of the i th particle at time t . η repre-sents the strength of the noise and can vary from 0 to1. w i ( t ) is the normalization factor, which reduces theright-hand side of the Eq.2 to a unit vector.For zero self-propulsion speed model reduces to the equi-librium random bond XY -model [25, 26]. However, for (cid:15) = 0, the model reduces to the clean polar flock. Wenumerically update the Eqs.1 and 2 for all SPPs sequen-tially. One simulation step is counted after the update ofEqs.1 and 2 once for all the particles. Periodic boundarycondition (PBC) is used for a system of size L × L , and L is taken as 100 , ρ N = NL × L . We fix the density at ρ N = 1 . v = 0 .
5. Since for the samedensity for the clean polar flock, the critical noise is closeto η ∼ .
6, we limit our study near to the critical point,and the noise strength is varied from η = 0 . . η is fixed 0.62 to charac-terised the properties of polar flock near to critical point.The study deep in the ordered state η < . and 20 independent realizations to study thesteady-state results. III. RESULTSA. Disorder-to-order transition
First, we study the disorder-to-order transition in thesystem for different disorder strengths (cid:15) . Ordering in thesystem is characterized by the mean orientation orderparameter, χ ( t ) = 1 N | N (cid:88) i n i ( t ) | (3)In the ordered state, i.e., when majority of particles aremoving in the same direction, then χ will be closer to 1, χ η V ξ η η (a) (b)(c) (d) P ( χ ) χ χ η FIG. 1. (color online) (a) Plot of the mean orientation orderparameter χ vs. noise strength η , inset : zoomed plot showsenhance ordering on increasing (cid:15) . (b) Variation of susceptibil-ity ξ vs. (cid:15) . (c) The probability distribution function of orderparameter P ( χ ) vs. χ at the transition point ( η c ( (cid:15) ) = 0.625,0.640, 0.654 for (cid:15) = 0 , V vs. η . Different symbols im-plies different values of disorder strength (cid:15) = 0 (circles), 1 . . ρ = 1 . and of the order of √ N for a random disordered state.In Fig.1(a) we have shown the variation of χ ( t ) withthe noise strength η for three different (cid:15) = (0 , , (cid:15) = 0, the variation of χ shows a sharp change from χ ∼ ∼
0. This kind of change is a common feature offirst-order phase transition [10–14]. Whereas for (cid:15) = 2, χ varies continuously, and the transition has a signatureof second-order phase transition. The variation of χ for (cid:15) = 1, it shows the intermediate behaviour. Theplot of order parameter fluctuation or the susceptibility ξ = (cid:112) < χ > − < χ > , is shown in Fig. 1(b), where < . > denotes the average over steady-state time. Thecritical noise η c is obtained from the maximum of ξ .The η c ( (cid:15) ) shifts towards right on increasing (cid:15) = 0 , P ( χ ) vs. χ in Fig.1(c) atthe critical noise η c ( (cid:15) ) = 0 . , . , .
654 for three (cid:15) = 0 , (cid:15) = 0, there is aclear bimodal nature of P ( χ ) gradually changes tounimodal on increasing (cid:15) . To further characterise thenature of the transition for (cid:15) = (0 , ,
2) in Fig.1 (d),we calculate the fourth-order cumulant or the Bindercumulant V = 1 − <χ > <χ > vs. η . We plot V ( η ) versus η in Fig.1(d). It shows strong discontinuity between V = 1 / V = 2 / η c for (cid:15) = 0; however, itsmoothly goes between a disordered state ( V = 1 /
3) toan ordered state ( V = 2 /
3) for (cid:15) = 2. Now we try tounderstand the shift in critical noise strength η c towardshigher values, as well as enhanced ordering near η = 0 . χ ( t ) P ( θ ) θ t(a) (b) FIG. 2. (color online) (a) Time series of χ ( t ) for disorder (cid:15) = 1 . (cid:15) = 2 . ± π are introducedat time t = 10 , P ( θ ) vs. mean orieantation θ , for (cid:15) = 1 . (cid:15) =2 . N = 10 , η = 0 .
