Derivation and Analysis of Amplitude Equation for Generalized AMB+ in Presence of Chemical Reaction
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2026
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| author | Mondal, Sayantan Das, Prasenjit |
| author_facet | Mondal, Sayantan Das, Prasenjit |
| contents | 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 $g$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_07231 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Derivation and Analysis of Amplitude Equation for Generalized AMB+ in Presence of Chemical Reaction Mondal, Sayantan Das, Prasenjit Statistical Mechanics Soft Condensed Matter 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 $g$. |
| title | Derivation and Analysis of Amplitude Equation for Generalized AMB+ in Presence of Chemical Reaction |
| topic | Statistical Mechanics Soft Condensed Matter |
| url | https://arxiv.org/abs/2601.07231 |