text:
Collective motion of self-propelled particles interacting without cohesion
Hugues Chaté CEA – Service de Physique de l'État Condensé, Centre d'Etudes de Saclay, 91191 Gif-sur-Yvette, France Francesco Ginelli CEA – Service de Physique de l'État Condensé, Centre d'Etudes de Saclay, 91191 Gif-sur-Yvette, France Guillaume Grégoire Matière et Systèmes Complexes, CNRS UMR 7057, Université Paris–Diderot, Paris, France Franck Raynaud CEA – Service de Physique de l'État Condensé, Centre d'Etudes de Saclay, 91191 Gif-sur-Yvette, France Matière et Systèmes Complexes, CNRS UMR 7057, Université Paris–Diderot, Paris, France
We present a comprehensive study of Vicsek-style self-propelled particle models in two and three space dimensions. The onset of collective motion in such stochastic models with only local alignment interactions is studied in detail and shown to be discontinuous (first-order like). The properties of the ordered, collectively moving phase are investigated. In a large domain of parameter space including the transition region, well-defined high-density and high-order propagating solitary structures are shown to dominate the dynamics. Far enough from the transition region, on the other hand, these objects are not present. A statistically-homogeneous ordered phase is then observed, which is characterized by anomalously-strong density fluctuations, superdiffusion, and strong intermittency.
pacs:
64.60Cn, 05.70Ln
Collective motion phenomena in nature have attracted the interest of scientists and other authors for quite a long time Pliny . The question of the advantage of living and moving in groups, for instance, is a favorite one among evolutionary biologists Parrish1997 . In a different perspective, physicists are mostly concerned with the mechanisms at the origin of collective motion, especially when it manifests itself as a true, non-trivial, emerging phenomenon, i.e. in the absence of some obvious cause like the existence of a leader followed by the group, a strong geometrical constraint forcing the displacement, or some external field or gradient felt by the whole population. Moreover, the ubiquity of the phenomenon at all scales, from intra-cellular molecular cooperative motion to the displacement in group of large animals, raises, for physicists at least, the question of the existence of some universal features possibly shared among many different situations.
One way of approaching these problems is to construct and study minimal models of collective motion: if universal properties of collective motion do exist, then they should appear clearly within such models and thus could be efficiently determined there, before being tested for in more elaborate models and real-world experiments or observations. Such is the underlying motivation of recent studies of collective motion by a string of physicists Albano ; Couzin2003 ; Czirok1997 ; Gregoire2001 ; Huepe ; Toner1995 . Among them, the group of Tamas Vicsek has put forward what is probably the simplest possible model exhibiting collective motion in a non-trivial manner.
In the "Vicsek model" Vicsek1995 , point particles move off-lattice at constant speed v0, adjusting their direction of motion to that of the average velocity of their neighbors, up to some noise term accounting for external or internal perturbations (see below for a precise definition). For a finite density of particles in a finite box, perfect alignment is reached easily in the absence of noise: in this fluctuation-less collective motion, the macroscopic velocity equals the microscopic one. On the other hand, for strong noise particles are essentially non-interacting random walkers and their macroscopic velocity is zero, up to statistical fluctuations.
Vicsek et al. showed that the onset of collective motion occurs at a finite noise level. In other words, there exists, in the asymptotic limit, a fluctuating phase where the macroscopic velocity of the total population is, on average, finite. Working mostly in two space dimensions, they concluded, on the basis of numerical Vicsek1995 ; Czirok1997 simulations, that the onset of this ordered motion is well described as a novel non-equilibrium continuous phase transition leading to long range order, at odds with equilibrium where the continuous XY symmetry cannot be spontaneously broken in two space dimensions and below Mermin . This brought support to the idea of universal properties of collective motion since the scaling exponents and functions associated to such phase transitions are expected to bear some degree of universality, even out of equilibrium.
The above results caused a well-deserved stir and prompted a large number of studies at various levels Albano ; Huepe ; Bertin ; Birnir ; Bussem ; Chate2006 ; Couzin2002 ; Couzin2003 ; Couzin2005 ; Csahok1995 ; Czirok1999 ; Czirok1999_2 ; Duparcmeur ; Comment_gg ; Gregoire2001 ; Gregoire2003 ; Hemmingson ; Levine ; Mikhailov ; Mogilner ; Oloan ; dOrsogna ; Ken ; Simha2002 ; Simha2002_2 ; Szabo ; Toner1995 ; Toner1998 ; Toner1998_2 ; Topaz2004 ; Topaz2006 ; Vicsek1999 . In particular, two of us showed that the onset of collective motion is in fact discontinuous Gregoire2004 , and that the original conclusion of Vicsek et al. was based on numerical results obtained at too small sizes Vicsek1995 ; Czirok1997 . More recently, the discontinuous character of the transition was challenged in two publications, one by the Vicsek group Nagy and another by Aldana et al. Aldana2007 .
Here, after a definition of the models involved (Section II), we come back, in Section III, to this central issue and present a rather comprehensive study of the onset of collective motion in Vicsek-style models. In Section IV, we describe the ordered, collective motion phase. Section V is devoted to a general discussion of our results together with some perspectives. Most of the numerical results shown were obtained in two space dimensions, but we also present three-dimensional results. Wherever no explicit mention is made, the default space dimension is two. Similarly, the default boundary conditions are periodic in a square or cubic domain.
Ii The models
ii.1 Vicsek model: angular noise
Let us first recall the dynamical rule defining the Vicsek model Vicsek1995 . Point particles labeled by an integer index i move off-lattice in a space of dimension d with a velocity →vi of fixed modulus v0=|→vi|. The direction of motion of particle i depends on the average velocity of all particles (including i) in the spherical neighbourhood Si of radius r0 centered on i. The discrete-time dynamics is synchronous: the direction of motion and the position of all particles are updated at each timestep Δt, in a driven, overdamped manner:
where ϑ is a normalization operator (ϑ(→w)=→w/|→w|) and Rη performs a random rotation uniformely distributed around the argument vector: in d=2, Rη→v is uniformely distributed around →v inside an arc of amplitude 2πη; in d=3, it lies in the solid angle subtended by a spherical cap of amplitude 4πη and centered around →v. The particles positions →ri are then simply updated by streaming along the chosen direction as in
→ri(t+Δt)=→ri(t)+Δt→vi(t+Δt). (2)
Note that the original updating scheme proposed by Vicsek et al. in Vicsek1995 defined the speed as a backward difference, although we are using a forward difference. The simpler updating above, now adopted in most studies of Vicsek-style models, is not expected to yield different results in the asymptotic limit of infinite size and time.
ii.2 A different noise term: vectorial noise
The "angular" noise term in the model defined above can be thought of as arising from the errors committed when particles try to follow the locally-averaged direction of motion. One could argue on the other hand that most of the randomness stems from the evaluation of each interaction between particle i and one of its neighbors, because, e.g., of perception errors or turbulent fluctuations in the medium. This suggests to replace Eq.(1) by:
→vi(t+Δt)=v0ϑ⎡⎣∑j∈Si→vj(t)+ηNi→ξ⎤⎦ (3)
where →ξ is a random unit vector and Ni is the number of particles in Si. It is easy to realize that this "vectorial" noise acts differently on the system. While the intensity of angular noise is independent from the degree of local alignment, the influence of the vectorial noise decreases with increasing local order.
ii.3 Repulsive force
In the original formulation of the Vicsek model as well as in the two variants defined above, the only interaction is alignment. In a separate work Gregoire2003 , we introduced a two-body repulsion/attraction force, to account for the possibility of maintaining the cohesion of a flock in an infinite space (something the Vicsek model does not allow). Here, we only study models without cohesion. Nevertheless, we have considered, in the following, the case of a pairwise repulsion "force", to estimate in particular the possible influence of the absence of volume exclusion effects in the basic model, which leaves the local density actually unbounded. We thus introduce the short ranged, purely repulsive interaction exerted by particle j on particle i:
→fij=−→eij×[1+exp(|→rj−→ri|/rc−2)]−1, (4)
where →eij is the unit vector pointing from particle i to j and rc<r0 is the typical repulsion range. Equations (1) and (3) are then respectively generalized to
→vi(t+Δt)=v0ϑ⎡⎣∑j∈Si→vj(t)+β∑j∈Si→fij+ηNi→ξ⎤⎦, (6)
where β measures the relative strength of repulsion with respect to alignment and noise strength.
ii.4 Control and order parameters
The natural order parameter for our polar particles is simply the macroscopic mean velocity, conveniently normalized by the microscopic velocity v0
→φ(t)=1v0⟨→vi(t)⟩i, (7)
where ⟨⋅⟩i stands for the average over the whole population. Here, we mostly consider its modulus φ(t)=|→φ(t)|, the scalar order parameter.
In the following, we set, without loss of generality, Δt=1 and r0=1, and express all time- and length-scales in terms of these units. Moreover repulsive force will be studied by fixing rc=0.127 and β=2.5.
This leaves us with two main parameters for these models: the noise amplitude η and the global density of particles ρ. Recently, the microscopic velocity v0 has been argued to play a major role as well Nagy . All three parameters (η, ρ, and v0) are considered below.
Iii Order-disorder transition at the onset of collective motion
Figure 1: (color online) Typical behavior across the onset of collective motion for moderate size models (ρ=2, v0=0.5, L=64) with angular noise (black circles) and vectorial noise (red triangles). The reduced noise amplitude ε=1−η/ηt is shown in abscissas (transition points estimated at ηt=0.6144(2) – vectorial noise – and ηt=0.478(5) – angular noise). (a) Time-averaged order parameter ⟨φ(t)⟩t. (b) Binder cumulant G. (c) Variance σ of φ; (d) Order parameter distribution function P at the transition point. Bimodal distribution for vectorial noise dynamics (red dashed line), unimodal shape for angular noise (black solid line). Time averages have been computed over 3⋅105 timesteps.
As mentioned above, the original Vicsek model attracted a lot of attention mostly because of the conclusions drawn from the early numerical studies Vicsek1995 ; Czirok1997 : the onset of collective motion was found to be a novel continuous phase transition spontaneously breaking rotational symmetry. However, it was later shown in Gregoire2004 that beyond the typical sizes considered originally the discontinuous nature of the transition emerges, irrespective of the form of the noise term. Recently, the discontinuous character of the transition was argued to disappear in the limit of small v0 Nagy . We now address the problem of the nature of the transition in full detail.
Figure 2: (color online) FSS analysis of angular noise dynamics (ρ=2, v0=0.5, time averages computed over 2⋅107 timesteps). Time-averaged order parameter (a) and Binder cumulant (b) as a function of noise for various system sizes L. (c) Piece of an order parameter time series close to the transition point (L=256, η=0.476).
Even though there is no rigorous theory for finite-size-scaling (FSS) for out-of-equilibrium phase transitions, there exists now ample evidence that one can safely rely on the knowledge gained in equilibrium systems Lubeck ; Marcq ; Marcq2 . The FSS approach Binder ; Privman involves the numerical estimation of various moments of the order parameter distribution as the linear system size L is systematically varied. Of particular interest are the variance
σ(η,L)=Ld(⟨φ2⟩t−⟨φ⟩2t)
and the so-called Binder cumulant
G(η,L)=1−⟨φ4⟩t3⟨φ2⟩2t, (8)
where ⟨⋅⟩t indicates time average. The Binder cumulant is especially useful in the case of continuous phase transitions, because it is one of the simplest ratio of moments which takes a universal value at the critical point ηt, where all the curves G(η,L), obtained at different system sizes L, cross each other. At a first-order transition point, on the other hand, the Binder cumulant exhibits a sharp drop towards negative values Binder1997 . This minimum is due to the simultaneous contributions of the two phases coexisting at threshold. Moreover, it is easy to compute that G(η,L)≈2/3 in the ordered phase, while for a disordered state with a continuous rotational simmetry one has G(η,L)≈1/3 in d=2 and G(η,L)≈4/9 in d=3.
Figure 3: (color online) Transition to collective motion with short range repulsive interactions. Left panels: angular noise. Right panels: vectorial noise. (a,b): order parameter vs noise amplitude at different system sizes. (c,d): Binder cumulant G as a function of noise amplitude. (ρ=2, v0=0.3, time averages carried over 107 timesteps).
iii.1 Overture
As an overture, we analyze systems of moderate size in two dimensions (N≈104 particles) at the density ρ=2, typical of the initial studies by Vicsek et al., but with the slightly modified update rule (2) and for both angular and vectorial noise. The microscopic velocity is set to v0=0.5.
