[Paper Review] The transition to collective motion in nonreciprocal active matter: coarse graining agent-based models into fluctuating hydrodynamics

D.G. Martin, Daniel S. Seara+3 more|arXiv (Cornell University)|Jul 17, 2023

Material Dynamics and PropertiesMaterials Science3 citations

TL;DR

This paper introduces an exactly coarse-grained fluctuating hydrodynamics model for nonreciprocal active matter, derived from a minimal agent-based model with two species exhibiting antialigned interactions. It reveals a novel 'Chase & Rest' phase under strong nonreciprocity—unpredictable by linear models—while demonstrating that finite-size fluctuations enable detection of microscopic nonreciprocity via macroscopic correlation functions.

ABSTRACT

Two hallmarks of non-equilibrium systems, from active colloids to animal herds, are agents motility and nonreciprocal interactions. Their interplay creates feedback loops leading to complex spatiotemporal dynamics crucial to understand and control the nonlinear response of active systems. Here, we introduce a minimal model that captures these two features at the microscopic scale, while admitting an exact hydrodynamic theory valid also in the fully-nonlinear regime. Our goal is to account for the fact that animal herds and colloidal swarms are rarely in the thermodynamic limit where particle number fluctuations can be completely ignored. Using statistical mechanics techniques we exactly coarse-grain a nonreciprocal microscopic model into a fluctuating hydrodynamics and use dynamical systems insights to analyze the resulting equations. In the absence of motility, we find two transitions to oscillatory phases occurring via distinct mechanisms: a Hopf bifurcation and a Saddle-Node on Invariant Circle (SNIC) bifurcation. In the presence of motility, this rigorous approach, complemented by numerical simulations, allows us to quantitatively assess the hitherto neglected impact of inter-species nonreciprocity on a paradigmatic transition in active matter: the emergence of collective motion. When nonreciprocity is weak, we show that flocking is accelerated and bands tend to synchronize with a spatial overlap controlled by nonlinearities. When nonreciprocity is strong, flocking is superseded by a Chase & Rest dynamical phase where each species alternates between a chasing state, when they propagate, and a resting state, when they stand still. Finally, we demonstrate how fluctuations in finite systems can be harnessed to characterize microscopic non-reciprocity from macroscopic time-correlation functions, even in phases where nonreciprocal interactions do not affect the thermodynamic steady-state.

Motivation & Objective

• To develop an exact hydrodynamic description of nonreciprocal active matter by coarse-graining agent-based models with nonreciprocal interactions.
• To analyze how nonreciprocity and motility jointly influence the transition to collective motion in finite systems.
• To quantify the role of density and polarization fluctuations in shaping emergent spatiotemporal dynamics beyond the thermodynamic limit.
• To demonstrate that fluctuation-induced correlations can reveal microscopic nonreciprocity even when steady-state properties are unaffected.
• To identify and characterize a novel 'Chase & Rest' dynamical phase absent in linear nonreciprocal models.

Proposed method

• Derive exact fluctuating hydrodynamics equations (1) from microscopic agent dynamics via statistical mechanics techniques, valid in the fully nonlinear regime.
• Use the form $\partial_t\boldsymbol{\psi} = \mathbf{F}(\boldsymbol{\psi}, \nabla\boldsymbol{\psi}) + \sqrt{a}\,\mathbf{\mathcal{M}}(\boldsymbol{\psi}, \nabla)\cdot\boldsymbol{\xi} + \mathcal{O}(a)$, where $a$ is lattice spacing, $\mathbf{F}$ is deterministic, and $\boldsymbol{\xi}$ is Gaussian white noise.
• Apply dynamical systems analysis to identify bifurcations: Hopf and Saddle-Node on Invariant Circle (SNIC) in the nonmotile limit.
• Perform agent-based simulations with random sequential updates and small time steps to validate hydrodynamic predictions.
• Use semi-spectral methods with anti-aliasing and semi-implicit Euler schemes to numerically integrate the PDEs.
• Compare theoretical predictions of inter-species magnetization correlations with numerical simulations to validate the model.

Experimental results

Research questions

• RQ1 How does nonreciprocal interaction between two active species affect the onset and nature of collective motion?
• RQ2 What novel dynamical phases emerge when nonreciprocity is strong, and how do they differ from those predicted by linear models?
• RQ3 Can finite-size fluctuations in non-equilibrium active systems be used to infer microscopic nonreciprocal interactions from macroscopic observables?
• RQ4 What are the distinct bifurcation mechanisms (Hopf vs. SNIC) governing oscillatory transitions in nonreciprocal active matter?
• RQ5 How do nonlinearities and density fluctuations influence the synchronization and spatial overlap of polar bands in flocking systems?

Key findings

• In the absence of motility, two distinct oscillatory transitions occur: one via a Hopf bifurcation and another via a Saddle-Node on Invariant Circle (SNIC) bifurcation.
• With motility, weak nonreciprocity accelerates flocking and promotes spatial synchronization of bands, with overlap controlled by nonlinearities.
• Strong nonreciprocity induces a novel 'Chase & Rest' phase where species alternate between propagating and standing still, a phase absent in linear nonreciprocal models.
• Theoretical prediction (201) for the ratio of inter-species to intra-species magnetization correlations matches numerical simulations quantitatively in the disordered phase.
• Fluctuations in finite systems allow detection of microscopic nonreciprocity through macroscopic time-correlation functions, even when steady-state properties are unchanged.
• The exact coarse-graining procedure enables rigorous analysis of non-reciprocal correlation functions and density fluctuations, previously neglected in analytical treatments.