PHYSICAL REVIEW E 89, 012718 (2014)

Flocking with minimal cooperativity: The panic model Kevin R. Pilkiewicz and Joel D. Eaves* University of Colorado Boulder, Boulder, Colorado 80309, USA (Received 17 August 2013; published 24 January 2014; corrected 29 January 2014) We present a two-dimensional lattice model of self-propelled spins that can change direction only upon collision with another spin. We show that even with ballistic motion and minimal cooperativity, these spins display robust flocking behavior at nearly all densities, forming long bands of stripes. The structural transition in this system is not a thermodynamic phase transition, but it can still be characterized by an order parameter, and we demonstrate that if this parameter is studied as a dynamical variable rather than a steady-state observable, we can extract a detailed picture of how the flocking mechanism varies with density. DOI: 10.1103/PhysRevE.89.012718

PACS number(s): 87.18.Gh, 05.65.+b, 05.70.Ln, 87.10.Hk

Flocking is ubiquitous in nature, exhibited by a diverse host of organisms ranging from bacteria in the human body to wildebeests on the Serengeti [1–4]. While “strength in numbers” can be seen as an evolutionary impetus for flocking behavior in animals, the spectrum of flocking fauna runs the gamut of intelligence and biological complexity, and they have disparate capacities to sense and react to their external environments. This suggests that flocking must have its origin in simple mechanical tendencies that do not depend upon some sort of survival instinct. The observation that similar behaviors occur in a variety of nonliving systems such as driven colloids [5,6] and granular media [7–10], active polar gels [11–13], and robot swarms [14,15] further supports this notion. Many theoretical models have been developed to study flocking, including self-propelled particle models in continuous space [16–25] and on a lattice [26–30] as well as hydrodynamic models [31–35]. Most of these models agree that the principal driving force for flocking is a local, isotropic interaction between flockers that favors alignment of their velocities. This putative isotropy is problematic, however, in that it allows members of a flock to react to the motion of the members behind them just as they do to those in front of them. This effectively precludes any sort of “follow the leader” behavior, expected in the flocking of animals like ants and sheep. Since most organisms cannot sense their environment equally well in all directions, robust flocking cannot depend on such a high level of cooperativity. In this paper, we demonstrate that flocking can occur with high fidelity, even with minimal cooperativity between agents. We present a two-dimensional lattice model where the interactions between flockers are markedly anisotropic and extremely short-ranged. It consists of self-propelled spins that move ballistically, exclude volume, and can only “see” directly in front of them, effectively minimizing the cooperativity present in the dynamics. We call this model the panic model because the spins mimic the phenomenology of panicked animals, whose hurried motion and finite cone of vision do not leave them time to survey their surroundings. We will show that this model exhibits a spontaneous flocking transition for a broad range of densities, and that this transition can be couched in the language of thermodynamic phase transitions by introducing and analyzing an order parameter that

*

Corresponding author: [email protected]

1539-3755/2014/89(1)/012718(5)

distinguishes structurally disordered from ordered dynamical phases. Because flocking is a global phenomenon spawned by local interactions, large, system-spanning structures must coalesce out of a smaller-scale coarsening; and we will show that this same order parameter can be studied as a dynamical variable to elucidate how the local coarsening of the system proceeds to the global steady state as a function of density. Our model consists of an L × L, periodic square lattice with a fixed fraction, ρ, of its sites occupied by spins that can take one of four orientations: up, down, left, or right, such that no two spins occupy the same site. Each spin can move only in the direction of its orientation, and it will move ballistically in that direction, one lattice spacing per time step, until another spin blocks its path. When this happens, the spin will attempt to change its orientation to match that of the obstructing spin, although it will fail to do so with a specified probability, given by Eq. (1). Because spins can change their orientation only if another spin is directly adjacent to them, this model has some characteristics of a facilitated kinetic model [36–38], although it does not exhibit glassy behavior. The dynamics of the panic model proceeds according to the following algorithm: First, the system is initialized in a random configuration with density ρ. At the beginning of each time step, a spin is chosen at random to attempt a move of one lattice spacing in the direction of its orientation. This move is always accepted unless it would place two spins on the same site. In that case the move fails and the selected spin instead changes its orientation to that of the spin obstructing its movement plus a clockwise angle θ that can be either 0, ±π/2, or π . The probability that this angle takes a given value is determined by a distribution Pθ that we parametrize by an error parameter 0   < ∞ that is fixed for each simulation and acts as an ersatz temperature: P0 () =

3/4 1/4 1 1 + , P±π/2 ()= − , 4 1 + (1/10) 2 4 1 + (3/25) 2

Pπ () = 1 − P0 () − 2P±π/2 ().