For angular noise, the transition looks indeed continuous, as found by Vicsek et al.. On the other hand, the time averaged scalar order parameter ⟨φ⟩t displays a sharp drop for vectorial noise, and the Binder cumulant exibits a minimum at the transition point, indicating a discontinuous phase transition (Fig. 1a-b). Simultaneously, the variance is almost delta-peaked. The difference betwen the two cases is also recorded in the probability distribution function (PDF) of φ which is bimodal (phase coexistence) in the vectorial noise case (Fig. 1c-d).
Figure 4: (color online) Transition to collective motion in three spatial dimensions. Left panels: angular noise. Right panels: vectorial noise. (a,b): time-averaged order parameter vs. noise amplitude at different system sizes. (c,d): Binder cumulant G as a function of noise amplitude. (ρ=0.5, v0=0.5, time averages carried over 105 timesteps).
The qualitative difference observed upon changing the way noise is implemented in the dynamics is, however, only a finite-size effect. As shown in Gregoire2004 , the transition in the angular noise case reveals its asymptotic discontinuous character provided large-enough system sizes L are considered (Fig. 2a-b). Remaining for now at a qualitative level, we show in Fig. 2c a typical time-series of the order parameter for the angular noise case in a large system in the transition region. The sudden jumps from the disordered phase to the ordered one and vice-versa are evidence for metastability and phase coexistence.
Note that the system size beyond which the transition reveals its discontinuous character for the angular noise case at density ρ=2 and velocity v0=0.5 — the conditions of the original papers by Vicsek et al.— is of the order of L=128, the maximum size then considered. It is clear also from Fig. 1 that the discontinuous nature of the transition appears earlier, when increasing system size, for vectorial noise than for angular noise. Thus, finite-size effects are stronger for angular noise. The same is true when one is in presence of the repulsive interactions (Fig.3). Finally, the same scenario holds in three space dimensions, with a discontinous phase transition separating the ordered from the disordered phases for both angular and vectorial noise (Fig. 4).
Before proceeding to a study of the complete phase diagram, we detail now how a comprehensive FSS study can be performed on a particular case.
iii.2 Complete FSS analysis
Figure 5: Correlation time τ of the order parameter near the transition point for vectorial noise dynamics with repulsion. System parameters are ρ=2, v0=0.5 and η≈ηt. (a) Time correlation function C(t) at L=128. The lin-log inset shows the exponential decay. (b) Correlation time τ as a function of system size L. The dashed line marks linear growth with L. Correlation functions were computed on samples of ≈106 realizations for typically 103 timesteps.
For historical reasons, the following study has been performed on the model with vectorial noise and repulsive force (Eq. 6). It has not been repeated in the simpler case of the "pure" Vicsek model because its already high numerical cost would have been prohibitive due to the strong finite size effects.
As a first step, we estimated the correlation time τ(L), whose knowledge is needed to control the quality of time-averaging: the duration T of numerical simulations has been taken much larger than τ(L) (T=100τ in the largest systems, but typically 10000τ for smaller sizes). Moreover, τ is also useful to correctly estimate the statistical errors on the various moments (as ⟨φ⟩t, σ, and G ) of the PDF of the order parameter, for which we used the Jackknife procedure Efron . The correlation time was estimated near the transition (where it is expected to be largest) as function of system size L measuring the exponential decay rate of the correlation function (Fig. 5a)
C(t)=⟨φ(t0)φ(t0+t)⟩t−⟨φ(t0)⟩2t0∼exp(−tτ). (9)
We found τ to vary roughly linearly with L (see Fig. 5b). It is interesting to observe that at equilibrium, one would expect τ to scale as Zinn
τ=Ld2exp(κLd−1)
where κ is the surface tension of the metastable state. Therefore, our result implies a very small or vanishing surface tension κ≪1/L, a situation reminiscent of observations made in the cohesive case Gregoire2003 , where the surface tension of a cohesive droplet was found to vanish near the onset.
Following Borgs and Koteckỳ Borgs , the asymptotic coexistence point ηt (i.e. the first order transition point) can be determined from the asymptotic convergence of various moments of the order parameter PDF. First, the observed discontinuity in ⟨φ(t)⟩t, located at ηφ(L), is expected to converge exponentially to ηt with L. Second, the location of the susceptibility peak ησ(L) — which is the same as the peak in σ provided some fluctuation-dissipation relation holds, see the Appendix — also converges to ηt, albeit algebraically with an exponent γσ. Third, the location of the minimum of G, ηG(L), is also expected to converge algebraically to ηt with with an exponent γG=γσ.
Interestingly, the value taken by these exponents actually depends on the number of phases and of the dimension d of the system: for two-phase coexistence one has γG=γσ=2d, while for more than two phases γG=γσ=d. In Figure 6, we show that our data are in good agreement with all these predictions. The three estimates of ηt are consistent with each other within numerical accuracy. Moreover ηφ(L) is found to converge exponentially to the transitional noise amplitude, while both ησ(L) and ηG(L) show algebraic convergence with an exponent close to 2. This agrees with the fact that due to the continuous rotational symmetry, the ordered phase is degenerate and amounts to an infinite number of possible phases.
Figure 6: (color online) FSS analysis of vectorial dynamics with short range repulsive force (ρ=2, v0=0.3). Convergence of the finite-size transition points measured from different moments of the order parameter FSS to the asymptotic transition point ηt (see fig. 3). Upper panels: finite size transition points estimated from (a) time average, (b) variance and (c) Binder cumulant. The horizontal dashed line marks the estimated asymptotic threshold ηt=0.5544(1). Lower panels: scaling of the finite size reduced noise ε=1−η/ηt transition point. (d) Exponential convergence for the jump location in the time-averaged order parameter. (e) Power-law behavior of the variance peak position. (f) Power-law behavior of the Binder cumulant minimum. The dashed lines in (e-f) mark the estimated exponents γσ=γG=2.
iii.3 Hysteresis
One of the classical hallmarks of discontinuous phase transitions is the presence, near the transition, of the hysteresis phenomenon: ramping the control parameter at a fixed (slow) rate up and down through the transition point, a hysteresis loop is formed, inside which phase coexistence is manifest (see Fig. 7a for the d=3 case with vectorial noise). The size of such hysteresis loops varies with the ramping rate. An intrinsic way of assessing phase coexistence and hysteresis is to study systematically the nucleation time τ↑ needed to jump from the disordered phase to the ordered one, as well as τ↓ the decay time after which the ordered phase falls into the disordered one. Fig. 7b shows, in three space dimensions, how these nucleation and decay vary with η at two different sizes. A sharp divergence is observed, corresponding to the transition point. At a given time value τ, one can read, from the distance between the "up" and the "down" curve, the average size of hysteresis loops for ramping rates of the order of 1/τ.
Figure 7: (color online) Hysteresis in three spatial dimensions with vectorial noise. (a) order parameter vs noise strength along the hysteresis loop observed with a ramp rate of 2⋅10−6 per time step (ρ=1/2, v0=0.5, L=32). Empty circles mark the path along the adiabatic increase of noise amplitude, full triangles for adiabatic decrease. (b) nucleation times from the disordered phase to the ordered phase (τ↑, left curves) and vice-versa (τ↓, right curves) for two system sizes (other parameters as in (a)). Each point is averaged over 1000 realizations.
iii.4 Phase diagram
The above detailed FSS study would be very tedious to realize when varying systematically the main parameters η, ρ, and v0, as well as the nature of the noise and the presence or not of repulsive interactions. From now on, to characterize the discontinuous nature of the transition, we rely mainly on the presence, at large-enough system sizes L, of a minimum in the variation of the Binder cumulant G with η (all other parameters being fixed). We call L∗ the crossover size marking the emergence of a minimum of G(η).
We are now in the position to sketch the phase diagram in the (η, ρ, v0) parameter space. The numerical protocol used is, at given parameter values, to run a large-enough system so that the discontinuous character of the transition is seen (i.e. L>L∗). For larger sizes, the location of the transition point typically varies very little, so that for most practical purposes, locating the (asymptotic) transition point from systems sizes around L∗ is satisfactory.
Figure 8: (color online) Asymptotic phase diagrams for the transition to collective motion. (a) Two space dimensions: threshold amplitude ηt for angular noise as a function of density ρ at v0=0.5. Inset: Log-log plot to compare the low density behavior with the mean field predicted behavior ηt∼√ρ (dashed red line). (b) As in panel (a), buth with vectorial noise dynamics. (c) Noise-density phase diagram in three dimensions for vectorial noise dynamics at fixed velocity v0=0.5. In the log-log inset the transition line can be compared with the predicted behavior ηt∼ρ1/3 (dashed red line). (d) Two space dimensions: threshold amplitude ηt for angular noise as a function of particle velocity v0 at fixed density ρ=1/2 (black circles) and ρ=1/8 (red triangles). The horizontal dashed line marks the noise amplitude considered in Ref. Nagy (see Section III.6).
Figure 9: (color online) First-order transition for angular noise dynamics at high (left panels) and low velocity v0 (right panels). Typical averaging time is ≈106 timesteps. (a) Time-averaged order parameter and (c) Binder cumulant at large particle velocity for angular noise in two spatial dimensions at increasing velocities and L≳L∗ (ρ=2). (b) Time averaged order parameter and (d) Binder cumulant for v0=0.05 and two increasing system sizes (ρ=1/2).
The results presented below are in agreement with simple mean-field-like arguments in the diluted limit: in the small-ρ regime, one typically expects that the lower the density, the lower the transitional noise amplitude ηt. Indeed, for Δtv0 of the order of or not much smaller than the interaction range r0 and in the low-density limit ρ≪1/r0d, the system can be seen as a dilute gas in which particles interact by short range ordering forces only. In this regime, the persistence length of an isolated particle (i.e. the distance travelled before its velocity loses correlation with its initial direction of motion) varies like v0/η. To allow for an ordered state, the noise amplitude should be small enough so that the persistence length remains larger than the average inter-particle distance, i.e. 1/ρ1/d. Thus the transition noise amplitude is expected to behave as
ηt∼v0ρ1d. (10)
In Czirok1997 , it was indeed found that ηt∼ρα with α≃12 in two dimensions. Our own data (Fig. 8a-c) now confirm Eq. (10) for both the angular and vectorial noise in two and three spatial dimensions, down to very small ρ values. The data deviate from the square-root behavior as the average inter-particle distance gets of the order of or smaller than the interaction range.
Finally, we also investigated the transition line when v0 is varied (Fig. 8d). For the vectorial noise case, at fixed density, the threshold noise value ηt is almost constant (data obtained at ρ=12, not shown). For the angular noise, in the small v0 limit where the above mean-field argument does not apply, we confirm the first-order character of the phase transition down to v0≈0.05 for both angular and vectorial noise (Fig. 9b,d). For even smaller values of v0, the investigation becomes numerically too costly (see section below). Note that ηt seems to be finite when v0→0+, a limit corresponding to the XY model on a randomly connected graph. Still for angular noise, the large velocity limit is also difficult to study numerically. Again, we observe that the transition is discontinuous as far as we can probe it, i.e. v0=20 (Fig. 9a,b).
iii.5 Special limits and strength of finite-size effects
We now discuss particular limits of the models above together with the relative importance of finite-size effects. Recall that these are quantified by the estimated value of the crossover size L∗ beyond which the transition appears discontinuous. All the following results have been obtained for d=2. Partial results in three dimensions indicate that the same conclusions should hold there. Keep in mind that in all cases reported, the transition is discontinuous. We are just interested here in how large a system one should use in order to reach the asymptotic regime.
Figure 10: (color online) Crossover system size L∗ above which the discontinuous character of the transition appears (as testified by the existence of a minimum in the G(η) curve). Black circles: angular noise. Red triangles: vectorial noise. (a) L∗ vs ρ for v0=0.5; Inset: the low density behavior in log-log scales; the dashed line marks a power-law divergence proportional to 1/√ρ. (b) L∗ vs v0 at fixed density (open symbols: ρ=1/2; filled symbols ρ=2.0).
