[论文解读] Nonreciprocal pattern formation of conserved fields

Fridtjof Brauns, Mauro Marchetti|arXiv (Cornell University)|Jun 15, 2023

Nonlinear Dynamics and Pattern Formation参考文献 83被引用 4

一句话总结

本文提出了一种通用规范型——非互惠Cahn-Hilliard（NRCH）方程——统一描述了非互惠系统中守恒场的动力学，揭示了质量守恒与详细平衡破缺如何驱动行进波及复杂的时空模式。研究证明，行进波的速度与结构精确由界面处的局域色散关系决定，从而将非互惠模式形成理论推广至先前模型之外。

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.

研究动机与目标

• 识别一个最小化、通用化的模型，用于描述不同物理系统中守恒场的非互惠模式形成。
• 解释在质量守恒与详细平衡破缺条件下，尽管存在扩散弛豫，行进波如何涌现。
• 将不同系统——活性凝胶、反应-扩散系统、多孔弹性介质——统一于同一动力学框架下。
• 证明行进波的速度与结构由界面处的局域色散关系决定，而非全局模态耦合。
• 通过引入守恒场中的流体动力学模式，推广先前的非互惠相变模型。

提出的方法

• 通过非互惠交叉扩散系数D12 ≠ D21将守恒标量场φ与纯扩散的守恒场ψ耦合，推导出最小化的非互惠Cahn-Hilliard（NRCH）方程。
• 通过色散关系σ(q)分析均匀态的线性稳定性，显示不稳定的模态带边缘存在非零虚部。
• 将NRCH模型映射为FitzHugh-Nagumo模型的空间扩展形式，实现统一的动力系统视角。
• 利用梯度展开与力平衡方程的解析解，推导出活性凝胶与多孔弹性介质的有效方程，将其简化为NRCH形式。
• 应用微扰与数值方法研究二维系统中的波传播、粗化动力学及横向波动。
• 通过验证表明，波速与波长由远离均匀态的局域色散关系精确预测。

Figure 1: (a) Dispersion relations of the NRCH equations showing the eigenvalue crossing in the uncoupled case ( $D_{12}D_{21}=0$ , left) which becomes an avoided crossing for reciprocal coupling (center) and gives rise to a band of propagating modes for anti-reciprocal coupling (right). (b), (c) Ky

实验结果

研究问题

• RQ1 守恒场之间的非互惠耦合如何在具有扩散弛豫的系统中导致行进波的形成？
• RQ2 在特定模型之外，非互惠模式形成的普遍机制是什么？
• RQ3 为何在守恒标量场系统中，尽管缺乏显式的激活-抑制动力学，行进波仍会涌现？
• RQ4 质量守恒与平动不变性产生的流体动力学模式如何协同生成传播结构？
• RQ5 多样化的活性与非平衡系统在多大程度上可由单一规范型描述？

主要发现

• NRCH方程为守恒场中的非互惠模式形成提供了通用规范型，准确捕捉了活性凝胶、反应-扩散系统与多孔弹性介质的关键特征。
• 当色散关系在不稳定模态带边缘获得非零虚部时，行进波会普遍涌现，即使不存在显式振荡项。
• 波速与结构由远离均匀态的界面处局域色散关系精确预测，而非全局模态耦合。
• 系统表现出行进波粗化过程的中断，且无特征波长的选择，表明传统粗化定律的失效。
• 在二维系统中，由于非互惠耦合，行进波前缘发展出横向波动，这是守恒系统中非厄米动力学的特征。
• 最小NRCH模型通过将其简化为共同的有效描述，捕捉了复杂活性物质系统（包括Min蛋白系统与粘弹性活性凝胶）的本质物理机制。

Figure 2: Phase portrait of the FHN model ( 2 ) with $a=0$ . (a) For reciprocal coupling ( $c_{12}=c_{21}$ ), the $v$ -nullcline is always sloped such that the system is bistable. (b) For sufficiently strong anti-reciprocal coupling, limit-cycle oscillations emerge.