(1)

Another spin is then chosen at random from the remaining spins, and the previous steps are repeated. Once every spin has had a chance to move, the clock advances one time step. At  = 0, all moves are deterministic. A spin will always align with a spin that blocks its movement. As  → ∞, there is no orientational preference, and the system tends towards

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©2014 American Physical Society

KEVIN R. PILKIEWICZ AND JOEL D. EAVES

PHYSICAL REVIEW E 89, 012718 (2014)

FIG. 1. (Color online) Steady-state snapshots from panic model simulations at various points in (ρ,) parameter space. In each snapshot, up, down, left, and right spins are represented by black, blue (dark gray), red, and green (both lighter gray).

1.0 Order Parameter σ

disorder. The functional forms in Eq. (1) ensure that for 0 <  < ∞, P0 > P±π/2 > Pπ and all probabilities sum to unity. We simulated the panic model for L = 100 over a broad range of (ρ,) parameter space. At sufficiently large , the system remains disordered for all time; but for all densities exceeding roughly ρ = 0.07, there was a density-dependent, threshold value of  below which the system self-organized into bands of either vertical or horizontal stripes, spontaneously breaking the discrete rotational symmetry of the lattice. Importantly, because the dynamical rules of the panic model destabilize a host of ordered steady-state phases observed in other models of flocking, such as asters, waves, vortices, and gliders [23,25,29], these mutually aligned striped domains are the only stable ordered phase observed at steady state. In other flocking models, system spanning, nematic bands like these usually coexist with a dilute, disordered phase [19,21,23,29]. While a dilute phase does coexist with the nematic bands of our model, it is predominantly ordered along the same direction as the bands. This is because at low densities, the spins seldom change their orientation, so that clusters of spins aligned with the bands move parallel to them, while orthogonally oriented clusters inevitably run back into one of the bands and get absorbed into them. Snapshots of typical steady-state configurations of the system for a range of ρ and  can be seen in Fig. 1. In these images, the colored points indicate spins, with black, blue, red, and green points representing up, down, left, and right spins, respectively. To show the divide between ordered and disordered states more clearly, we depict only vertically striped steady states in this figure, where all spins are oriented either up or down. It is evident from the figure that the threshold value of  separating disordered and ordered phases is a monotonically increasing function of the density. Just like a thermodynamic phase transition, the panic model structural transition can be characterized by an order parameter that is zero in the disordered phase and unity in a perfectly

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FIG. 2. (Color online) The order parameter σ at steady state (t = 20 000 time steps) plotted as a function of the error parameter  for densities of 0.15 (red circles), 0.45 (green squares), and 0.85 (blue crosses). We define the critical  to be where the dashed line σ = 0.5 crosses each curve.

striped one: σ (ρ,,t) = |[χ↑ (ρ,,t) + χ↓ (ρ,,t)] − [χ← (ρ,,t) + χ→ (ρ,,t)]|.

(2)

In this expression, each χ is the fraction of spins with the indicated orientation at time t for a given configuration of the system, and the absolute value bars inside the angular brackets ensure that both horizontal and vertical spin alignments make equal contributions to the order parameter. The angular brackets themselves denote an average over all configurations of the system attainable at density ρ and error parameter  after a time t, assuming all configurations are equally likely at time t = 0. In principle, this is a very difficult average to compute, but the distribution of σ is sufficiently peaked about its mean that it is adequate in practice to average over only a couple thousand configurations. Figure 2 is a plot of the order parameter σ at steady state as a function of  for three different densities. For low densities, σ behaves just like the order parameter of a finite-sized thermodynamic system as temperature decreases across a phase boundary. The order parameter saturates near unity for small , approaches zero for large , and changes very sharply over a narrow range of  somewhere in between. For larger densities, the behavior is the same, except that for very small  (roughly  < 0.05), the order parameter falls sharply away from saturation. This anomaly comes from the contribution of frustrated configurations in which a single band of stripes that is orthogonal to the others persists at steady state either because it is system spanning itself or because it is trapped between two system-spanning bands perpendicular to it. When the order parameter is too small, these configurations are effectively frozen in and cannot break down. At lower densities, gaps in the stripes preclude this kind of frustration from occurring. Since our system is finite in size, the structural transition to a striped steady state actually occurs over a small range of  rather than below a single critical value, but to a good approximation we can define a critical error parameter c as the value of  for which σ = 0.50 at steady state. Plotting c versus the density yields a phase diagram for the panic model (Fig. 3) consisting of a single phase curve above which the system has a disordered steady state and below which it has an ordered, striped steady state. The phase curve terminates at