Fig. 10a shows that finite-size effects are stronger for angular noise than for vectorial noise for all densities ρ at which we are able to perform these measurements. Note in particular that at ρ=2, the density originally used by Vicsek et al., L∗∼128 for angular noise, while it is very small for vectorial noise, confirming the observation made in Section III.1.
In the small-ρ limit, the discontinuous character of the transition appears later and later, with L∗ roughly diverging as 1/√ρ (inset of Fig. 10a). Note that this means that in the small-ρ limit one needs approximately the same number of particles to start observing the discontinuity.
The large-ρ limit reveals a difference between angular and vectorial noise: while L∗ remains small for vectorial noise, it seems to diverge for angular noise (Fig. 10a), making this case difficult to study numerically.
We also explored the role the microscopic velocity v0 in the strength of finite-size effects. Qualitatively, the effects observed are similar to those just reported when the density is varied (Fig. 10b). In the small v0 limit, we record a strong increase of L∗ as v0→0 for both types of noise. In the large velocity limit, L∗ decreases for vectorial noise, whereas it increases for angular noise.
iii.6 Summary and discussion
The summary of the above lengthy study of the order-disorder transition in Vicsek-like models is simple: for any finite density ρ, any finite velocity v0, and for both types of noise introduced, the transition is discontinuous. This was observed even in the numerically-difficult limits of large or small ρ or v0. These results contradict recent claims made about the angular noise case (original Vicsek model). We now comment on these claims.
Vicsek and co-workers Nagy , showed that keeping the density and the noise intensity fixed, a qualitative change is observed when decreasing v0: for not too small v0 values, in the ordered phase, particles diffuse anisotropically (and the transition is discontinuous), while diffusion becomes isotropic at small v0, something interpreted as a sign of a continuous transition in this region. Rather than the convoluted arguments presented there, what happens is in fact rather simple: decreasing v0 at fixed ρ and η, one can in fact cross the transition line, passing from the ordered phase (where particles obviously diffuse anisotropically due to the transverse superdiffusive effects discussed in Section IV.3) to the disordered phase. Our Figure 8d, obtained in the same conditions as in Nagy (apart from harmless change of time updating rule), shows that if one keeps η=0.1 (as in Nagy ), one crosses the transition line at about v0≃0.1, the value invoked by Vicsek and co-workers to mark a crossover from discontinuous to "continuous" transitions.
In a recent letter Aldana2007 , Aldana et al. study order-disorder phase transitions in random network models and show that the nature of these transitions may change with the way noise is implemented in the dynamics (they consider the angular and vectorial noises defined here). Arguing that these networks are limiting cases of Vicsek-like models, they claim that the conclusions reached for the networks carry over to the transition to collective motion of the VM-like systems. They conclude in particular that in the case of "angular" noise the transition to collective motion is continuous. We agree with the analysis of the network models, but the claim that they are relevant as limits of Vicsek-like models is just wrong: the data presented there (Figure 1 of Aldana2007 ) to substantiate this claim is contradicted by our Figures 9a,c (see also Chate2007 ) obtained at larger system sizes. Again, for large-enough system sizes, the transition is indeed discontinuous. Thus, at best, the network models of Aldana et al. constitute a singular v0→∞ limit of Vicsek-like models.
Iv Nature of the ordered phase
We now turn our attention to the ordered, symmetry-broken phase. In previous analytical studies, it has often been assumed that the density in the ordered phase is spatially homogeneous, albeit with possibly large fluctuations (see, e.g. Toner1995 ). This is indeed what has been reported in early numerical studies, in particular by Vicsek et al. Vicsek1995 . In the following, we show that this is not true in large enough systems, where, for a wide range of noise amplitudes near the transition point, density fluctuations lead to the formation of localized, travelling, high-density and high-order structures. At low enough noise strength, though, a spatially-homogeneous ordered phase is found, albeit with unusually strong density fluctuations.
iv.1 Traveling in bands
Figure 11: (color online) Typical snapshots in the ordered phase. Points represent the position of individual particles and the red arrow points along the global direction of motion. (a)-(c): Angular noise, ρ=1/2, v0=0.5, η=0.3 and increasing system sizes - respectively L=64, L=256 and L=1024. Sharp bands can only be observed if L is larger than the typical band width w. (d) Vectorial noise: ρ=1/2, v0=0.5, η=0.55 and L=64: bands appear at relatively small system sizes for this type of noise. For clarity reasons, only a representative sample of 10000 particles is shown in (b) and (c). Boundary conditions are periodic.
Figure 12: (color online) Same as Fig. 11 but in different geometries/boundary conditions or space dimensions. (a)-(b) vectorial noise (η=0.325, ρ=1/8, and v0=0.5), boundary conditions are periodic along the y (vertical) axis and reflecting in x. (a): a long single band travels along the periodic direction. (b): the domain size along the periodic direction is too small to accomodate bands, and a single band bouncing back and forth along the non-periodic direction is observed. (c) Angular noise, repulsive force, and periodic boundary conditions (ρ=2, η=0.23 and v0=0.3). (d) High-density sheet traveling in a three-dimensional box with periodic boundary conditions (angular noise with amplitude η=0.355, ρ=1/2 and v0=0.5).
Numerical simulations of the ordered phase dynamics (η<ηt), performed at large enough noise amplitudes, are characterized by the emergence of high-density moving bands (d=2) or sheets (d=3). Typical examples are given in Figs. 11–12. These moving structures appear for large-enough systems after some transient. They extend transversally with respect to the mean direction of motion, and have a center of mass velocity close to v0. While particles inside bands are ordered and, in the asymptotic regime, move coherently with the global mean velocity, particles lying outside bands —in low density regions— are not ordered and perform random walks.
As shown in Fig. 11a-c for angular noise dynamics (1), there exists a typical system size Lb, below which the bands or sheets cannot be observed. Numerical simulations indicate that Lb depends only weakly on the noise amplitude and is of the same order of magnitude as the crossover size marking the appearance of the discontinuous character of the transition: Lb≈L∗. It is therefore numerically easier to observe bands in the ordered phase of vectorial noise dynamics (3), as in Fig 11d.
Bands may be observed asymptotically without and with a repulsive interaction (Fig 12c) and for both kinds of noise. They appear for various choices of boundary conditions (see for instance Fig. 12a-b, where reflecting boundary conditions have been employed), which may play a role in determining the symmetry-broken mean direction of motion. For instance, bands travelling parallel to one of the axis are favoured when periodic boundary conditions are employed in a rectangular box (they represent the simplest way in which an extended structure can wrap around a torus, and are thus reached more easily from disordered initial conditions), but bands travelling in other directions may also appear, albeit with a smaller probability.
Bands can be described quantitatively through local quantities, such as the local density ρℓ(→x,t), measured inside a domain V(→x) centered around →x, and the local order parameter
φℓ(→x,t)=1v0∣∣⟨→vi(t)⟩→ri∈V(→x)∣∣. (11)
Further averaging these local quantities perpendicularly to the mean velocity (7) one has the density profile ρ⊥(x∥,t)=⟨ρℓ(→x,t)⟩⊥ and the order parameter profile φ⊥(x∥,t)=⟨φℓ(→x,t)⟩⊥, where x∥ indicates the longitudinal direction w.r.t. mean velocity. Bands are characterized by a sharp kink in both the density and the order parameter profiles (see Fig. 13a and 13c-d). They are typically asymmetric, as it can be expected for moving structures, with a rather sharp front edge, a well defined mid-height width w – which typically is of the same order as Lb – and an exponentially decaying tail with a characteristic decay length of the order of the w (Fig. 13b).
Figure 13: (color online) (a) Typical density (black line) and order parameter (dashed red line) profiles for bands in two dimensions (vectorial noise, ρ=2, η=0.6 and v0=0.5). (b) Tail of the density profile shown in (d) (black line) and its fit (blue dashed line) by the formula: ρ⊥(x∥,t)≈a0+a1(t)exp(−x∥/w), with w≈6.3 (lin-log scales). (c-d) Traveling sheet in three dimensions (angular noise, ρ=1/2, η=0.355, and v0=0.5). (c) projection of particle positions on a plane containing the global direction of motion (marked by red arrow). (d) Density (black line) and order parameter (dashed red line) profiles along the direction of motion x∥.
Large systems may accomodate several bands at the same time, typically all moving in the same direction (see for instance Fig. 11c and the density profile in Fig. 14e). However, they do not form well-defined wave trains, but rather a collection of solitary objects, as hinted by the following numerical experiments.
Figure 14: (color online) Emergence of high-density high-order traveling bands (d=2) from a spatially-homogeneous (uniformely distributed random positions) initial condition with all particle velocities oriented along the major axis of a 196×1960 domain with periodic boundary conditions. Vectorial noise of amplitude η=0.6, density ρ=2, and v0=0.5. (a): space-time plot of the density profile (time is running from left to right from t=0 to t=12000, while the longitudinal direction is represented in ordinates. Colour scale from blue (low values) to red (high values).). (b): same as (a) but at later times (from t=148000 to t=160000). (c): spatial Fourier power spectrum S of an early density profile (t=12000). (d): early time evolution of selected Fourier modes (chosen between spectrum peaks of (c)), k=10,23,28,33,41,76,121 (inset: average over 50 different runs). (e): density profile at a late time (t=160000, final configuration of (b)). (f): same as (c) but for the late time density profile of (e). The quality of this figure has been reduced to meet the ArXiv size constraint.
We investigated the instability of the density-homogeneous, ordered state in a series of numerical simulations starting from particles uniformly distributed in space but strictly oriented along the major axis in a large rectangular domain. Figure 14a,b show space-time plots of the density profile: initially flat, it develops structures with no well-defined wavelength (Fig. 14c). Density fluctuations destroy the initially-ordered state in a rather unusual way: a dynamical Fourier analysis of the density profile show a weakly-peaked, wide band of wavelengths growing subexponentially (Fig. 14d). This is at odds with a finite-wavelength supercritical instability, which would lead to a wavetrain of traveling bands. Furthermore, the asymptotic (late time) power spectra of the density profiles are not peaked around a single frequency either, but rather broadly distributed over a large range of wavenumbers (Fig. 14f). In the asymptotic regime, bands are extremely long-lived metastable (or possibly stable) objects, which are never equally-spaced (a typical late-time configuration is shown in Fig. 14e).
To summarize, the emerging band or sheet structure in the asymptotic regime is not a regular wave train characterized by a single wavelength, but rather a collection of irregularly-spaced localized traveling objects, probably weakly interacting through their exponentially decaying tails.
iv.2 Low-noise regime and giant density fluctuations
As the noise amplitude is decreased away from the transition point, bands are less sharp, and eventually disappear, giving way to an ordered state characterized by an homogeneous local order parameter and large fluctuations of the local density.
A quantitative measure of the presence, in the ordered phase, of structures spanning the dimension transverse to the mean motion (i.e. bands or sheets) is provided by the variances of the density and order parameter profiles:
Δρ2⊥(t)=⟨(ρ⊥(x∥,t)−⟨ρ⊥(x∥,t)⟩∥)2⟩∥Δφ2⊥(t)=⟨(φ⊥(x∥,t)−⟨φ⊥(x∥,t)⟩∥)2⟩∥ (12)
where ⟨⋅⟩∥ indicates the average of the profile in the longitudinal direction with respect to mean velocity. Indeed, these profile widths vanish in the infinite-size limit except if band/sheet structures are present.
Figure 15: (color online) (a-b) Time-averaged profile width for both density (a) and order parameter (b) as a function of noise amplitude in the ordered phase (angular noise, 1024×256 domain, global motion along the major axis, ρ=2, and v0=0.5). The dashed vertical blue line marks the order-disorder transition. (c-d) Typical instantaneous profiles along the long dimension of the system described in (a-b) for intermediate noise value ((b) η=0.4) and in the bandless regime ((c) η=0.15).
In Fig 15a,b, we plot these profile widths averaged over time as a function of noise amplitude. Both quantities present a maximum close to the transition point in the ordered phase, and drop drastically as soon as the disordered phase is entered. Lowering the noise away from the transition point, these profiles decrease steadily: bands/sheets stand less sharply out of the disordered background (Fig 15c). At some point (η≈0.3 for the parameters values considered in Fig 15a,b), bands rather abruptly disappear and are no more well-defined transversal objects. It is difficult to define this point accurately, but it is clear that for lower noise intensities the local order parameter is strongly homogeneous in space. Nevertheless, fluctuations in the density field are strong (Fig 15d), but can no more give rise to (meta)stable long-lived transverse structures.
