[Paper Review] Bulk topological states in a new collective dynamics model

Pierre Degond, Antoine Diez+1 more|arXiv (Cornell University)|Jan 22, 2021

Distributed Control Multi-Agent Systems98 references6 citations

TL;DR

This paper introduces a new collective dynamics model for self-propelled rigid bodies that exhibit topological order, demonstrating the emergence of bulk topological states in a macroscopic model derived from an individual-based model (IBM). It shows that non-trivial topological solutions—such as milling orbits and helical traveling waves—arise in the macroscopic limit, but the IBM transitions to trivial states via a phase of maximal disorder, with topological indicators revealing a transfer of winding number between structures.

ABSTRACT

In this paper, we demonstrate the existence of topological states in a new collective dynamics model. This individual-based model (IBM) describes self-propelled rigid bodies moving with constant speed and adjusting their rigid-body attitude to that of their neighbors. In previous works, a macroscopic model has been derived from this IBM in a suitable scaling limit. In the present work, we exhibit explicit solutions of the macroscopic model characterized by a non-trivial topology. We show that these solutions are well approximated by the IBM during a certain time but then the IBM transitions towards topologically trivial states. Using a set of appropriately defined topological indicators, we reveal that the breakage of the non-trivial topology requires the system to go through a phase of maximal disorder. We also show that similar but topologically trivial initial conditions result in markedly different dynamics, suggesting that topology plays a key role in the dynamics of this system.

Motivation & Objective

• To establish the existence of non-trivial topological states in a novel macroscopic model derived from an individual-based model (IBM) of self-propelled rigid bodies.
• To investigate how topological order emerges and breaks down in the IBM, particularly through transitions involving maximal disorder.
• To quantify topological properties using indicators such as winding numbers and roll polarization curves (RPZ/RPX), linking macroscopic solutions to IBM dynamics.
• To assess the robustness and reproducibility of topological phase transitions under perturbations and varying initial conditions.
• To clarify the role of topology in shaping collective dynamics by contrasting topologically non-trivial and trivial initial states.

Proposed method

• Formulate an individual-based model (IBM) where self-propelled rigid bodies align their attitude to neighbors' orientations, with constant speed and local interaction rules.
• Derive a macroscopic model via a hydrodynamic limit, yielding a system of partial differential equations describing mean-field density and orientation fields.
• Construct explicit solutions of the macroscopic model, including milling orbits (MO), helical traveling waves (HW), and generalized topological states, characterized by non-zero winding numbers.
• Define topological indicators: winding number $w_z$, center-of-mass distance $d_z$, and mean radius $\bar{r}_z$ of the roll polarization (RPZ) curve to track topological evolution.
• Use numerical simulations of the IBM with large $N \approx 1.5 \times 10^6$ particles to compare dynamics with macroscopic solutions and track topological transitions.
• Apply order parameters such as global order parameter (GOP) and mean angular velocity $\bar{\Omega}$ to monitor collective behavior and detect transitions between phases.

Experimental results

Research questions

• RQ1 Can non-trivial topological states—such as helical traveling waves or milling orbits—emerge in a macroscopic model derived from an individual-based model of self-propelled rigid bodies?
• RQ2 How do topological indicators (e.g., winding number, roll polarization curve morphology) evolve during the transition from topologically non-trivial to trivial states in the IBM?
• RQ3 Does the breakdown of non-trivial topology in the IBM occur through a phase of maximal disorder, and is this phase consistently observed across simulations?
• RQ4 To what extent do topologically distinct initial conditions (e.g., MO vs. HW) lead to qualitatively different dynamical behaviors, even when initial order parameters are similar?
• RQ5 Are the observed topological transitions robust to small perturbations in initial conditions, and can transient topological structures (e.g., unstable HW) be stabilized under specific domain geometries?

Key findings

• Explicit solutions of the macroscopic model, including milling orbits and helical traveling waves, exhibit non-trivial topology characterized by a winding number $w_z = 1$.
• The IBM approximates the macroscopic solutions well initially, but eventually transitions to topologically trivial states, with the transition occurring via a phase of maximal disorder.
• The winding number $w_z$ drops to zero only after the system passes through a state of high disorder, as indicated by large fluctuations in $\bar{r}_z$ and $d_z$.
• A transfer of non-trivial topology occurs from a milling orbit (MO) to a helical traveling wave (HW), where $w_z$ decreases to 0 while $w_x$ builds up from undefined to 1, indicating a topological reconfiguration.
• Topologically trivial initial conditions lead to markedly different dynamics compared to non-trivial ones, even with similar initial order parameters, demonstrating that topology fundamentally influences collective behavior.
• Transient helical wave states can emerge during transitions (e.g., from MO to flocking), but are unstable due to periodic boundary conditions unless the wave vector aligns with domain symmetries, such as $\xi = \sqrt{2}\pi/L$.