Vortex Arrays and Mesoscale Turbulence of Self-Propelled Particles

Abstract

Inspired by the Turing mechanism for pattern formation, we propose a simple self-propelled particle model with short-ranged alignment and anti-alignment at larger distances. It is able to produce orientationally ordered states, periodic vortex patterns as well as meso-scale turbulence. The latter phase resembles observations in dense bacterial suspensions. The model allows a systematic derivation and analysis of a kinetic theory as well as hydrodynamic equations for density and momentum fields. A phase diagram with regions of such pattern formation as well as spatially homogeneous orientational order and disorder is obtained from a linear stability analysis of these continuum equations. Microscopic Langevin simulations of the self-propelled particle system are in agreement with these findings.The term active matter refers to non-equilibrium systems of interacting, self-propelled entities which are able to take up energy from their environment and convert it into motion [1,2]. Examples, such as cytoskeletal filaments [3], chemically driven colloids [4] or flocks of birds [5] have recently received a lot of attention in physics, chemistry and biology. They exhibit a wide range of collective phenomena which are absent in systems at thermodynamic equilibrium, for example large-scale travelling bands and polar clusters [6,7] as well as arrays of vortices [8,9]. In this context, bacteria represent important model systems, which have been used to investigate such different aspects as clustering [10] and rheological properties [11] of active matter systems.Recently, irregular vortex structures were experimentally observed in dense bacterial suspensions [11][12][13][14][15]. In addition, a phenomenological model was proposed which describes the observed behavior including the power spectrum of the bacterial dynamics [14,16]. The spectrum at large wave numbers as well as the dynamic vortex patterns in these experiments and simulations are reminiscent of a turbulent state which led the authors to denominate this new phenomenon as "meso-scale turbulence".We aim to formulate a model at the level of individual particles which is capable to produce such mesoscopic spatiotemporal patterns. Inspired by the Turing mechanism of short-range activation and long-range inhibition in reaction-diffusion systems [17,18], we propose interacting self-propelled particles with local alignment at short length scales and anti-alignment at larger distances. Such a type of interactions may be realized by a competition of local alignment and large-scale hydrodynamic back-flow effects [19,20], e. g. in suspensions of bacterial microswimmers, which leads to preferential alignment of neighboring cells and anti-alignment with more distant swimmers.Here, we abstain from considering a detailed model of individual swimmers immersed and interacting through a surrounding fluid. Our model includes effective interactions of self-propelled particles of the type first formulated by Vicsek et al. [21]. The original Vicsek model displays surprisingly complex sp...

Gaussian theory for spatially distributed self-propelled particles

“…Truncating the series of angular Fourier coefficients of particles distribution is vastly used in active matter to obtain the continuum equations [34,[59][60][61]. This method has a reasonable accuracy in determining the phase boundaries.…”

Theoretical approaches to the steady-state statistical physics of interacting dissipative units

“…It is tempting to describe such a system through a local mean-field approach [77,78,79,80]. Note that similar approaches have also been used for self-propelled particles with nematic interactions [81,82,78,83].…”

“…In a similar way, Γ(r, t) denotes the average torque resulting from the average over the positions of the swimmers. Thanks to the linearity of the Stokes equation (80), the average velocity field u(r, t) is obtained by solving the average Stokes equation…”

Getting drowned in a swirl: Deformable bead-spring model microswimmers in external flow fields

“…This includes the individual dynamics of particles with complex shape [7][8][9], as well as cases of self-rotation [9][10][11][12][13][14][15]. Furthermore, the collective behavior of many such interacting particles has been explored [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Collections of self-propelled particles in liquid environment exhibit fascinating and complex nonequilibrium phenomena emerging from self-organization, where hydrodynamic interactions can play a significant role [11,[36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52].…”