Density fluctuations in the bandless regime are in fact anomalously strong: measuring number fluctuations in sub-systems of linear size ℓ, we find that their root mean-square Δn does not scale like the square root of n=ρℓd, the mean number of particles they contain; rather we find σ(n)∝nα with α≈0.8 both in d=2 and in d=3 (Fig. 16a). This is reminiscent of the recent discovery of "giant density fluctuations" in active nematics Mishra ; Narayan ; Chate2006 . However, the theoretical argument which initially predicted such fluctuations Ramaswamy cannot be invoked directly in the present case. (Indeed, the above value of α, although needing to be refined, does not seem to be compatible with the prediction α=12+1d made in Ramaswamy .) More work is needed to fully understand under which circumstances the coupling between density and order in systems of "active", self-propelled particles gives rise to such anomalous density fluctuations.
Figure 16: (color online) Giant density fluctuation and transverse superdiffusion in the bandless ordered phase. (a) anomalous density fluctuations (see text): Δn scales approximately like n0.8 (the dashed line has slope 0.8) both in two dimensions (black circles, L=256, ρ=2, v0=0.5, angular noise amplitude η=0.25) and in three dimensions (red triangles, L=64, ρ=1/2 v0=0.5, vectorial noise amplitude η=0.1, values shifted for clarity). (b) average doubling time τ2 of the transverse (w.r.t. mean velocity) interparticle distance δ⊥. Black circles: ordered bandless regime (ρ=4, angular noise amplitude η=0.2 in a rectangular box of size 1024×256). The black dashed line marks the expected growth τ2∼δ3/2⊥. Red squares: same but deep in the disordered phase (ρ=4, angular noise amplitude η=1, L=512). The dot dashed red line shows normal diffusive behavior: τ2∼δ2.
iv.3 Transverse superdiffusion
According to Toner and Tu predictions Toner1995 ; Toner1998 ; Toner1998_2 , the dynamics of the symmetry-broken ordered phase of polar active particles should be characterized by a superdiffusive mean square displacement
Δx⊥=√⟨(x⊥(t)−x⊥(0))2⟩i (13)
in the direction(s) transversal to mean velocity. In particular, in d=2 one has Toner1998_2
Δx2⊥∼tν (14)
with ν=4/3. While this analytical result has been succesfully tested by numerical simulations of models with cohesive interactions Toner1998_2 ; Gregoire2004 , numerical simulations in models without cohesion present substantial difficulties, mainly due to the presence of continuously merging and splitting sub-clusters of particles moving coherently (as discussed in Section IV.4). As a consequence, an ensemble of test particles in a cohesionless model is exposed to different "transport" regimes (w.r.t. center of mass motion) which are not well separated in time. When the mean displacement is averaged at fixed time, this tends to mask the transverse superdiffusion.
To overcome this problem, we chose to follow Vulpiani2000 and to measure τ2, the average time taken by two particles to double their transverse separation distance δ⊥. From Eq. (14) one immediately has
τ2∼δ2/ν⊥ (15)
with 2/ν=3/2 in d=2. In order to easily separate the transverse from the parallel component, we considered an ordered system in a large rectangular domain with periodic boundary conditions and the mean velocity initially oriented along the long side. The mean direction of motion then stays oriented along this major axis, so that we can identify the transverse direction with the minor axis. Furthermore, a high density and a small (angular) noise amplitude (corresponding to the bandless regime) have been chosen to avoid the appearance of large, locally disordered patches.
Our results (Fig. 16b) confirm the prediction of Toner and Tu: transverse superdiffusion holds at low-enough noise, while normal diffusion is observed in the disordered, high-noise phase. Note that the systematic deviation appearing in our data at some large scale is induced by large fluctuations in the orientation of the global mean velocity during our numerical simulations (not shown here).
We take the opportunity of this discussion to come back to the superdiffusive behavior of particles observed in the transition region Comment_gg . There, sub-clusters emerge propagate ballistically and isotropically due to the absence of a well-established global order. Particles trajectories consist in "ballistic flights", occurring when a particle is caught in one of these coherently moving clusters, alternated with ordinary diffusion in disordered regions. The mean square displacement of particles exhibits the scaling Δx2=⟨|→x(t)−→x(0))|2⟩i∝t5/3 Comment_gg . In view of our current understanding of the discontinuous nature of the transition, we now tend to believe that this isotropic superdiffusion is probably not asymptotic.
iv.4 Internal structure of the ordered region
We now turn our attention to the internal structure of the ordered regimes. As we noted in the previous section, these regimes do not consist of a single cluster of interacting particles moving coherently. Even in the case where high density bands/sheets are present, these are in fact dynamical objects made of splitting and merging clusters. Note that for the models considered here, clusters are unambiguously defined thanks to the strictly finite interaction range r0.
As noticed first by Aldana and Huepe Huepe , clusters of size n are distributed algebraically in the ordered region, i.e. P(n)∼n−μ. But a closer look reveals that the exponent μ characterizing the distribution of cluster sizes changes with the distance to the transition point. For noise intensities not too far from the threshold, when bands are observed, we find μ values larger than 2, whereas μ<2 in the bandless regimes present at low noise intensities (Fig. 17a,b).
Thus, bands are truly complex, non-trivial structures emerging out of the transverse dynamics of clusters with a well defined mean size (since μ>2). It is only in the bandless regime that one can speak, as Aldana and Huepe, of "strong intermittency". We note in passing that the parameter values they considered correspond in fact to a case where bands are easily observed (at larger sizes than those considered in Huepe ). Thus, clusters do have a well-defined mean size in their case. Consequently, the probability distribution P(φ) of the order parameter φ does not show the behavior reported in Fig. 1 of Huepe as soon as the system size is large enough. Whether in the band/sheet regime or not, P(φ) shows essentially Gaussian tails, is strongly peaked around its mean, and its variance decreases with system size (Fig. 17c,d).
Although the picture of intermittent bursts between "laminar"intervals proposed by Aldana and Huepe has thus to be abandoned, the anomalous density fluctuations reported in the previous section are probably tantamount to the strong intermittency of cluster dynamics in the bandless regime. Again, these phenomena, reported also in the context of "active" nematics Chate2006 ; Mishra ; Ramaswamy , deserve further investigation.
Figure 17: (a-b) Cluster size distributions (arbitrary units) for domain sizes L=32 (black), 128 (red) , and 512 (green) from left to right (d=2, ρ=2, v0=0.5, angular noise). (c-d) Probability distribution functions of the order parameter φ (arbitrary units) for the same parameters and system sizes as in (a-b). (The most peaked distributions are for the largest size L=512.) Left panels (a,c): η=0.1, bandless regime; Right panels (b,d), regime with bands at η=0.4.
iv.5 Phase ordering
Figure 18: (color online) Phase-ordering from disordered initial conditions (d=2, angular noise amplitude η=0.08, ρ=1/8, v0=0.5, system size L=4096). Snapshots of the density field coarse-grained on a scale ℓ=8 at times t=160, 320, and 640 from left to right.
Figure 19: (color online) Phase ordering as in Fig. 18 (L=4096 ρ=1/2, v0=0.5). (a) two-point density correlation function C(r,t)=⟨ρℓ(→x,t)ρℓ(→x+→r,t)⟩→x (coarse grained over a scale ℓ=4) as a function of distance r=|→r| at different time steps: from left to right t=50,100,200,400,800,1600. Noise amplitude is η=0.25, data have been further averaged over ≈40 different realizations. Inset: log scales reveal the intermediate near-algebraic decay and the quasi-exponential cut-off. (b) Lengthscale ξ, estimated from the exponential cut-off positions, as a function of time. Empty black circles: η=0.25 as in (a) (i.e. regime in which bands are observed asymptotically). Red full triangles: η=0.1 (i.e. in the bandless regime). The dashed black line marks linear growth.
The ordered regimes presented above are the result of some transient evolution. In particular, the bands/sheets are the typical asymptotic structures appearing in finite domains with appropriate boundary conditions. In an infinite system, the phase ordering process is, on the other hand, infinite, and worth studying for its own sake.
Numerically, we have chosen to start from highly disordered initial conditions which have a homogeneous density and vanishing local order parameter. In practice, we quench a system "thermalized" at strong noise to a smaller, subcritical, η value. Typical snapshots show the emergence of structures whose typical scale seems to increase fast (Fig. 18). During this domain growth, we monitor the two-point spatial correlation function of both the density and velocity fields. These fields are defined by a coarse-graining over a small lengthscale ℓ (typically 4). These correlation functions have an unusual shape (Fig. 19a): after some rather fast initial decay, they display an algebraic behavior whose effective exponent decreases with time, and finally display a near-exponential cut-off. As a result, they cannot be easily collapsed on a single curve using a simple, unique, rescaling lengthscale. Nevertheless, using the late exponential cut-off, a correlation length ξ can be extracted. Such a lengthscale ξ grows roughly linearly with time (Fig. 19b). Qualitatively similar results are obtained whether the noise strength is in the range where bands/sheets appear in finite boxes or not.
We note that the above growth law is reminiscent of that of the so-called model H of the classification of Halperin and Hohenberg Hohenberg . Since this model describes, in principle, the phase separation in a viscous binary fluid, the fast growth observed could thus be linked to the hydrodynamic modes expected in any continuous description of Vicsek-like models Toner1995 ; Toner_rev .
V General discussion and outlook
v.1 Summary of main results
We now summarize our main results before discussing them at a somewhat more general level.
We have provided ample evidence that the onset of collective motion in Vicsek-style models is a discontinuous (first-order) phase transition, with all expected hallmarks, in agreement with Gregoire2004 . We have made the (numerical) effort of showing this in the limits of small and large velocity and/or density.
We have shown that the ordered phase is divided in two regions: near the transition and down to rather low noise intensities, solitary structures spanning the directions transverse to the global collective velocity (the bands or sheets) appear, leading to an inhomogeneous density field. For weaker noise, on the other hand, no such structures appear, but strong, anomalous density fluctuations exist and particles undergoes superdiffusive motion transverse to the mean velocity direction.
Finally, we have reported a linear growth (with time) of ordered domains when a disordered configuration is quenched in the ordered phase. This fast growth can probably be linked to the expected emergence of long wavelength hydrodynamic modes in the ordered phase of active polar particles models.
v.2 On the role of bands/sheets
The high-density high-order traveling bands or sheets described here appear central to our main findings. They seem to be intimately linked to the discontinuous character of the transition which can, to some extent, be considered as the stability limit of these objects. In the range of noise values where they are observed, the anomalous density fluctuations present at lower noise intensities are suppressed.
One may then wonder about the universality of these objects. Simple variants of the Vicsek-style models studied here (e.g. restricting the interactions to binary ones involving only the nearest neighbor) do exhibit bands and sheets Bertin . Moreover, the continuous deterministic description derived by Bertin et al. Bertin does possess localized, propagating solitary solutions rather similar to bands Bertin_prep . Although the stability of these solutions need to be further investigated, these results indicate that these objects are robust and that their existence is guaranteed beyond microscopic "details". However, the emergence of regular, stable, bands and sheets is obviously conditioned to the shape and the boundary conditions of the domain in which the particles are allowed to move. In rectangular domains with at least one periodic direction, these objects can form, span across the whole domain, and move. But in, say, a circular domain with reflecting boundary conditions, they cannot develop freely, being repeatedly "frustrated". Nevertheless, simulations performed in such a geometry indicate that the transition remains discontinuous, with the ordered phase consisting of one or several dense packets traveling along the circular boundary. Note, though, that these packets intermittently emit elongated structures (bands) traveling towards the interior of the disk before colliding on the boundary. To sum up, bands appear as the "natural" objects in the transition region, but they may be prevented by the boundaries to develop into full-size straight objects.
