Quantitative Assessment of the Toner and Tu Theory of Polar Flocks

We present a quantitative assessment of the Toner and Tu theory describing the universal scaling of fluctuations in polar phases of dry active matter. Using large-scale simulations of the Vicsek model in two and three dimensions, we find the overall phenomenology and generic algebraic scaling predicted by Toner and Tu, but our data on density correlations reveal some qualitative discrepancies. The values of the associated scaling exponents we estimate differ significantly from those conjectured in 1995. In particular, we identify a large crossover scale beyond which flocks are only weakly anisotropic. We discuss the meaning and consequences of these results.

Figure 1

Equal-time correlations in 2D (left) and 3D (right). Insets contain the same data as their main panel but rescaled by ${(q/2\pi )}^{\sigma }$, where $\sigma$ is an estimated exponent. The purple vertical lines mark the crossover scale $q_c/(2\pi )$ (see the text). Symbol code for linear system size: in 2D, squares, triangles, and dots for $L=2000$, 4000, and 8000, respectively; in 3D, squares, triangles, diamonds, and dots for $L=100$, 200, 500, and 960, respectively. (a),(b) Velocity correlations in the transverse (red, lower data) and longitudinal (blue, upper data) directions. Insets: $\sigma =1.33$ (2D, $\perp$), 1.40 (2D, $\parallel$), and 1.77 (3D, $\perp$ and $\parallel$). (c),(d) Density correlations in the transverse (red, lower data) and longitudinal directions for different values of $q_{\perp }^{*}$ (upper curves, shifted upward for clarity). In (c), Roman numerals indicate the three longitudinal scaling regimes discussed in the text (upper curves). Insets: $\sigma =1.33$ (2D, $\perp$) and 1.73 (3D, $\perp$). (e),(f): Longitudinal data of (c),(d) rescaled by ${q_{\perp }^{*}}^{-\mu }$ with $\mu =1$ and $\frac{1}{2}$ in 2D and 3D, respectively. Points in the regime $q_{\perp }\gg q_{\parallel }$ are shown in thin dashed lines for clarity. Insets: $\sigma =2.40$ (2D) and 2.27 (3D).

Figure 2

(a) 2D peak widths as functions of $q$ in the transverse (red, upper curves) and longitudinal (blue, lower curves) directions; insets: the same data rescaled by ${(q/2\pi )}^{\sigma }$ with, respectively, $\sigma =1.33$ and 1.40. (Diamonds, squares, and triangles, respectively, correspond to system sizes $L=1000$, 2000, and 4000.) (b) The same as (a) but in 3D and only in the transverse direction for the peak related to ${\mathbf{v}}_T$ (see [33]) at sizes $L=100$ (squares) and 200 (dots).