Entropic effects in a nonequilibrium system: Flocks of birds

When European starlings come together to form a flock, the distribution of their individual velocities narrows around the mean velocity of the flock. We argue that, in a broad class of models for the joint distribution of positions and velocities, this narrowing generates an entropic effect that opposes the cohesion of the flock. The strength of this effect depends strongly on the nature of the interactions among birds: If birds are coupled to a fixed number of neighbors, the entropic forces are weak, while if they couple to all other birds within a fixed distance, the entropic effects are sufficient to tear a flock apart.

Figure 1

Graphs corresponding to a two-dimensional flock with $N=4$ birds and $J=1$. The number of connected components, the adjacency and Laplacian matrices, and the Laplacian eigenvalues are also shown. (a) Topological case: Here $n_c=2$, and the graph $G$ is a directed graph with one connected component. The Laplacian has only one null eigenvalue, corresponding to one connected component in the graph [38]. (b) Metric case: $r_c$ is the radius of the dashed circle centered around bird 4, and the graph $G$ is an undirected graph with two connected components. The first connected component is given by bird 1 (in red), and the second connected component is given by birds $2,3,4$ (in green). The adjacency and Laplacian matrices are block matrices composed of two blocks corresponding to the first and second connected component depicted in red and green, respectively. The Laplacian has two null eigenvalues, corresponding to the two connected components in $G$.

Figure 2

Probability of finding a connected flock. Connection probabilities $p_c^{\mathrm{ran}},p_c^{\mathrm{mot}}$, from random positions and from the motional potential, respectively, as functions of the number of neighbors $n_c$, both in the topological case (a) and in the metric case (b), where the continuous curves are shown to guide the eye. In the metric case, the average number of neighbors $n_c$ is obtained from the interaction range $r_c$ from the relation $n_c=\rho \frac{4\pi }{3}{r}_c^3$, where $\rho$ is the average density of the flock. Error bars have been estimated with the bootstrap method [39].

Figure 3

Fraction of birds in the largest connected component of a flock. The fraction $\varphi$ from the motional potential and from random positions is plotted as a function of the number of neighbors $n_c$, both in the topological case (a) and in the metric case (b). In the metric case, the average number of neighbors $n_c$ is obtained as in Fig. 2.

Figure 4

Probability of finding a connected flock with the simplified Monte Carlo method on graphs in the metric case. Connection probabilities ${\mathcal{p}}_c^{\mathrm{ran}},{\mathcal{p}}_c^{\mathrm{mot}}$ from random graphs and from the motional potential, respectively, as functions of the number of neighbors $n_c$. In the topological case, ${\mathcal{p}}_c^{\mathrm{mot}}$ is equal to 1 for all $n_c>1$, and it is not shown here.

Figure 5

Crossover values $n_c^{*}$ of the number of neighbors in the metric case, given by ${\mathcal{p}}_c(n_c^{*},N)=1/2$, for random graphs and for graphs drawn from the motional potential, as functions of the flock size $N$. The fitting functions $n_c^{*}=alogN+b$ are shown to guide the eye.

Figure 6

Fraction of birds in the largest connected component of a flock with the simplified Monte Carlo method on graphs in the metric case. The fraction $\phi$ is plotted as a function of the number of neighbors $n_c$ from the motional potential and from random positions, respectively.