Quantitative Assessment of the Toner and Tu Theory of Polar Flocks

Abstract

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. arXiv:1908.03794v1 [cond-mat.stat-mech]

Small Obstacle in a Large Polar Flock

“…This term refers to the homogeneous collective motion, resulting from spontaneous rotational symmetry breaking, of self-propelled particles locally aligning their velocities, as in the Vicsek model [14][15][16]. Remarkably, polar flocks exhibit true long-range polar order even in two space dimensions (2D), as argued by Toner and Tu, who also predicted the scaling structure of their space-time fluctuations [5,[17][18][19]. Since these early works, a wealth of results have been obtained on particle-level models, hydrodynamic theories, and even experimental realizations of polar flocks [20][21][22][23][24][25].…”

“…In numerical and theoretical studies of active matter, 'polar flocks' continue to play a central role (see, e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13] for recent examples). This term refers to the homogeneous collective motion, resulting from spontaneous rotational symmetry breaking, of self-propelled particles locally aligning their velocities, as in the Vicsek model [14][15][16].…”

“…This may seem an odd choice for modeling collisions with an obstacle since particles have no physical size, and discrete-time dynamics look inconvenient. Our motivation here, though, is to be able to reach the asymptotic regime of polar flocks, which is known to require very large system sizes, even for Vicsek-like systems [5], and remains inaccessible with more realistic particles.…”

Long-Range Nematic Order in Two-Dimensional Active Matter

“…Whereas it is now well known, notably thanks to the seminal work by Toner and Tu, that long-range polar order can arise in 2D active systems [3][4][5][6][7][8][9][10][11], the debate has remained opened for active nematics: On the one hand, theoretical results conclude that nematic order can at best be quasi-long-range [12], as in equilibrium, albeit with important differences [13][14][15][16][17][18][19]. On the other hand numerical and experimental results obtained on self-propelled particles without spontaneous velocity reversals yielded convincing data demonstrating true longrange nematic order over a large range of scales [20,21].…”

“…At the linear fixed point, 9 of the 10 nonlinear terms in Eqs. (8) scale like b (4−d)/2 , i.e. are relevant in d ≤ d c ≡ 4 (the exception is ω 5 ).…”