Disorder to Order Transition in 1D Nonreciprocal Cahn-Hilliard Model

TL;DR

This work addresses how nonreciprocal coupling in a conserved one-dimensional Cahn-Hilliard system organizes defects and drives ordering. It develops and analyzes the NRCH model under periodic and non-periodic boundaries, identifying defect sources/sinks and a characteristic wave number $k_{\infty}$ selected by defects. A disorder-to-order transition occurs at $\alpha_c\approx0.6$, with a crossover $\alpha_{\times}\approx0.62$ from wave-number considerations, while boundary conditions qualitatively alter defect dynamics, sometimes yielding intermittent patchy order or two-domain polarization. The results shed light on defect dynamics in conserved, nonreciprocal systems and highlight the crucial role of boundary conditions in non-equilibrium pattern formation.

Abstract

We extensively study the phenomenology of one dimensional Nonreciprocal Cahn Hilliard model for varying nonreciprocity $(α)$ and different boundary conditions. At small $α$, a perturbed uniform state evolves to a defect laden configuration that lacks global polar order. Defects are the sources and sinks of travelling waves and nonreciprocity selects defects with a unique wave number that increases monotonically with $α_c$. A critical threshold $α_c$ marks the onset of a transition to states with finite global polar order. For periodic boundaries, above $α_c$, the system shows travelling waves that are completely ordered. In contrast, travelling waves are incompatible with Dirichlet and Neumann boundaries. Instead, for $α\gtrsim α_c$, we find fluctuating domains that show intermittent polar order and at large $α$, the system partitions into two domains with opposite polar order.

Figures (7)

• Figure 1: Kymograph plot of the polar order parameter $J(x,t)$ highlighting the evolution of disordered states. Defects are located on zeroes of $J(x,t)$, which appear white in the kymograph. We also mark the exact defect positions (small black points) at late times. Initially, numerous source-sink pairs are formed, which merge and the system settles into a stable configuration, which persists till the end of the simulations.
• Figure 2: Sources and sinks in the 1D NRCH model. (a, b) Representative plot of $\text{Re}\left(\phi\right)$ and the polar order parameter $J$ (normalised by it's maximum value), respectively. Sources (blue circles) and sinks (orange circles) are arranged in an alternating manner on the 1D domain. $J$ vanishes at defect centres and is constant far from defects, where the wavefront is that of a plain wave. (c, d) Plot of $R(x)$ and $k(x)$ for sources (left column) and sinks (right column) for different values of $\alpha$. For these plots, $x$ is measured from the centre of the defects. At defect centres, $\partial_x R$ and $k(x)$ vanish. Far from the defects, $R(x)$ and $k(x)$ approach their plain wave forms.
• Figure 3: Wave number selection in 1D. For a given $\alpha$, sources and sinks with a particular wave number are selected. Using a least-square fit to power law forms, we obtain $k_{\infty} \sim \alpha^{0.6}$ (Solid black line). The selected wave number is comparable to the $k_{\infty}$ obtained for the targets in 2D (Orange circles) rana2024. Inset: $k_{\infty}$ for a single source-sink pair is identical to the $k_{\infty}$ for a multi-defect disordered configuration.
• Figure 4: (a) Defect density $\rho_D$ versus $\alpha$ for P-BC shows nontrivial behaviour. It is low at certain values of $\alpha \simeq 0.08, 0.2, 0.3$, which we call "resonances" (shaded green). Overall it first increases with $\alpha$, peaks around $\alpha \sim 0.2$ and then starts to decrease. Above $\alpha_{c} \sim 0.6$, $\rho_D$ vanishes marking the onset of the travelling wave phase. $\alpha_{c}$ is close to the crossover threshold $\alpha_{\times} \sim 0.62$ predicted by the wavenumber selection. (b) Average polar order $\bar{J}$ versus $\alpha$ in the long time steady state. For small $\alpha$, $\bar{J} \sim 0$, at large $\alpha$, $\bar{J} \sim 1$. Near $\alpha_{c}$ (grey shaded region), $\bar{J}$ shows large fluctuations due to finite size effects. Inset: The fraction of simulations $(F_{T})$ that show defect states for $\alpha$ close to $\alpha_{c}$. Below $\alpha_{c}$, $F_{T} = 1$, above $\alpha_{c}$, all simulations transition to travelling waves, thus $F_{T} = 0$. (c) Time ($t_{\text{st}}$) after which the system reaches the steady-state value of $\rho_{D}$. At resonances, defect merger events continue for long times, thus $t_{\text{st}}$ is orders of magnitude larger. To verify that the resonances are not the effect of domain size, we also plot results for $L=256\pi$ (black points), which shows similar behaviour. (d) Minimum inter-defect separation $d_{\mathrm{min}}$ versus $\alpha$. As defects keep merging for long times, we find sharp jumps in $d_{\mathrm{min}}$ in the resonance regions, which we also observe in 2D rana2024a. All plots share the same legend keys, and the location of the three resonances that are shaded green are at the same values of $\alpha \simeq 0.08, 0.2, 0.3$ for panels (a), (c), and (d).
• Figure 5: Comparison of $k_{l}$ for different $\alpha$ for $\alpha < \alpha_{c}$ for different boundary conditions. For a given $\alpha$, $k_{\infty}$ is higher for both N-BC and D-BC when compared to the $k_{\infty}$ for the P-BC. As a check, we also plot the $k_{\infty}$ obtained from Dedalus for the P-BC, which matches with our own simulations (dashed black line). $\alpha_{\times} = 0.35$.