Fluctuations and Pattern Formation in Self-Propelled Particles Shradha Mishra, 1 Aparna Baskaran, 1 and M. Cristina Marchetti 1, 2 $^{1}$Physics Department, Syracuse University, Syracuse, NY 13244 $^{2}$Syracuse Biomaterials Institute, Syracuse University, Syracuse, NY 13244 (Dated: October 18, 2018) We consider a coarse-grained description of a system of self-propelled particles given by hydrody- namic equations for the density and polarization fields. We find that the ordered moving or flocking state of the system is unstable to spatial fluctuations beyond a threshold set by the self-propulsion velocity of the individual units. In this region, the system organizes itself into an inhomogeneous state of well-defined propagating stripes of flocking particles interspersed with low density disor- dered regions. Further, we find that even in the regime where the homogeneous flocking state is stable, the system exhibits large fluctuations in both density and orientational order. We study the hydrodynamic equations analytically and numerically to characterize both regimes. Large collections of living organisms exhibit a highly coherent collective dynamics at large scales [1]. This be- havior, often referred to as “flocking”, spans an enormous range of length scales and is seen in diverse systems, in- cluding mammalian herds [2], crowds of pedestrians [3, 4], bird flocks [5], fish schools [6], insect swarms [7], bac- terial suspensions [8], extracts of cytoskeletal filaments and molecular motor proteins [9, 10], and motility assays [11]. Many of these systems can be unified under the theoretical paradigm of collections of self-propelled par- ticles. Their intriguing collective behavior has received considerable attention in recent years. A number of different theoretical approaches have proved fruitful in understanding the dynamics of collec- tions of self-propelled units. Starting with the seminal work of Vicsek [12], rule-based models have been in- vestigated numerically and have been shown to exhibit nonequilibrium transitions between disordered and or- dered (flocking or moving) states. Subsequent work has focused on characterizing the nature of the order-disorder transition, its dependence on the noise, and pattern for- mation in the ordered state, both in the context of rule- based Vicsek-type models [12–14] and of models of bac- terial swarming [27–29]. Continuum hydrodynamic theo- ries have been used to describe the behavior of the system at large scales [1, 16, 18, 19]. Self-propelled particles are typically elongated and move along one direction of their long body axis. They can exhibit orientational order at high concentration. The ordered state is characterized by a vector order parameter, the polarization, which is also proportional to the mean velocity of the system. Hence the ordered state is a macroscopically moving state. The continuum theory has been developed phenomenologi- cally on the basis of general symmetry arguments by drawing on analogies with magnetic systems and with liquid crystals [1]. In fact active or self-propelled sys- tems have been likened to “living liquid crystals” [15]. This work has yielded several important results, includ- ing the possibility of long range order in 2D [16] and the prediction and observation of giant number fluctua- In a recent paper we derived the hydrodynamic equa- tions for a collections of self propelled hard rods mov- ing on a frictional substrate and interacting through ex- cluded volume interactions [23]. Although self-propulsion and steric effects alone are not sufficient to yield a ho- mogeneous polarized moving state in bulk, the hydro- dynamic equations are easily modified to incorporate a mean-field continuous transition from an isotropic state at low concentration of rods to a polar state at high den- sity. In the present paper we examine analytically and numerically the coupled nonlinear hydrodynamic equa- tions for density and polarization to characterize the large-scale structures that replace the linearly unstable homogeneous ordered state. The main results of our work are summarized in Fig. 1 that represents a “phase dia- gram” in terms of the density ρ of rods and their self- propulsion speed, v$_{0}$ . In the absence of self-propulsion ( v$_{0}$ = 0) the model considered exhibits a mean-field continuous transition at the critical density ρ$_{c}$ from an isotropic state of zero polarization for ρ$_{0}$ < ρ$_{c}$ to an ordered moving state, with uniform density and macro- scopic polarization P = 0. In a uniform ordered state at finite v$_{0}$ , all rods would move with uniform mean ve- locity ∼ v$_{0}$ P . We find however that the moving state for ρ$_{0}$ > ρ$_{c}$ exhibits more complex behavior. For v$_{0}$ be- low a critical value v$_{c}$ ( ρ$_{0}$ ) the steady state of the sys- tem is still macroscopically polarized on average, but ex- hibits anomalous density and polarization fluctuations. We refer to this state as the “fluctuating flocking state”. The anomalous density fluctuations are the giant num- ber fluctuations predicted by Toner and coworkers [17] and observed experimentally in active nematics [31]. An additional feature of this regime, is a very slow tempo- ral approach to this noisy steady state, with some fea- tions in the ordered state[18, 20]. The continuum theory has also been derived by systematic coarse-graining of specific microscopic models, including rule-based [21, 22] and physically motivated [23, 30] models. These deriva- tions yields (model-dependent) estimates for the param- eters in the hydrodynamic equations and have provided insight into the microscopic origin of of the large-scale collective physics. arXiv:1001.3334v2 [cond-mat.stat-mech] 16 Apr 2010 I. INTRODUCTION