Hydrodynamics and phases of flocks

Abstract

We review the past decade's theoretical and experimental studies of flocking: the collective, coherent motion of large numbers of self-propelled "particles" (usually, but not always, living organisms). Like equilibrium condensed matter systems, flocks exhibit distinct "phases" which can be classified by their symmetries. Indeed, the phases that have been theoretically studied to date each have exactly the same symmetry as some equilibrium phase (e.g., ferromagnets, liquid crystals). This analogy with equilibrium phases of matter continues in that all flocks in the same phase, regardless of their constituents, have the same "hydrodynamic" - that is, long-length scale and long-time behavior, just as, e.g., all equilibrium fluids are described by the Navier-Stokes equations. Flocks are nonetheless very different from equilibrium systems, due to the intrinsically nonequilibrium self-propulsion of the constituent "organisms." This difference between flocks and equilibrium systems is most dramatically manifested in the ability of the simplest phase of a flock, in which all the organisms are, on average moving in the same direction (we call this a "ferromagnetic" flock; we also use the terms "vector-ordered" and "polar-ordered" for this situation) to exist even in two dimensions (i.e., creatures moving on a plane), in defiance of the well-known Mermin-Wagner theorem of equilibrium statistical mechanics, which states that a continuous symmetry (in this case, rotation invariance, or the ability of the flock to fly in any direction) can not be spontaneously broken in a two-dimensional system with only short-ranged interactions. The "nematic" phase of flocks, in which all the creatures move preferentially, or are simply oriented preferentially, along the same axis, but with equal probability of moving in either direction, also differs dramatically from its equilibrium counterpart (in this case, nematic liquid crystals). Specifically, it shows enormous number fluctuations, which actually grow with the number of organisms faster than the √N "law of large numbers" obeyed by virtually all other known systems. As for equilibrium systems, the hydrodynamic behavior of any phase of flocks is radically modified by additional conservation laws. One such law is conservation of momentum of the background fluid through which many flocks move, which gives rise to the "hydrodynamic backflow" induced by the motion of a large flock through a fluid. We review the theoretical work on the effect of such background hydrodynamics on three phases of flocks - the ferromagnetic and nematic phases described above, and the disordered phase in which there is no order in the motion of the organisms. The most surprising prediction in this case is that "ferromagnetic" motion is always unstable for low Reynolds-number suspensions. Experiments appear to have seen this instability, but a quantitative comparison is awaited. We conclude by suggesting further theoretical and experimental work to be done.