62 and the density ρ = 1 . (cid:15) = 0, 1 and 2 respectively. colorbarshows intensity with respect to particle clusters. All the pa-rameters are same as in Fig. 2. for the higher disorder, as shown in the zoomed-in plotin Fig. 1(a).To understand the enhanced ordering mechanism, weperform a small perturbative study on the system.Since we find enhanced ordering near η ∼ .
6, theperturbation is imposed at η = 0 .
62 for finite disorder (cid:15) = 1 and 2.In the perturbative study, the system is waited to reachto the steady-state ( t = 10 ) and once the steady-stateis reached; as shown in Fig. 2(a), we choose 5% of theparticles randomly and out of which the direction of2 .
5% particles with
J > π and another 2 .
5% with
J < − π . Once this perturbation is applied, thesystem will respond to it and mean order parameter χ ( t ) shows a dip and then relaxes to a new steady statewith relatively lower value of χ ( t ) as shown in Fig.2(a). Very clearly before perturbation, χ is lower for (cid:15) = 2, hence a more ordered state for the lower disorder.But after perturbation, which is selectively for particleswith higher and lower J values, the response is differentfor (cid:15) = 1 and 2. For (cid:15) = 2, after perturbation χ islarger compare to (cid:15) = 1. Hence more ordered statefor the larger disorder. In the plot of Fig.2(b) we plotthe orientation probability distribution function (PDF) P ( θ ) of the particles orientation θ . For (cid:15) = 1, the P ( θ )shows two distinct peaks for θ = ± π/
2, but peak for π/ J is more. For (cid:15) = 2, themean of the total P ( θ ) shifts towards the non-zero θ ,hence the system’s response happens globally, and thewhole system is polarised in the direction of quenchedparticles with larger J values.Now we further study the consequence of such enhancedordering for larger disorder on the polar flock. Moreover,this dominated alignment is responsible for shifting oftransition point η c towards higher values. B. Properties of polar flock δφ ε (b)n P ( n ) (b’) (a) P ( n ) P ( n ) ε nn N c (c) (d) FIG. 4. (color online) (a) Plot of density phase separationorder parameter δφ vs. (cid:15) with blue squares. Blue dotted lineshows liner decay of δφ (b) P(n) vs. n for (cid:15) = 0 , . , , . (cid:15) = 0 , . , , . N c vs. (cid:15) whereblue dotted line shows linear decay of N c . (c) and (d) showthe zoom plot of (b) near to the head and tail where tailpart are fitted with exponential function with dotted lines.Symbols with black(circles), red(squares), green(diamonds),blue(triangles up) and magenta(triangles left) colors are for (cid:15) = 0 , . , , . How disorder affects the density fluctuations in the sys-tem? We plot realspace snapshot of the local density(calculated in small region of unit size square sub-shell),in Fig.3(a)-(c) for three values of (cid:15) = (0 , t = 10 . For clean polar flock, (cid:15) = 0, particlesform isolated clusters. Whereas with a non-zero (cid:15) , theseisolated clusters break, and the system gets into a morehomogeneous state. To further confirm this, we calcu-late the density phase separation order parameter, δφ vs. (cid:15) (where δφ ( (cid:15) ) is the deviation of the number ofparticles among the sub-cells), Fig.4(a). We calculate δφ by dividing the whole L × L system into unit sizedsub-cells, δφ ( (cid:15) ) = (cid:113) L (cid:80) L j =1 ( φ j ( (cid:15) )) − ( L (cid:80) L j =1 φ j ( (cid:15) )) where φ j is the number of particles in the j th sub-cell and (cid:104) .... (cid:105) represents averaging over 20 realisations. We notethat δφ decreases in a linear fashion with increasing (cid:15) asshown in Fig. 4(a). Hence system becomes more homo-geneous with increasing the random bond disorder (cid:15) inthe system. Furthermore, in Fig.4(b) we plot the prob-ability distribution function (PDF) of number of neigh-bours P ( n ) for different values of (cid:15) = 0 , . , , . n for higher valueof (cid:15) = 2 shows that the system is approaching towardsmore homogeneous state or clusters of smaller size whilelonger tail, for lower values of (cid:15) = 0 hence bigger clusters.In the inset of Fig.4(b) we plot the mean number of par-ticles N c with (cid:15) where N c is obtained by fitting the tailof the main plot by the exponential function exp( − nN c ).This shows that N c decreases linearly with an increasein the value of (cid:15) . Similarly when zoomed for smaller n asshown in Fig. 4(c), P ( n ) for larger (cid:15) is higher as compareto smaller (cid:15) . Hence small clusters have more probabilityfor larger disorder. Fig. 4 (d) the zoomed tail of the P ( n ). t(a) t/t c t c ε (b) C ( t ) C ( t ) FIG. 5. (color online) (a) Plot of OACF C ( r, t ) vs. t . for (cid:15) =0 . . . . . C ( r, t ) vs. scaled time t/t c ;and t c vs. (cid:15) (inset) where the dashed lines is linear fit to thedata. All other parameters are same as in Fig. 2 C. Accelerated response to external perturbation