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FIG. 3. (Color online) The phase diagram for the panic model. The critical error parameter c plotted as a function of density separates the disordered and ordered phases. Dashed vertical lines at ρ = 0.07, 0.225, and 0.50 partition the structural transition into distinct dynamical regimes, labeled by Roman numerals.

ρ = 0.07 because below this density the order parameter does not saturate towards unity. It is important to note that while this structural transition bears many similarities to thermodynamic phase transitions, it is in fact a purely dynamical one. While most flocking models have an underlying thermodynamic description in the absence of self-propulsion that results from either explicit energetic interactions between participants [30,35] or an aligning tendency that can be formally reexpressed as a finite-ranged ferromagnetic interaction [16,17,23,29], the panic model is distinct in being more like a facilitated model [39] with trivial thermodynamics and intrinsically dynamical interactions that cannot be characterized by a Hamiltonian or turned off by an external agent. Since the structural transition exhibited by the panic model has no thermodynamic component, it is more useful to shift our focus from a standard thermodynamic analysis to a study of how the dynamical characteristics of this structural transition vary across the broad range of densities over which it is observed. To probe the dynamics of the transition, we study the time evolution of σ as the system evolves from a random configuration to a striped steady state. Four qualitatively distinct types of behaviors are observed, allowing us to divide the phase diagram into four dynamical regimes (demarcated by vertical dashed lines in Fig. 3 and labeled by Roman numerals). The characteristic time evolution of the order parameter in each of these regimes is shown in Fig. 4. Figure 4(a) shows the order parameter at zero  plotted as a function of time for three densities in regime I of the phase diagram. The order parameter grows steadily for roughly the first thousand time steps, corresponding to an initial coarsening of the system into small unidirectional clusters, as shown in the lower panel for one particular simulation at ρ = 0.07 and t = 100 time steps. As these clusters begin to grow in size, the density exhibits large fluctuations in space, making it possible to end up with orthogonally moving clusters that never collide, thus preventing the transition to a fully striped steady state and causing σ to taper off to a value less than unity. In Fig. 4(b), the time evolution of the order parameter in region II is depicted, again at  = 0 for three densities. Due to the higher density, the clusters that initially form in this regime are larger in size, as seen from the lower

PHYSICAL REVIEW E 89, 012718 (2014)

