Dynamical self-regulation in self-propelled particle flows

Arvind Gopinath, Michael F. Hagan, M. Cristina Marchetti, and Aparna Baskaran

Phys. Rev. E 85, 061903 – Published 1 June 2012

Abstract

We study a continuum model of overdamped self-propelled particles with aligning interactions in two dimensions. Combining analytical theory and computations, we map out the phase diagram for the parameter space covered by the model. We find that the system self-organizes into two robust structures in different regions of parameter space: solitary waves composed of ordered swarms moving through a low density disordered background, and stationary radially symmetric asters. The self-regulating nature of the flow yields phase separation, ubiquitous in this class of systems, and controls the formation of solitary waves. Self-propulsion and the associated active convection play a crucial role in aster formation.

Received 17 December 2011

DOI:https://doi.org/10.1103/PhysRevE.85.061903

Authors & Affiliations

Arvind Gopinath¹, Michael F. Hagan¹, M. Cristina Marchetti², and Aparna Baskaran¹

¹Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts, USA
²Physics Department and Syracuse Biomaterials Institute, Syracuse University, Syracuse, New York 13244, USA

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Vol. 85, Iss. 6 — June 2012

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Images

Figure 1
Left: Phase diagram in the $(λ, w_{0})$ plane for $ρ_{0} = 1.05$ . The dashed (blue) and dash-dotted (red) lines are the neutral stability curves for the L [Eq. (5)] and T [Eq. (6)] modes, respectively. The shaded regions describe the stable states obtained numerically: a homogenous polar state (H), a regime of propagating solitary stripes (S), a regime of stationary asters (A), and moving localized polar clusters (B). The domain size is (in dimensionless units) $128 \times 128$ and the equations were integrated up to $5 \times 10^{4}$ diffusion times. The moving polar clusters are a result of finite system size and the finite time of integration. Right: Snapshot of a propagating stripe (bottom) and a stationary aster (top), the contours (blue, low; and red, high) denoting density.Reuse & Permissions
Figure 2
Profiles of $ρ$ , the density (upper curve, pink) and $τ_{x}$ and $τ_{y}$ the polarization fields (lower curves, blue and green) are plotted for a striped state when $w_{0} = 0.4$ , $λ = 0$ , and $ρ_{0} = 1.05$ . The system size is $1024 \times 32$ and results shown are obtained after integrating to $10^{4}$ diffusion times. The arrow at the top right (red) denotes the direction of motion of the stripes. The vertical lines demarcate the trailing (T) and leading (L) edges of the stripe. The inset shows the details of the profile of density (solid line, blue) and polarization (dashed line, green) for one stripe.Reuse & Permissions
Figure 3
Illustration of the phase separation in the stripe state. Left: Computational results obtained at $T = 10^{4}$ for a $128 \times 128$ domain with $λ = 0$ . The shaded (red striped) region indicates the domain in parameter space when the polar state is unstable due to the convection mediated density instability. The density in the propagating stripe $ρ_{h}$ and the low density background $ρ_{ℓ}$ for various values of the base density and $w_{0}$ are shown. The lines are a guide to the eye. Right: Plot of the polarization $τ_{h}$ as a function of $w_{0}$ from the numerical simulation. The solid line is the analytical prediction $\frac{ρ_{h} (w_{0}) - 1}{ρ_{h} (w_{0}) + 1}$ .Reuse & Permissions
Figure 4
Radial density (top curve, green) and ${| τ |}^{2}$ (bottom curve, blue) profiles of an aster for $ρ_{0} = 1.07$ , $λ = 0.8$ , and $w_{0} = 0.05$ . The system size is $128 \times 128$ and the equations have been integrated up to $10^{4}$ diffusion times; these profiles remained constant even after integrating to $5 \times 10^{4}$ diffusion times. The density field $ρ (r)$ exhibits a point of inflection; at the same radial position the order parameter $| τ |$ attains a maximum value $τ_{\max}$ (indicated by the vertical lines). The aster is characterized by a core of size $ℓ_{co}$ and a region where both density and polarization decay over a characteristic length scale $ℓ_{\infty}$ . The dashed curves overlaying the lower curve of ${| τ |}^{2}$ are the theoretical result obtained by the asymptotic analysis based on Eq. (7) and described in the text.Reuse & Permissions
Figure 5
Steady inhomogeneous phase-separated structures, which we term streamers, obtained at fixed domain size $128 \times 128$ when $λ_{1} = 0$ , $λ_{2} = λ_{3} = 0.8$ , $w_{0} = 0.1$ , and $ρ_{0} = 1.05$ . (a) The central high density (red) region is also highly polar and ordered with the direction of polarity along the neutral direction. (b) We quantify the polar variation by plotting the two components of $τ$ as a function of $y$ . Note that $τ_{x}$ (dashed curve, green) is a single valued function of $y$ while $τ_{y}$ (solid curve, blue) changes sign as we traverse the phase-separated ordered region from one side to the other.Reuse & Permissions

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