Turbulence and pattern formation in continuum models for active matter

Abstract

Living and nonliving active matter, ranging from flocks of birds to active colloids, exhibit a fascinating range of physical phenomena such as order-disorder transitions and density waves in flocking phases, chaotic states and pattern formation. While the properties of flocking phases have received considerable attention, other active matter phases are relatively less explored. In this dissertation, we theoretically and computationally investigate turbulence and crystalline patterns, as well as transitions between these phases, in two-dimensional active matter. In the first half, we study turbulence in active fluids. Important statistical quantities such as probability density functions of velocity and vorticity as well as velocity correlations and energy spectra are analyzed. We show that active turbulence, in contrast to hydrodynamic turbulence, is characterized by a strong length-scale selection. We develop a statistical closure theory for velocity correlations based on the eddy-damped quasi-normal Markovian approximation from hydrodynamic turbulence theory. This theory captures the statistical features of active turbulence across a range of activity values, suggesting the applicability of classical hydrodynamic theory in investigating the properties of active fluids. In the second half of this dissertation, we investigate the properties of a spontaneously emerging crystalline phase. We show that this nonequilibrium crystal preserves some of the properties of their equilibrium counterparts. The melting of such active vortex crystals may proceed with a hysteretic transition region, or through an intermediate hexatic phase, depending on the values of the control parameters. Interestingly, we observe that the duration of crystallization increases with the domain size. As we approach the thermodynamic limit, superstructures of vortex crystal domains emerge leading to a supertransient phase. These superstructures form domains of vortex crystals of opposite polarity spins, demarcated by a turbulent active fluid. We also discuss generalizations to the continuum equations used in this work. Advected Swift-Hohenberg equations form a wider class of models that can qualitatively describe active fluids. Taken together, the results in this dissertations present an analysis of crystallization and turbulent dynamics in active matter within one uniform framework.