panel, which depicts a typical configuration at ρ = 0.15 and t = 100 time steps. Larger clusters can collide more easily and will swiftly coalesce into stripes, reflected by the rapid saturation of the order parameter. Although there is a monotonic increase in the growth rate of σ with density, its ultimate saturation begins to occur more slowly as regime III is approached. The analogous plot for regime III is shown in Fig. 4(c), this time for  = 0.05, where σ has its maximum steady-state value at these densities (see Fig. 2). In this regime, the order parameter rises very sharply over the first roughly 100 time steps, which corresponds to the system rapidly coalescing into large, competing domains of horizontally and vertically oriented spins, as seen in the lower panel for a simulation at ρ = 0.40 and t = 100 time steps. The large, often system-spanning, size of these domains makes them difficult to break down, frustrating the completion of the structural transition and resulting in a drastic slowdown in the growth of σ at longer times. As density increases, the transition between the fast- and slow-growth regimes occurs at smaller values of σ . The rate of growth in the slow regime does increase with density, but lower-density systems still saturate faster overall. The growth of the order parameter in the highest-density regime is plotted in Fig. 4(d), once again at  = 0.05 for three different densities. A snapshot of the system at ρ = 0.75 and t = 100 time steps (see the lower panel) shows that the domains that initially emerge in this regime are smaller than those of the previous regime. Smaller domains are easier to break down, so the overall saturation is faster in this regime, although still much slower than in regime II. In this regime σ increases monotonically with density and fits a power law with exponent less than 1. This means that saturation occurs more quickly for denser systems, a result of the domains having more frequent collisions at their boundaries due to being packed more tightly together. This effect is also present in regime III, but there it is offset by a steady increase in the size of the domains with higher density. Although the single structural transition of the panic model is not a thermodynamic one, its varying dynamical character is nonetheless reminiscent of thermodynamic phase transitions in other systems. For example, the Blume-Capel model [40,41], an Ising model variant, exhibits a single phase transition to a ferromagnetic state, but depending on where the phase curve is crossed, the transition can be either first or second order. By analogy, the structural transition of the panic model occurs through a nucleation process in regime II that is reminiscent of the sorts of activated processes present in freezing liquids (a first-order phase transition), whereas in regimes III and IV the transition occurs through the formation of competing domains, similar to the second-order ferromagnetic transition of the standard Ising model. The point on the phase curve that separates regime II from regime III (ρ ≈ 0.225) is thus something akin to a tricritical point connecting a line of firstorder-like transitions to one of second-order-like transitions. The panic model illustrates that self-propelled particles with a minimal amount of cooperativity can still exhibit robust flocking behavior. The sort of collision-based flocking that this model describes depends heavily upon excluded volume interactions, and as such it is not surprising that the dynamics of this flocking is highly sensitive to the density of the system. It

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FIG. 4. (Color online) The order parameter σ plotted as a function of time (a) in regime I, for densities of 0.03 (red, bottom curve), 0.05 (green, middle curve), and 0.07 (blue, top curve)  = 0; (b) in regime II, for densities of 0.10 (red, bottom curve), 0.15 (green, middle curve), and 0.20 (blue, top curve) with  = 0; (c) in regime III, for densities of 0.25 (red, top curve), 0.35 (green, middle curve), and 0.40 (blue, bottom curve) with  = 0.05; (d) and in regime IV, for densities of 0.60 (red, bottom curve), 0.75 (green, middle curve), and 0.95 (blue, top curve) with  = 0.05. The lower half of each panel depicts a typical configuration of the system after 100 time steps in the corresponding dynamical regime (spins are color coded as in Fig. 1).

is likely that this sensitivity is also present in more cooperative flocking models with excluded volume interactions, but in those models the larger variety of steady-state structures that can emerge at different densities occludes it. It is tempting to focus on the steady-state structural features of flocking models because they can be related to better understood equilibrium phenomena like phase transitions, but to truly understand spontaneous ordering phenomena like flocking, it will be necessary to develop new tools for

studying and describing these systems at times other than zero and infinity. The regimented dynamical behavior of the order parameter in our model suggests that such parameters, traditionally studied only as steady-state observables, may be a useful tool for studying the dynamics of coarsening and self-assembly in other, more complex systems as well.

[1] F. Heppner in Animal Groups in Three Dimensions, edited by J. K. Parrish and W. M. Hamner (Cambridge University Press, New York, 1997). [2] J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller, and S. J. Simpson, Science 312, 1402 (2006). [3] E. B. Steager, C.-B. Kim, and M. J. Kim, Phys. Fluids 20, 073601 (2008). [4] D. B. Kearns, Nat. Rev. Microbiol. 8, 634 (2010). [5] J. Drocco, L. M. Lopatina, C. Reichhardt, and C. J. Olson Reichhardt, Proc. SPIE 8458, 84581l (2012). [6] J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, and P. M. Chaikin, Science 339, 936 (2013). [7] V. Narayan, N. Menon, and S. Ramaswamy, J. Stat. Mech.: Theory Exp. (2006) P01005. [8] I. S. Aranson, A. Snezhko, J. S. Olafsen, and J. S. Urbach, Science 320, 612 (2008). [9] A. Kudrolli, G. Lumay, D. Volfson, and L. S. Tsimring, Phys. Rev. Lett. 100, 058001 (2008). [10] J. Deseigne, O. Dauchot, and H. Chat´e, Phys. Rev. Lett. 105, 098001 (2010). [11] F. Backouche, L. Haviv, D. Groswasser, and A. BernheimGroswasser, Phys. Biol. 3, 264 (2006). [12] J. Joanny and J. Prost, HFSP J. 3, 94 (2009). [13] T. Toyota, D. A. Head, C. F. Schmidt, and D. Mizuno, Soft Matter 7, 3234 (2011).