We claim that enhanced ordering near-critical region andhomogeneous density clusters promote faster informa-tion transfer among the flock. To confirm the same weperform another perturbation to the well-ordered flock inthe steady state and calculate its response. We randomlyselect a fraction of particles 1% and quench their direc-tion to a randomly selected fixed orientation. With timeall other particles will rotate in that direction. Their re-sponse to the direction of quench is measured by calcu-lating the orientation auto-correlation function (OACF) C ( t ) = (cid:104) cos θ i ( t ) − θ i (0) (cid:105) − (cid:104) cos θ i ( T ) − θ i (0) (cid:105) .Where θ i ( t ) and θ i (0) are the orientation of the i th par-ticle at time t and 0 from the time of quench and T is the late time when approximately all the particles areoriented in the direction of the quench. (cid:104) .... (cid:105) denotesaveraging over all the SPPs and 30 independent realiza-tions.In the Fig.5 (a) OACF C ( t ) decay exponentially andshow the sharper decay with increase in the strengthof disorder (cid:15) . Therefore, the response of the flock toexternal perturbation becomes faster, with the increasein (cid:15) . In Fig. 5(b) we plot the C ( t ) vs. scaled time t/t c , where t c is obtained from the fitting of C ( t ) toexp( − t/t c ). The inset of Fig. 5(b), shows the variation of t c vs. (cid:15) . t c shows linear decay with (cid:15) , which confirms thefaster response of the flock towards external perturbationwith an increase in the value of (cid:15) . D. Disorder increases system’s informationentropy -10000010000-10000010000 -10000010000
100 20000.050.10.15 t χ (t) C n (r , t ) (a)(b)(c)(d) (e)t t t n ε X(t) (f) (g) t t/t n C n (r , t ) ∆ S ( t ) FIG. 6. (color online)(a), (b) and (c) shows the variation of X ( t ) vs. t . Black, blue and magenta colors are for (cid:15) = 0 , C n ( r, t ) vs. t. Symbols with black (circles),red (squares), green (diamonds), blue (triangles up) and ma-genta (triangles left) colors are for (cid:15) = 0 , . , , . t n vs. (cid:15) shows linear decay with (cid:15) ; inset; plot of corre-lation C n ( r, t ) vs. scaled time t/t n . (f) Plot of informationentropy ∆ S (t) vs. t . (g) Time evolution of χ ( t ) with time t where colors black, blue and magenta are for (cid:15) = 0 , Further, we claim that the accelerated response to exter-nal perturbation is due to neighbors’ frequent updatesfor high disorder strength.We define the update in the neighbour list of the SPPs as X ( t ) = N (cid:80) Ni =1 (( < N iR ( t ) × N/ > ) − (cid:80) j ∈ R j ). where N iR is the number of SPPs inside the interaction radiusof the i th particle, N is the total number of particles inthe system, and the second term on the right-hand sideis the sum over all the particle indices j inside the inter-action radius of the i th particle. The time series of X ( t )oscillates around 0 for different values of (cid:15) , as shown inFig.6(a),(b) and (c). The frequency of oscillation of X ( t )increases with increasing (cid:15) . The increase in the oscilla-tion frequency of X ( t ) suggests more frequent updatesof the neighbor list, and the decrease in the magnitudeof X ( t ) implies a lesser number of neighbors inside theinteraction radius of an SPP. Furthermore, we calculatethe neighbor autocorrelation function, C n ( t ) = (cid:10) (cid:80) T − tt (cid:48) =1 ( X ( t (cid:48) ) − X )( X ( t (cid:48) + t ) − X ) (cid:80) Tt (cid:48) =1 ( X ( t (cid:48) ) − X ) (cid:105) (4)where X is the mean value of X ( t ) over the total time T and t < T . (cid:104) .... (cid:105) represents averaging over 20 indepen-dent realisations. In Fig.6(d) faster decay of C n ( t ) withincrease the disorder strength (cid:15) , suggest more frequentupdate of neighour list. Also in the inset of