At any rate, time-series of the order parameter such as the one presented in Fig. 2 clearly show that the transition is discontinuous irrespective of the geometry and boundaries of the domain, and thus of whether bands/sheets can develop into stable regular structures or not: the sudden, abrupt, jumps from the disordered state to some ordered structure are tantamount to a nucleation phenomenon characteristic of a discontinuous transition.
v.3 A speculative picture
We would now like to offer the following speculative general picture: the key feature of the Vicsek-like models studied here —as well as of other models for active media made of self-propelled particles Csahok2002 ; BenJacob ; Bertin ; Topaz2004 ; Toner_rev — is the coupling between density and order. Particles are forced to move, and, since they carry informations about the order, advection, density fluctuations and order are intimately linked. High-density means strong local order (if noise is low enough) because the many particles in a given neighborhood will adopt roughly the same orientation. The reverse is also true: in a highly-ordered region, particles will remain together for a long time and thus will sweep many other particles leading to a denser and denser group.
Figure 20: (color online) Scatter plots of local order parameter vs local density in the ordered phase (angular noise, ρ=2, v0=0.5 in a domain of size 1024×256, (same parameters as in Fig. 15a). The local quantities were measured in boxes of linear size ℓ=8. Here, the local order is represented by the angle between the orientation Θlocal of the local order parameter and the global direction of motion Θglobal. The black solid lines are running averages of the scatter plots. The red solid lines indicate the global density ρ=2 (and thus marks the percolation threshold in a two-dimensional square lattice). (a,b): η=0.1 and 0.2: in the bandless regime, the ordered plateau starts below ρ=2, i.e. ordered regions percolate. (c) approximately at the limit of existence of bands: the start of the plateau is near ρ=2. (d) at higher noise amplitude in the presence of bands.
At a given noise level, one can thus relate, in the spirit of some local equilibrium hypothesis, local density to local order. In practice, such an "equation of state" approach can be justified by looking, e.g., at a scatter plot of local order parameter vs local density. Fig. 20 reveals that, in the ordered bandless regime, such a scatter plot is characterized by a plateau over a large range of local density values corresponding to order, followed, below some crossover density, by more disordered local patches. The regions in space where local density is below this crossover level do not percolate in the bandless regime, and order can be maintained very steadily in the whole domain (this is corroborated by the fact that in spite of the large, anomalous density fluctuations, the order parameter field is, on the other hand, rather constant, see Fig. 15c). The noise intensity at which bands emerge roughly corresponds to the value where the low-density disordered regions percolate. The reamining disconnected, dense patches then eventually self-organize into bands/sheets. The emergence of these elongated structures is rather natural: moving packets elongate spontaneously because they collect many particles; superdiffusion in the directions perpendicular to the mean motion endow these nascent bands/sheets with some "rigidity". At still stronger noise, the bands/sheets are destroyed and global order disappears.
The above features are at the root of the approach by Toner and Tu Toner1995 ; Toner1998 ; Toner1998_2 . Their predictions of strong density fluctuations, transverse superdiffusion, and peculiar sound propagation properties are correct as long as bands/sheets do not exist, i.e. for not too strong noise intensities. This is indeed in agreement with their assumption that the density field is statistically homogeneous in the frame moving at the global velocity (albeit with strong fluctuations), which is only true in the bandless regime.
v.4 Outlook
The results presented here are almost entirely numerical. Although they were obtained with care, they need to be ultimately backed up by more analytical results. A first step is the derivation of a continuous description in terms of a density and a velocity field (or some combination of the two), which would allow to go beyond microscopic "details". In that respect the deterministic equation derived by Bertin et al. from a Boltzmann description in the dilute limit Bertin is encouraging. However, one may suspect that intrinsic fluctuations are crucial in the systems considered here if only because some of the effective noise terms will be multiplicative in the density. A mesoscopic, stochastic equation description is thus a priori preferable. This is especially true in view of the "giant" anomalous fluctuations present in the bandless ordered phase. These fluctuations clearly deserve further investigation, all the more so as they seem to be generic features of active particle models Mishra ; Chate2006 .
Ongoing work is devoted to both these general issues.
Acknowledgements.
Most of this research has been funded by the European Union via the FP6 StarFLAG project. Partial support from the French ANR Morphoscale project is acknowledged. We thank A. Vulpiani and M. Cencini for introducing us to the method outlined in Ref. Vulpiani2000 .
Appendix A Fluctuation-dissipation relation
Figure 21: (color online) Test of the fluctuation-dissipation relation on the vectorial model with repulsive force (ρ=2, v0=0.3, L=128). (a): susceptibility vs time at reduced noise amplitude ε=1−η/ηt=0.005, |→h|=10−2(dashed red line) and |→h|=10−3(plain black line). (b): susceptibility vs. correlation at |→h|=10−3 and ε=0.08, 0.26 and 0.44 from top to bottom. (c): effective temperature vs noise amplitude. The vertical dashed line marks the transition point.
In Czirok1997 , Vicsek et al. also studied the validity of the fluctuation-dissipation theorem and concluded, from numerical analysis, that it is violated. Here we approach this question again, using the "dynamical" approach put forward in Cuku , rather than the equilibrium used in Czirok1997 . The fluctuation-dissipation relation is expressed as:
R(t−t0)=1Teff∂C(t−t0)∂t (16)
where R is the response, C the associated correlation function, and Teff is some "effective temperature".
In both cases, at any rate, the key point is to investigate the effect of an external field on the ordering process. Since here one cannot rely on any Hamiltonian structure, the external field remains somewhat arbitrary, as it cannot be unambiguously defined as the conjugate variable of the order parameter.
In Czirok1997 , the field →h was directly confronted to the local average velocity, and the governing equation was replaced by:
This manner of introducing →h leads to an effective intensity which depends on the local ordering: →h is comparatively stronger in disordered regions (an important effect at the early stages of ordering) than in ordered regions. This could in fact prevent the necessary linear regime to occur, even at very low field values (see, for instance, Fig. 6 of Czirok1997 ). A mean-field analysis has confirmed this view, showing a logarithmic variation of the response with the field intensity 111Private communication from E. Bertin..
To bypass this problem, we have preferred to use the following equation:
→vi(t+Δt)=v0ϑ⎡⎣∑j∈Vi→vj(t)+∣∣ ∣∣∑j∈Vi→vj(t)∣∣ ∣∣→h+ηNi→ξ⎤⎦ (18)
where the vectorial noise was chosen because, as shown above, it leads more easily to the asymptotic regime. The effective intensity of the field is now proportional to the local order.
Two (scalar) response functions can be defined in our problem: the longitudinal response R∥ along the field direction, and the transverse response R⊥. We consider the former. In practice, we quenched, at time t0, a strong-noise, highly-disordered system φ(t0)≈0 to a smaller noise value and started applying the constant, homogeneous field →h immediately. We then followed the response of the system by monitoring the growth of the order parameter. We measured the susceptibility χ∥ which is nothing but the integrated response function: χ∥(t,t0)=∫tt0R∥(t,t′)dt′. In practice, we have:
χ∥(t−t0)=1|→h|→φ(t)⋅ϑ[→h]. (19)
In a well-behaved system, the susceptibility should be independent of the amplitude of the field, at least at small enough values ("linear" regime). This is what we observed, as shown in Fig. 21a.
Correspondingly, the correlation function is defined as:
C(t−t0)=1v20⟨→vi(t0)⋅→vi(t)⟩i. (20)
The fluctuation-dissipation relation (16) can then be written in its integrated form:
χ∥(t−t0)=1Teff(C(0)−C(t−t0)) (21)
In Fig. 21b, we show that χ∥ and C are related linearly in time, confirming the validity of this relation and allowing an estimation of Teff.
This well-defined —although not uniquely defined— effective temperature varies as expected in parameter space. In particular, it increases systematically with the noise strength η (Fig. 21c), although this variation is not linear. Note also, that, intriguingly, there is a small jump of Teff at the noise value corresponding to the transition in this case (ηt≃0.55).
(1) Pliny the Elder, Naturalis historia, 79.
(2) J. K. Parrish and W. M. Hamner, editors, Animal Groups in Three Dimensions (Cambridge University Press, Cambridge, 1997).
(3) E. V. Albano, Phys. Rev. Lett. 77, 2129 (1996).
(4) I. D. Couzin and J. Krause, Advances in the Study of Behavior 32, 1 (2003).
(5) A. Czirók, H. E. Stanley, and T. Vicsek, J. Phys. A 30, 1375 (1997).
(6) G. Grégoire, H. Chaté, and Y. Tu, Phys. Rev. E 64, 011902 (2001).
(7) C. Huepe and M. Aldana, Phys. Rev. Lett. 92, 168701 (2004).
(8) J. Toner and Y. Tu, Phys. Rev. Lett. 75, 4326 (1995).
(9) T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. 75, 1226 (1995).
(10) N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
(11) E. Bertin, M. Droz, and G. Grégoire, Phys. Rev. E 74, 022101 (2006).
(12) B. Birnir, J. Stat. Phys. 128, 535 (2007).
(13) H. J. Bussemaker, A. Deutsch, and E. Geigant, Phys. Rev. Lett. 78, 5018 (1997).
(14) H. Chaté, F. Ginelli, and R. Montagne, Phys. Rev. Lett. 96, 180602 (2006).
(15) I. D. Couzin, J. Theor. Biol. 218, 1 (2002).
(16) I. D. Couzin, J. Krause, N. Franks, and S. Levin, Nature 433, 513 (2005).
(17) Z. Csahók and T. Vicsek, Phys. Rev. E 52, 5297 (1995).
(18) A. Czirók, A.-L. Barabási, and T. Vicsek, Phys. Rev. Lett. 82, 209 (1999).
(19) A. Czirók, M. Vicsek, and T. Vicsek, Physica A 264, 299 (1999).
(20) Y. L. Duparcmeur, H. Herrmann, and J. P. Troadec, J. Phys. I France 5, 1119 (1995).
(21) G. Grégoire, H. Chaté, and Y. Tu, Phys. Rev. Lett. 86, 556 (2001).
(22) G. Grégoire, H. Chaté, and Y. Tu, Physica D 181, 157 (2003).
(23) J. Hemmingsson, J. Phys. A 28, 4245 (1995).
(24) H. Levine, W.-J. Rappel, and I. Cohen, Phys. Rev. E 63, 017101 (2000).
(25) A. S. Mikhailov and D. H. Zanette, Phys. Rev. E 60, 4571 (1999).
(26) A. Mogilner and L. Edelstein-Keshet, J. Math. Biol. 38, 534 (1999).
(27) O. J. O'Loan and M. R. Evans, J. Phys. A 32, 99 (1999).
(28) M. R. d'Orsogna, Y. Chuang, A. Bertozzi, and L. Chayes, Phys. Rev. Lett. 96, 104302 (2006).
(29) N. Shimoyama, K. Sugawara, T. Mizuguchi, Y. Hayakawa, and M. Sano, Phys. Rev. Lett. 76, 3870 (1996).
(30) R. A. Simha and S. Ramaswamy, Physica A 306, 262 (2002).
(31) R. A. Simha and S. Ramaswamy, Phys. Rev. Lett. 89, 058101 (2002).
(32) B. Szabó et al., Phys. Rev. E 74, 061908 (2006).
(33) J. Toner and Y. Tu, Phys. Rev. E 58, 4828 (1998).
(34) J. Toner, Y. Tu, and M. Ulm, Phys. Rev. Lett. 58, 4828 (1998).
(35) C. M. Topaz and A. L. Bertozzi, SIAM J. Appl. Math. 65, 152 (2004).
(36) C. Topaz, A. Bertozzi, and L. M.A., Bull. Math. Bio. 68, 1601 (2006).
(37) T. Vicsek, A. Czirók, I. J. Farkas, and D. Helbing, Physica A 274, 182 (1999).
(38) G. Grégoire and H. Chaté, Phys. Rev. Lett. 92, 025702 (2004).
(39) M. Nagy, I. Daruka, and T. Vicsek, Physica A 373, 445 (2006).
(40) M. Aldana, et al., Phys. Rev. Lett. 98, 095702 (2007).
(41) S. Lübeck, Int. J. of Mod. Phys. B 18, 3977 (2004).
(42) P. Marcq, H. Chaté, and P. Manneville, Phys. Rev. Lett. 77, 4003 (1996).
(43) P. Marcq, H. Chaté, and P. Manneville, Phys. Rev. E 55, 2606 (1997).
(44) K. Binder, Monte carlo investigations of phase transitions and critical phenomena, in Phase transitions and critical phenomena, edited by C. Domb and M. S. Green, Academic Press, 1976.
(45) V. Privman, editor, Finite size scaling and numerical simulations of statistical systems (ed. World scientific, Singapore, 1990).
(46) K. Binder, Rep. Prog. Phys. 60, 487 (1997).
(47) B. Efron, The Jackknife, The Bootstrap and other Resampling Plans (SIAM, Philadelphia, 1982).
(48) J. C. Niel and J. Zinn-Justin, Nucl. Phys. B 280, 355 (1987).
(49) C. Borgs and R. Koteckỳ, J Stat. Phys. 61, 79 (1990).
(50) H. Chaté, F. Ginelli, and G. Grégoire, Phys. Rev. Lett. 99, 229601 (2007).
(51) S. Mishra and S. Ramaswamy, Phys. Rev. Lett. 97, 090602 (2006).
(52) V. Narayan, S. Ramaswamy, and N. Menon, Science 317, 105 (2007).
(53) S. Ramaswamy, R. A. Simha, and J. Toner, Europhys. Lett. 62, 196 (2002).
(54) G. Boffetta, A. Celani, M. Cencini, G. Lacorata, and A. Vulpiani, Chaos 10, 50 (2000).
(55) P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
(56) J. Toner, Y. Tu, and S. Ramaswamy, Annals Of Physics 318, 170 (2005).
(57) E. Bertin, M. Droz, and G. Grégoire, Hydrodynamic equations for self-propelled particles: a derivation from the microscopic dynamics, in preparation, 2007.
(58) Z. Csahòk and A. Cziròk, Physica A 243, 304 (2002).
(59) E. Ben-Jacob, I. Cohen, and H. Levine, Adv. in Phys. 49, 395 (2000).
(60) L. F. Cugliandolo, J. Kurchan, and L. Peliti, Phys. Rev. E 55, 3898 (1997).


