Nonreciprocal pattern formation of conserved fields

Abstract: In recent years, nonreciprocally coupled systems have received growing attention. Previous work has shown that the interplay of nonreciprocal coupling and Goldstone modes can drive the emergence of temporal order such as traveling waves. We show that these phenomena are generically found in a broad class of pattern-forming systems, including mass-conserving reaction--diffusion systems and viscoelastic active gels. All these systems share a characteristic dispersion relation that acquires a non-zero imaginary part at the edge of the band of unstable modes and exhibit a regime of propagating structures (traveling wave bands or droplets). We show that models for these systems can be mapped to a common "normal form" that can be seen as a spatially extended generalization of the FitzHugh--Nagumo model, providing a unifying dynamical-systems perspective. We show that the minimal nonreciprocal Cahn--Hilliard (NRCH) equations exhibit a surprisingly rich set of behaviors, including interrupted coarsening of traveling waves without selection of a preferred wavelength and transversal undulations of wave fronts in two dimensions. We show that the emergence of traveling waves and their speed are precisely predicted from the local dispersion relation at interfaces far away from the homogeneous steady state. Our work thus generalizes previously studied nonreciprocal phase transitions and shows that interfaces are the relevant collective excitations governing the rich dynamical patterns of conserved fields.

• The paper introduces a minimal nonreciprocal Cahn–Hilliard model that predicts traveling waves and multistable states in conserved systems.
• It employs stability analysis to show that nonreciprocity and Goldstone modes trigger complex propagating patterns and interface undulations.
• Numerical results reveal a direct correlation between local dispersion relations and wave speed, providing a new metric for pattern prediction.

Overview of "Non-reciprocal pattern formation of conserved fields"

The paper "Non-reciprocal pattern formation of conserved fields" by Fridtjof Brauns and M. Cristina Marchetti explores the phenomenon of non-reciprocally coupled systems and their intrinsic ability to form complex patterns, such as traveling waves and oscillatory states. This study is set within the context of various physical systems that embody non-reciprocity, a quality characterized by the absence of Newton’s third law, often manifesting through non-conservative forces that do not result from potential energy derivatives.

Key Findings

The authors demonstrate that the interplay between non-reciprocal coupling and Goldstone modes drives the emergence of propagating patterns across a spectrum of systems including mass-conserving reaction-diffusion systems and viscoelastic active gels. They present these systems as sharing a common dispersion relation which exhibits a non-zero imaginary component at the band edges of unstable modes. This characteristic predicts a regime of propagating structures, captured by a generalized dynamical model analogous to the FitzHugh–Nagumo (FHN) system, but extended to spatially distributed terms like the Cahn–Hilliard (CH) model.

Model and Methodology

The analysis focuses on a minimal non-reciprocal Cahn–Hilliard (NRCH) model:

$$\begin{align*} \partial_t \phi(x,t) &= \nabla^2 (D_{11} \phi + D_{12} \psi + \phi^3 - \kappa \nabla^2 \phi), \ \partial_t \psi(x,t) &= \nabla^2 (D_{21} \phi + D_{22} \psi). \end{align*}$$

Here $\phi$ and $\psi$ represent conserved scalar fields with non-reciprocal cross-diffusion terms $D_{12} \neq D_{21}$. Through stability analysis, the authors reveal that non-reciprocity induces interfaces to propagate, resulting in traveling waves, even when the fastest-growing mode in the dispersion relation is stationary.

Results and Implications

Noteworthy numerical results indicate that minimal NRCH equations can lead to complex behaviors such as interrupted coarsening of wave patterns and interface undulations in two dimensions. Traveling wave speed was shown to correlate with the local dispersion relation at interfaces, establishing a novel metric for the wave velocity prediction. Furthermore, these systems exhibit a range of stable wavelengths due to an arrested coarsening process, yielding multistable traveling wave states.

The study has significant implications for theoretical and applied physics, as it offers insight into pattern formation beyond equilibrium. The unifying model developed here serves to enhance our understanding of active systems' dynamics, pertinent to both biological and synthetic settings.

Future Directions

The theoretical framework and predictions suggest several avenues for further research, particularly involving density-dependent transport coefficients and non-conserved dynamics. Moreover, the stability mechanisms in two-dimensional systems could provide deeper insights into the dynamics of interfaces across different physical contexts. The paper's findings on traveling waves and their multistability hint at potential applications in controlling patterns in chemical and biological systems, and possibly inspiriting novel algorithms for wave-based computing architectures.

Overall, this work deepens our comprehension of non-reciprocal interactions in complex systems, providing a robust groundwork for exploring the rich tapestry of nonequilibrium pattern formation.