Derivation and Analysis of Amplitude Equation for Generalized AMB+ in Presence of Chemical Reaction

TL;DR

This work develops a unified amplitude-description for pattern formation in generalized Active Model B+ (AMB+) subjected to a reversible chemical reaction. By symmetry arguments and a detailed multiscale expansion, it derives a real Ginzburg–Landau-type amplitude equation with explicit nonlinear coefficient $\alpha_2(g,\lambda,\xi,u,K,q_c)$, and analyzes how the quadratic term $g$ and activity parameters control the transition from supercritical to subcritical. It also establishes the Eckhaus instability boundary and derives a phase-diffusion equation that confirms the same instability band, showing it is independent of $g$. The results connect generalized AMB+ to familiar models (AMB, Cahn–Hilliard) in various limits and provide quantitative criteria for pattern selection and stability near onset in active matter with reactions.

Abstract

We derive and analyze the amplitude equation for the roll patterns in case of generalized Active Model B+ (AMB+) in the presence of chemical reactions. The generalized AMB+ differs from the original AMB+ introduced by Tjhung \textit{et al.} [E. Tjhung \textit{et al.}, Phys. Rev. X \textbf{8}, 031080 (2018)] by the addition of a quadratic term, $gφ^2$, in the expression for the equilibrium part of the current. Also, the model includes a rotation-free active current of strength $λ$ and a rotational current of strength $ξ$. The inclusion of a chemical reaction with rate $Γ$ removes the conservation constraint and introduces a preferred wavenumber that governs the pattern formation below a critical reaction rate $Γ_c$. We argue for the analytical form of the amplitude equation based on symmetry considerations and explicitly derived it using multiscale analysis. By taking different limits of $g$, $λ$, and $ξ$, we recover amplitude equations for several well-known physical models as special cases and determine the nature transitions close to the onset of instability. We find that for $g = 0$, the transition is always supercritical, whereas for $g \ne 0$, the transition between the supercritical and subcritical regimes depends sensitively on the model parameters. Further, we derive the condition for the \textit{Eckhaus instability} from the stability analysis of the amplitude equation as well as from the phase diffusion equation, and find that it is independent of $g$.

Figures (5)

• Figure 1: Plot of $\sigma(q)$ versus $q$, given by Eq. (\ref{['apeqn6']}), for $a = 4$ and $K = 1$, at different values of $\Gamma$. Here, $\Gamma_c=4$ is the critical reaction rate below which $\sigma(q)$ becomes positive for a band of wave numbers $q\in[q_-, q_+]$.
• Figure 2: Plot of the stationary $\phi(x,t)$ versus $x$ at $t = 1000$ for a system of size $L = 128$, obtained by solving Eq. (\ref{['apeqn3a']}) in $d=1$ with parameters $a = 2$, $g = 0.4$, $\lambda = 0.05$, $\xi = 0.05$, $u=1$ and $K = 1$, for different values of $\Gamma$. For this choice of parameters, $\Gamma_c = 1$ denotes the critical reaction rate.
• Figure 3: Phase diagram marking the domains of supercritical-subcritical transition in the $(\lambda-\tfrac{\xi}{2})$ -- $g$ plane for $a=1$, $u=1$, and $K=1$. In the regions marked in red, supercritical transitions take place, while in the blue regions, subcritical transitions occur.
• Figure 4: Time evolution of $\phi(x,t)$ in $d = 1$ for a system of size $L = 128$, obtained by solving Eq. (\ref{['apeqn3']}) with parameters in (a) the supercritical regime: $a = 1$, $g = 0.5$, $\lambda = 0.2$, $\xi = 0.2$, $K = 1$, $u=1$, and $\Gamma = 0.15$; and (b) the subcritical regime: $a = 1$, $g = 2.0$, $\lambda = 0.2$, $\xi = 0.2$, $K = 1$, $u=1$, and $\Gamma = 0.15$. Lines of different colors indicate different times, as indicated.
• Figure 5: Plot of $|\phi|$ versus $\epsilon^2$ for parameters $a = 1$, $\lambda = 0.2$, $u=1$, $\xi = 0.2$, and $K = 1$ in (a) the supercritical regime with $g = 0.5$ and (b) the subcritical regime with $g = 1.7$. The critical value of $g$ is $g_c=1.275$ for the aforementioned parameters in the positive-$g$ domain.