Resting and traveling localized states in an active phase-field-crystal model

Abstract

The conserved Swift-Hohenberg equation (or phase-field-crystal [PFC] model) provides a simple microscopic description of the thermodynamic transition between fluid and crystalline states. Combining it with elements of the Toner-Tu theory for self-propelled particles, Menzel and Löwen [Phys. Rev. Lett. 110, 055702 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.055702] obtained a model for crystallization (swarm formation) in active systems. Here, we study the occurrence of resting and traveling localized states, i.e., crystalline clusters, within the resulting active PFC model. Based on linear stability analyses and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of periodic and localized, resting and traveling states in a one-dimensional active PFC model. This allows us, for instance, to explore how the slanted homoclinic snaking of steady localized states found for the passive PFC model is amended by activity. A particular focus lies on the onset of motion, where we show that it occurs either through a drift-pitchfork or a drift-transcritical bifurcation. A corresponding general analytical criterion is derived.

Two-dimensional localized states in an active phase-field-crystal model

“…9 the bifurcation occurring on the branch of asymmetric RLS does not exhibit the typical shape of a transcritical bifurcation as both branches of TLS seem to bifurcate towards larger v 0 . Our identification of this bifurcation as a drifttranscritical bifurcation is based on similar behavior observed in 1D [67] where more precise computations are possible, and for this reason we believe that one of two branches undergoes a fold very close to the transcritical bifurcation. Grid effects make it very hard to remain on branches of RLS and lead to rather blurred onsets of the drift velocity c vs v 0 .…”

“…We now systematically explore how LS in 2D respond to increasing activity by employing the activity parameter v 0 as the main control parameter. From results obtained for LS in 1D [67], we expect transitions from resting to traveling LS (RLS and TLS, respectively) associated with symmetry breaking between the two fields ψ and P, as centers of the density peaks shift with respect to +1 defects in P at a critical activity v c . For resting crystals, P points down the gradient of ψ, leading to a defect at the center of the density peak, similar to the vector field of a monopole.…”

“…For this reason, we investigate here how 2D active crystals described by the aPFC model evolve from a localized state (LS) consisting of a single peak into spatially extended states (crystals) under the influence of activity. As in [67], we focus on the mean density ψ and the activity parameter v 0 as the main control parameters. The activity parameter v 0 must, of course, be nonzero for the presence of traveling structures but its specific value will turn out to have a major influence not only on the transition from resting to traveling states but also on the structure of 2D crystals and associated pattern selection.…”

“…Thus far, the active PFC model has been employed mainly to investigate the linear stability of the liquid state with respect to the development of resting and traveling crystalline patterns and in the study of domainfilling resting and traveling crystals by direct time simulations in different geometries [38,39,65,66]. A bifurcation study of a spatially one-dimensional system was first provided in [67]. The main purpose of the present work is to investigate these states and the transitions between them in two spatial dimensions, focusing on both spatially extended and spatially localized states.…”

Orientational Order Parameters for Arbitrary Quantum Systems

“…These are typical variables of field theories for liquid crystals and active soft matter. [ 1–13 ] The standard definitions of and assume uniaxial (e.g., rodlike) particles and have to be modified for particles with low symmetry. [ 14–16 ] This modification allows to study a much richer phenomenology of phase transitions.…”