text:
US > Indiana > Parke County > Bloomingdale, IN 47832 > Places of Interest Near Indiana ZIP Code 47832 > Top 20 Popular Places
Indiana ZIP Code 47832 Most Popular Places
Most Popular Places near Indiana ZIP Code 47832
Search Results: 20 Popular Places near Indiana ZIP Code 47832 using a 16-mile search radius.
Changing the search radius will alter ranking order and number of search results.
Selecting a Search Caterory will also alter the ranking order and may change the number of search results.
Search results are in ranked order based on positive reviews, search queries, and other user data.
Search Results - Ranked by Popularity (All Categories)
Location: Indiana ZIP Code 47832. Search Type: All Categories. Found: 20 Results
(1.) Wheel House Donuts (6.0 miles)
(2.) Italian Pie & Bakery (12.0 miles)
(3.) Dairy Queen (5.5 miles)
Snacks/Fast food
(4.) Dairy Queen Grill & Chill (5.5 miles)
(5.) Burger King (5.9 miles)
(6.) Scooters Pub and Eatery (14.6 miles)
(7.) Burger King (12.5 miles)
(8.) Dairy Queen (14.7 miles)
(9.) Bloomin Barn Antiques (2.0 miles)
(10.) McDonald's (15.5 miles)
(11.) Taco Bell (15.6 miles)
(12.) Papa John's Pizza (15.6 miles)
(13.) Old Goat Trading Post (2.5 miles)
(14.) Up the Creek (3.4 miles)
(15.) Up the Creek Boat-Ique (3.6 miles)
(16.) Turkey Run Gas and Grill (3.8 miles)
Petrol Station
(17.) DTP Entertainment (4.1 miles)
Theatre, Music & Culture
(18.) Under the Arch Restaurant (4.2 miles)
(20.) Turkey Run State Park Inn (4.3 miles)
Change the Search Radius
NOTE: The above search results used a 16-mile search radius (SR).
Changing the SR will often change the rankings and may alter the number of results. If we can not find at least 8 popular places and the SR is not already set to 16, we'll automatically increase it to 16 and repeat the search.
2 miles 4 miles 8 miles 16 miles
Search for Popular Places near Indiana ZIP Code 47832 by CATEGORY
Eat and Drink (restaurants, coffee shops, etc.)
Going-Out / Entertainment (bars, nightclubs, theatres, cinemas)
Sights and Museums (special interest places, e.g., tourist attractions, museums, etc.)
Leisure and Outdoor parks, and places for sports, recreation and other outdoor activities
Search by Name for a Specific Business or Organization
Search the Indiana ZIP Code 47832, IN Area by Business or Organization Name:
The Data Source for Most Popular Places & Business Listings
The data for our Most Popular Places and business listings is supplied by HERE. They pioneered in-car navigation systems over 30 years ago - first as NAVTEQ, later as Nokia, and now as HERE.
HERE maps and information products provide in-vehicle navigation and location services to more than 25 leading automobile brands. Their products are in use in over 80% of the cars and trucks being driven today in North America.
HERE draws on more than 80,000 data sources and relies on local source data for information about local business and attractions.
View ALL Indiana ZIP Code 47832 Content