Fig.6(e) weplot the scaled correlation C n ( t ) vs. t/t n where t n isobtained by fitting the exponential function to exp( − tt n ).In the Fig.6(e) we have shown the variation of t n with (cid:15) ,which decays linearly. Now use the information entropy[44, 47] approach to show that the larger the disorder,the larger is the information entropy and hence the moreinformation transfer among the SPPs. The faster infor-mation transfer in more disorder system is due to thepossibility of more number of accessible states for theparticles. Each state can be defined as the new neigh-bors in the chosen particle’s contact list. If we denote P s , as the probability of being in the i th states from theset of all possible accessible states.If a particle changes its neighbors frequently, then itis exploring more number of neighboring particles andhence more number of states. Hence the neighbor au-tocorrelation, C n,s ( t ), ( the quantity inside the < ... > of Eq. 4) is the neighbour autocorrelation for one state.The subscript s denotes the different independent con-figurations and hence different sets can be generalised asdifferent independent configurations. And C n,s ( t ) and P s ( t ) are equivalent. One is the measure of probabilityof being in a given neighbour list and is the same as P s ( t ).As time progress C n,s ( t ) decreases hence more and morestates are accessed. Hence we define the information en-tropy of the system as ∆ S ( t ) = − (cid:80) s C n,s ( t ) ln C n,s ( t ),where summation s is over all possible realisations.Larger the information entropy larger the available mi-crostate for the particles and hence the more informationtransfer among the flocks. We plot the variation of in-formation entropy ∆ S ( t ) for different disorder in Fig.6(f). We note that ∆ S ( t ) increases with (cid:15) , which further FIG. 7. (color online) Top to bottom color plot shows theprobability of newly visited particles p for three (cid:15) = 2, 1 and0 with time. All other parameters are same as in Fig. 2 confirms that particles are exploring the more states forhigher disorder strength hence more information trans-fer. Also, in fig.6(g), we have plotted the time evolutionof order parameter in the early time, which shows thatfor larger (cid:15) , the system reaches the ordered state quickerin comparison to the lower (cid:15) .Further, we also show the quicker update of neighbourlist for a single particle. In Fig. 7, we plot the fractionof new particles p ( (cid:15), t ) in the neighbour list of a givenparticle from some reference time t = 0. At time t ,all the particles are labeled as old hence p = 0. Astime progress new particles come in the contact list ofthe given particle and p starts to increase. At very latetime all the old particles are gone out of the neighbourlist and hence p = 1. As shown in the figure, for largerdisorder new neighbours are updated faster than for thesmall disorder. IV. DISCUSSION
We introduced a minimal model for a collection of self-propelled particles with bond-disorder. Each particle hasa different ability (interaction strength) to influence itsneighbors. The varying interaction strength is obtainedfrom a uniform distribution, and it can be varied from[1 − (cid:15)/ (cid:15)/ (cid:15) is the disorder strength. For (cid:15) = 0, the model reduces to the constant interactionstrength model or the Vicsek-like model [7]. We stud-ied the steady-state characteristics for different strengthsof the disorder near to order-disorder transition. To oursurprise, bond disorder leads to faster information trans-fer within the flock, viz; the system’s information en-tropy gets increased.We also find that the disorder-to-order transition is dis-continuous in the disorder free system and changes tocontinuous type with an increase in disorder. Further-more, the transition point shifts towards the higher η forthe large disorder.Our study provides a new direction to understand theeffect of intrinsic inhomogeneity in many natural activesystems. It shows that how the bond-disorder in the sys-tem can enhance ordering and faster information trans-fer among the particles. Such properties can be usefulfor many applications: like the faster evacuation of ac-tive particles and also for crowd control in many socialgatherings [48, 49] V. ACKNOWLEDGEMENT
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