Incompressible polar active fluids in the moving phase in dimensions d > 2

Abstract

We study universal behavior in the moving (polar ordered) phase of a generic system of motile particles with alignment interactions in the incompressible limit for spatial dimensions d>2. Using a dynamical renormalization group analysis, we obtain the exact dynamic, roughness, and anisotropy exponents that describe the scaling behavior of such incompressible systems. This is the first time a compelling argument has been given for the exact values of the anomalous scaling exponents of a flock moving through an isotropic medium in d>2. We note, however, that these exponents are not continuous at d=2; that is, the exponents do not take on the values given by equations (2) in d=2 [12]. This is because, as we will show in this paper, and as also discussed in [12], the number of soft modes in incompressible flocks is d−2. Because this vanishes in d=2, the 2D case is special, and is not characterized by the exponents (2).Results identical to (2) have been conjectured for compressible polar active fluids [2, 4] and Malthusian flocks [15], but only in the latter case, and even there only in 2D, can a compelling argument for them be made. The results we present here make incompressible flocks the first polar active system with underlying isotropic symmetry in d>2 for which the exact scaling laws have been determined.

Giant number fluctuations in dry active polar fluids: A shocking analogy with lightning rods

“…Furthermore, it can also be shown that for both "Malthusian" flocks [12] (that is, polar ordered dry active fluids without number conservation due to " birth and death" of the active particles) and incompressible polar ordered dry active fluids [13] , for all dimensions 2 ≤ d ≤ 4, these exponents are given by…”

A novel nonequilibrium state of matter: a $d=4-ε$ expansion study of Malthusian flocks

“…In other dimensions, in particular d = 3, little beyond the existence of anomalous hydrodynamics can be said. Interestingly, one system about which more can be said is incompressible flocks [12,13]; i.e., polar ordered active fluids in which the density is fixed, either by an infinitely stiff equation of state, or by long-ranged forces. For these systems, it is possible to obtain exact exponents for all spatial dimensions; as for number conserving systems with density fluctuations, these prove to be anomalous for spatial dimensions d in the range 2 ≤ d ≤ 4.…”

“…Overall, the theoretical situation is therefore still quite unsatisfactory: we only have the scaling laws for flocks if they either are incompressible (which requires either infinitely strong, or infinitely ranged, interactions), or in d = 2. And in the cases in which we do know the exponents, their values are either the canonical ones (I.1) [12,14], or those from the (1+1)-dimensional KPZ model [13].…”

Phase ordering, topological defects, and turbulence in the 3D incompressible Toner-Tu equation

“…Although the exact biological or environmental factors that trigger such transition depend on the particular species, physicists have successfully used the theory of dry-active matter to study disorder-order phase transition in these systems [6,7]. Theoretical studies have revealed that for an incompressible flock, where we can ignore density fluctuations, the order-disorder phase transition is continuous [8][9][10].…”