redpajama_set_name: RedPajamaCommonCrawl


text:
Baby Looney Tunes is an American animated television series taking place in an alternate universe depicting the Looney Tunes characters as infant versions of themselves. It was produced by Warner Bros. Animation. The show is similar to Muppet Babies, with the main characters taken care of by Granny. The show premiered on WB stations usually before or after the Kids' WB! block from 2001 to 2002 and continued to air on Cartoon Network from 2002 until 2006. Reruns of the show returned August 17, 2015 to help promote the premiere of the new Uncle Grandpa special, Uncle Grandpa Babies. The program stayed on the network after that week. As of September 2015 "Baby Looney Tunes" airs on Boomerang.
"Cat-Taz-Trophy"/"Duck, Mother, Duck!" (1.7 [9/24/02]): Theme Song and Star...
"Taz in Toyland"/"A Secret Tweet" (9/16/02): Theme Song and Start of First ...
Brazilian Portuguese Intro
Original intro.
Granny: "My little ones."
Added By: RGSwell95
Taz: "Do it again! Do it again! Do it again!"
Bugs: "Surf's up, doc!"
Babies: "Hi Granny! How the (squeak squeak) are ya?!"
Locate The Duplicate Entry For Baby Looney Tunes
Noveltoons
Roger Rabbit: "PPPPPleese!"
Who Framed Roger Rabbit
dalmatianlover
Disney's 1 Saturday Morning
Take a look at this memorable moment of Saturday Morning Cartoon History.
Eletta2217
Why aren't there any good Toys to buy?
My 2014 Christmas Rant
drooks
My Favorite TV Stations Part 2
A comprehensive review of why Nickelodeon will forever reign supreme - Part 1 of 4
Major Video and the NES, part 6
Almost at the finish line
16 WORST TV Shows IMO
Or more than likely...the shows that annoyed me since I watched TV
ccann702
Honey, I shrunk the toons...
The growing trend of shrinking classic characters.
TreyVore
Great shows: Tiny Toons
TreyVore remembers Tiny Toons.
Nostalgia_Junkie
My Top 5 Happy Meal Toys!
A list of my top 5 favorite Happy Meal toys of all time!
retrofan01
My Favorite Happy Meal Toys from 1990 to 1998
Find Out What Happens...When Happy Meals Stop Being for XYers...And Start Getting Yer Geared
'80s and '90s Worst cartoons
My personal picks for worst cartoons of the 1980s and 1990s.
Nickelodeon - Nick Is Every Day promo


text:
The Republic of Honduras has many historical monuments relating to its pre-Columbian native past, as well as its Spanish colonial era and republican period. The law for the Protection of the Cultural Heritage of Honduras considers examples of historical monuments, fine furniture, and expressions of folk art (folclórica) worthy of preservation.

Background
The Historical Monument and National Heritage act was passed by the National Congress of Honduras. It is administered by the Honduran Institute of Anthropology and History (IHAH) with legal status and its own assets. Under Honduran law, all archaeological and historical sites and objects are the property of the people of Honduras, with the Instituto Hondureño de Antropología e Historia the guardian of this commonwealth. IHAH is charged with conducting and regulating research, with managing both the movable property and sites, and with disseminating knowledge of this patrimony to the people.

A special office, the Fiscalia de Etnias, currently occupied by Jany del Cid (2012), prosecutes violations of the law concerning national patrimony. Controversially, the state government of Honduras does not actually pay the bill for most of this important work. It pays less than a third of the budget. The rest comes mainly from admission fees.

List of historical monuments

Historical buildings of Tegucigalpa

Historical buildings of Juticalpa

References

Honduran culture
Historic sites in Honduras

redpajama_set_name: RedPajamaWikipedia


What are you trying to tackle? Browse or search the list below to see if a group of Innovative Farmers is working on it. You can read all about it, follow their progress or even join in. If you can't find what you're looking for, get in touch – we're always looking for ideas.
Reducing the use of synthetic pesticides is good for the environment and farmers' bottom lines. What different methods of tackling pests and diseases are available and how do they work in different situations?
Projects supporting the creation and development of rural communities.
Field labs in this topic are focussed on increasing resilience in dairy farming by using benchmarking and KPI tools to analyse performance and identify areas for improvement.
This topic is investigating techniques to improve hen welfare and production specifically by tackling injurious feather pecking in flocks and reducing stress.
The challenge is to find alternatives to peat that demonstrate comparable efficiency to provide growers with affordable, reliable and resilient growing mediums.
Soils are degrading and affecting the long-term ability for farmers to keep up food production. Reducing compaction and erosion and increasing soil organic matter are the key focus areas of this topic.
Encroachment of soft rush is a growing problem for many land managers. Rushes can take over grass/clover swards with knock-on effects for livestock productivity and profit margins.
Reducing the use of synthetic herbicides is good for the environment and farmers' bottom lines. What different methods of tackling weeds are available and how do they work in different situations?
Using the understory effectively in orchards makes the most out of the land. Improving floor management gives the potential to profit from two crops, increase nutrient enrichment, water management, natural pest control, pollination, or reduce spoiling.
The majority of crop varieties grown in organic and other low-input systems are bred for the situations where synthetic fertilisers and pesticides are used. Which varieties are most effective – on farm and at scale – in lower input systems?
Feed is one of the largest variable costs to farmers. An animal's health, stress, nutrition, weight and growth rate all affect feed efficiency. Finding ways to improve efficiency sustainably is crucial for better returns.
Using antibiotics is a major cost to farmers and frequent use contributes to bacterial resistance – a problem in farming and human medicine. Promoting animal health and reducing the need for treatment are important in making dairy production more profitable, safer and more resilient.
Products based on naturally occurring substances are appearing on the UK market claiming to be able to reduce the requirements for inputs for plant growth. How effective and practical are these products are in the field?
Laura liked Investigating alternative methods in terminating cover crops to reduce reliance on glyphosate.
Jasmine added an Image in Investigating alternative methods in terminating cover crops to reduce reliance on glyphosate.
Jasmine commented on Investigating alternative methods in terminating cover crops to reduce reliance on glyphosate.
Laura followed Investigating alternative methods in terminating cover crops to reduce reliance on glyphosate.
Katherine shared Can music improve hen welfare?
Katherine liked Can music improve hen welfare?