[14] Y. Hanada, G. Lee, and N. Y. Chong, in Proceedings of the IEEE Swarm Intelligence Symposium (Curran Associates Inc., Red Hook, NY, 2007). [15] A. E. Turgut, H. C ¸ elikkanat, F. G´okc¸e, and E. S¸ahin, Swarm Intell. 2, 97 (2008). [16] T. Vicsek, A. Czir´ok, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. 75, 1226 (1995). [17] A. Czir´ok and T. Vicsek, Physica A 281, 17 (2000). [18] F. Peruani, A. Deutsch, and M. B¨ar, Phys. Rev. E 74, 030904(R) (2006). [19] H. Chat´e, F. Ginelli, G. Gr´egoire, and F. Raynaud, Phys. Rev. E 77, 046113 (2008). [20] T. Ohkuma and T. Ohta, Chaos 20, 023101 (2010). [21] F. Ginelli, F. Peruani, M. B¨ar, and H. Chat´e, Phys. Rev. Lett. 104, 184502 (2010). [22] Y. Yang, V. Marceau, and G. Gompper, Phys. Rev. E 82, 031904 (2010). [23] F. D. C. Farrell, M. C. Marchetti, D. Marenduzzo, and J. Tailleur, Phys. Rev. Lett. 108, 248101 (2012). [24] S. R. McCandlish, A. Baskaran, and M. F. Hagan, Soft Matter 8, 2527 (2012). [25] J. L. Silverberg, M. Bierbaum, J. P. Sethna, and I. Cohen, Phys. Rev. Lett. 110, 228701 (2013). [26] H. J. Bussemaker, A. Deutsch, and E. Geigant, Phys. Rev. Lett. 78, 5018 (1997).

J.D.E. thanks the University of Colorado for startup funds.

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FLOCKING WITH MINIMAL COOPERATIVITY: THE PANIC MODEL [27] O. J. O’Loan and M. R. Evans, J. Phys. A 32, L99 (1999). [28] J. R. Raymond and M. R. Evans, Phys. Rev. E 73, 036112 (2006). [29] F. Peruani, T. Klauss, A. Deutsch, and A. Voss-Boehme, Phys. Rev. Lett. 106, 128101 (2011). [30] A. P. Solon and J. Tailleur, Phys. Rev. Lett. 111, 078101 (2013). [31] Y. Tu, Physica A 281, 30 (2000). [32] J. Toner, Y. Tu, and S. Ramaswamy, Ann. Phys. (NY) 318, 170 (2005). [33] K. Kruse, J. F. Joanny, F. J¨ulicher, J. Prost, and K. Sekimoto, Eur. Phys. J. E 16, 5 (2005). [34] E. Bertin, M. Droz, and G. Gr´egoire, Phys. Rev. E 74, 022101 (2006).

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[35] A. Gopinath, M. F. Hagan, M. C. Marchetti, and A. Baskaran, Phys. Rev. E 85, 061903 (2012). [36] G. H. Fredrickson and H. C. Andersen, Phys. Rev. Lett. 53, 1244 (1984). [37] W. Kob and H. C. Andersen, Phys. Rev. E 48, 4364 (1993). [38] P. Sollich and M. R. Evans, Phys. Rev. Lett. 83, 3238 (1999). [39] Y. J. Jung, J. P. Garrahan, and D. Chandler, Phys. Rev. E 69, 061205 (2004). [40] M. Blume, Phys. Rev. 141, 517 (1966). [41] M. Blume, V. J. Emery, and R. B. Griffiths, Phys. Rev. A 4, 1071 (1971).

012718-5

Flocking with minimal cooperativity: the panic model.

We present a two-dimensional lattice model of self-propelled spins that can change direction only upon collision with another spin. We show that even ...
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