Creating a perfectly manicured lawn takes time, and retaining all your hard work takes routine maintenance. Our lawn mower tips will teach you how to use a lawn mower, and provide the dos and don'ts of lawn mower operation. Learning how to use a snowblower and log splitter properly is an important task that requires special care and attention. With our easy-to-follow instructions you'll be able to review safety precautions and tips to help make using your snowblower and log splitter a breeze.
Discover the difference between mulching blades and high-lift cutting blades. Use mulching lawn mower blades instead of high-lift blades with mulch kits.
Discover the difference between mulching blades and high lift cutting blades, and the advantages of each type of blade.
Holiday light stringing ideas that will highlight your yard and landscape beds, as well as quick pruning and trimming tips for neatening up your trees and shrubs.
Holiday decorating ideas to highlight your yard and landscape beds, and tips for using your outdoor power equipment to create the perfect setting.
Buy lawn mower parts online with the Parts Finder from MTD. The Parts Finder helps you quickly locate the outdoor power equipment parts you need to keep your machines running smoothly. Order MTD riding lawn mower parts and other product parts today.
Watch the video or follow the step-by-step instructions on how to seal a hole using Tire Science sealant, courtesy of MTD Parts.
Learn to care for your yard, lawn mower parts and other outdoor power equipment all year long with MTD. Our lawn care tips keep your yard looking its best and your equipment in working order. Get seasonal lawn care advice from MTD.
Learn how to use a garden tiller from MTD. From garden tiller parts to tiller operation tips, this list will help you maintain and use your garden tiller for years to come. Check out these tips for using a tiller and get started today.
Learn how to use a push mower from MTD. Whether you're wondering how to operate a push mower or what lawn mower parts you need to keep your machine in great shape, these tips will help get you started. Find out more about operating a push mower today.
Install a mulching kit with these easy steps. A mulching lawn mower blade or mulch kit is a riding lawn mower attachment used to cut grass, leaves and twigs on your lawn to compost to improve soil quality.
Protect yourself from winter elements. Follow these step-by-step instructions from MTD Genuine Parts for assembling and installing a universal snow cab on your snow blower.
Get tips on choosing a snow blower from MTD Genuine Parts. Before buying a snow thrower, watch our video to learn how single-stage, two-stage, and three-stage units differ and which is best for your snow removal needs.


I told my brother we were going to the middle of rural Normandy for 10 days and that we planned to do nothing but read. He laughed! I guess that's not his idea of a good vacation. Then we had a conversation about how his vacations are all about entertaining the kids. I sympathized and told him it wouldn't always be that way.
So what did I do on my summer vacation besides read?
Saw a 13th-century church in the village of Rugles on a gray (but not cold) day. You get those in Normandy, in the summer.
Contemplated seedpods that fascinated me (their wings) and delighted me (their color scheme, which I promptly decided I would do my dream living room in).
Pitted cherries. A tree planted by Vincent's dad about 20 years ago bore fruit for the first time this year. The best cherries I've ever had in my life, easily 20 years' worth of stockpiled cherry flavor in 'em. Can't you see it? We made a cobbler. There are no words.
Geek tip: If you have an image whose colors make you salivate, and if you have Photoshop, you can break the image down into a color palette you can use for web design, or home décor, or putting an outfit together, or whatever! Choose the Filter menu, then Pixelate, then Mosaic. Make the squares big till you get a workable palette.
Welcome to my living room in the little house in the little field by the little tree. This is all I have so far, but it'll do. I don't really need much.
I am pink with envy Pam.
David, you are someone who would like a vacation like this, I'm sure.


This tall mirror is going to give you lots of flexibility. You'll be able to hang it up like a normal mirror. However, due to its size, you'll also be able to rest this mirror on the ground at a slight angle to give you a full body reflection. It also means you won't have to worry about finding a stud for hanging it, or the mirror ever coming off the wall.
The outside frame of this mirror is polished metal which gives it a very nice chic appearance. It's also a nice compliment to the glass. This glass is high quality and it won't scratch easily. The modern appearance of this mirror makes it the perfect fit for any contemporary living room or bedroom. The beveled edges give it a craftsman appearance that is sure to gain you compliments the next time you have guests over.


Historians agree that in October of 1962, U.S. President John F. Kennedy and Soviet leader Nikita Khrushchev squared off in a showdown that brought the world to the brink of nuclear war.
Sergei Khrushchev, son of the Soviet leader and a professor at Brown University, says the crisis was ultimately resolved peacefully because both leaders were rational men.
"We were very lucky that the two leaders were balanced and reasonable and their policy was not shoot first then think, but first think, then, second time, think and maybe don't shoot at all," he said.
Shortly after the missile crisis, Kennedy and Khrushchev established a "hotline" between Washington and Moscow providing for direct communications between the White House and the Kremlin. It is in operation to this day. The two men also signed a test ban treaty, ending nuclear testing everywhere but underground.
"But then Kennedy was assassinated, then Khrushchev was ousted from power," said Professor Khrushchev. "I think that if these two leaders would have been in power longer, it is a big possibility that the Cold War would have been over. But history decided in a different way and we returned to the Cold War and all this crazy arms race until the Gorbachev time," he said.
Experts such as Graham Allison of Harvard University say there are lessons to be learned from the crisis, which even President Kennedy admitted.
"The president is going to be faced with an option between acquiescing in Iran acquiring a nuclear bomb - that's one option," he said. "Or alternatively, attacking Iran to prevent it acquiring a nuclear bomb - so attack or acquiesce.
Allison said the U.S. administration must search for a third option, as did President Kennedy during the Cuban Missile Crisis.
"I would hope that immediately after the election, the U.S. government will also turn intensely to the search for something that's not very good - because it won't be very good - but that is significantly better than attacking on the one hand or acquiescing on the other," he said.
For his part, Professor Khrushchev favors dialogue.
If the Iranian crisis cannot be resolved peacefully and it comes to war, says Professor Khrushchev, the United States would win. But, asks the son of the former Soviet leader, at what price?
Is The Tide Shifting?: Are We Witnessing A Sea Change In US/Iran Relations?


text:
40 Years of James Caldwell High School
Jessica Melnikoff
James Caldwell High School has changed immensely over the past 40 years. We all know that life was different in the 1980s compared to now. As society has changed, the dynamic of the school has also developed over the decades. In the past two years, Mrs. Callaghan and Mr. Edwards have been recognized for their 40 years of service at the district. These two teachers offer a unique, first-hand, perspective on how James Caldwell High School has evolved over the past 40 years.
Photo Courtesy of The Clarion 1984
The first interview with Mrs. Callaghan:
Jessica: "In a few words, how would you describe what James Caldwell was like in the 1980s?"
Mrs. Callaghan: "It was a little bit wilder and freer. Not that this was a good thing. Kids did things that we would consider pretty outrageous today."
Jessica: "How has the Cheerleading team changed over the past 40 years?"
Mrs. Callaghan: "Back when I first started working here, cheerleading was much more 'stomp it shake it.' In order to make the team, you had to be able to do a flying split. I can't tell you how many kids got hurt."
Jessica: "What is, in your opinion, the most significant change that our school has gone through?"
Mrs. Callaghan: "The inclusion of technology. Everything now is computer-based. To even think last year we had to teach through computers via zoom. Physically, the building has changed a lot. The library was totally different, there were no porticos, Bonnel field was not turf."
Jessica: "How has student behavior in the classroom changed?"
Mrs. Callaghan: "It's less extraverted and more introverted towards cell phones. Kids are more passive nowadays. Pre-cellphones, kids would talk to each other and pass notes. Now they text each other and share memes."
The second interview with Mr. Edwards:
Jessica: "How has the increase in technology over the past 40 years affected your teaching?"
Mr. Edwards: "Are our copy machines considered technology? You can definitely do more things now. It had a big impact on mathematics classrooms. Calculators are better now. The computers in the past few years changed and it got us through the COVID year. You almost had to program the old computers to get them to work. For me making my lessons up, computers have helped, and it's a lot less painful for students. Kids have grown into this technology. I have been able to see amazing things like the copy machine, cellphones, computers develop. Kids don't understand how awesome that stuff is. It makes school a lot easier."
Jessica: "What is your favorite teaching memory?"
Mr. Edwards: "I think that it's any time overall as a teacher when that kid accomplishes something they didn't think they can do. But you see them work hard and get what they want or get into the college that they wanted to go to. That's what makes my day."
Over the past 40 years our school has grown into what it is today. Technology has dramatically improved the quality of education. Our society has given people more of an opportunity to participate in sports, including the expansion of sports for female athletes. And overall, James Caldwell has evolved into a better learning environment for students.
#school
#teacher
JCHS Thanksgiving Fundraiser in Honor of Jim Brown


Saint Albert the Great School is pleased to announce that it has AGAIN been selected as one of Cleveland's Top Work Places!!
The Cleveland Plain Dealer will publish the complete list of Top Workplaces on Sunday, June 19, 2016. Please take time to celebrate the GREAT news!


The Federal Emergency Management Agency will conduct a nationwide test of the Emergency Alert System and Wireless Emergency Alert on Sept. 20.
LUXEMBURG — The Federal Emergency Management Agency, in coordination with the Federal Communications Commission, will conduct a nationwide test of the Emergency Alert System (EAS) and Wireless Emergency Alert (WEA) on Thursday, Sept. 20, according to a press release from the Kewaunee County Emergency Management Department.
The wireless test is scheduled to take place at 1:18 p.m., followed by the EAS test at 1:20 p.m. The backup date for these tests is Oct. 3.
"The test is intended to ensure public safety officials have the methods and system that will deliver urgent alerts and warnings to the public in times of an emergency or disaster" on a national level, the release said.
It is the first national WEA test, for emergency messages sent to cell phones, and the fourth nationwide EAS test.
WEA is used to warn the public about dangerous weather, missing children and other critical situations through alert on cell phones. The test will be sent through the Integrated Public Alert and Warning System as part of the infrastructure that automatically authenticates alerts.
Customers of wireless providers who participate in WEA and own a compatible wireless phone can receive targeted alerts of imminent threats to safety in their area through unique tones and vibrations. The national WEA test will use the same special tones and vibrations.
The EAS test is scheduled to last one minute on participating radio and television stations, including satellite radio and TV, and wireline video providers.
For more information about wireless emergency alerts, visit www.ready.gov/alerts.