Nonreciprocal Cahn–Hilliard Model

• The nonreciprocal Cahn–Hilliard model is a continuum framework for conserved scalar fields with asymmetric, non-action–reaction couplings that yield persistent currents and unique pattern formations.
• It employs antisymmetric coupling terms that break detailed balance, leading to oscillatory instabilities, traveling wave behavior, and defect-mediated dynamics.
• This model underpins research in active matter and chemically-driven materials, illuminating disorder–order transitions and enriching phase-separation theory with nonequilibrium effects.

The nonreciprocal Cahn–Hilliard (NRCH) model is a continuum description for conserved scalar fields exhibiting asymmetric (non-action–reaction) interactions, inspired by active matter, chemically-driven soft materials, and nonequilibrium statistical mechanics. Unlike the classical Cahn–Hilliard equation which is a gradient flow of a free energy, the NRCH model introduces antisymmetric coupling terms, causing deviation from detailed balance and enabling persistent currents, oscillatory states, and novel pattern formation mechanisms. Recent developments have elucidated its universal role in encoding the slow amplitude dynamics of conserved–Hopf instabilities in the hierarchy of pattern-forming systems (Frohoff-Hülsmann et al., 2023), provided sharp interface reduction methods (Gomez et al., 26 Sep 2025), and revealed disorder–order transitions controlled by nonreciprocity (Rana et al., 13 Oct 2025).

1. Mathematical Formulation and Symmetry Breaking

At its core, the NRCH model typically describes $N$ species with concentrations $\{\varphi_i\}$, evolving according to

$$\partial_t \varphi_i + \nabla \cdot \mathbf{j}_i = 0,$$

with the current

$$\mathbf{j}_i = -\nabla \mu_i^{\mathrm{eq}} - \sum_j \alpha_{ij} \nabla \varphi_j + \xi_i,$$

where $\mu_i^{\mathrm{eq}} = \delta F / \delta \varphi_i$ arises from an underlying Ginzburg–Landau energy functional $F$, and $\alpha_{ij}$ is an antisymmetric matrix ($\alpha_{ij} = -\alpha_{ji}$) quantifying nonreciprocity (Saha et al., 2020, Johnsrud et al., 10 Mar 2025). This construction breaks time-reversal symmetry and action–reaction symmetry, producing nonzero entropy production in steady states (Suchanek et al., 2023, Johnsrud et al., 4 Feb 2025).

For two fields, the effective chemical potentials become

$$\mu_1^{\mathrm{neq}} = \mu_1^{\mathrm{eq}} + \alpha\varphi_2,\quad \mu_2^{\mathrm{neq}} = \mu_2^{\mathrm{eq}} - \alpha\varphi_1,$$

with $\alpha$ controlling the degree of nonreciprocity and, at the linear level, causing the dynamical matrix to lose symmetry and induce oscillatory instabilities or traveling modes (Saha et al., 2020).

2. Phase Behavior: Static, Oscillatory, and Traveling States

The NRCH model exhibits fundamentally new states compared to its equilibrium counterpart. For small $\alpha$, phase separation proceeds via standard coarsening and Ostwald ripening; above a threshold, an oscillatory instability sets in (Saha et al., 2020, Rana et al., 13 Oct 2025). The system then displays propagating density waves or traveling bands:

• In disordered regimes (small $\alpha$), defects act as sources/sinks of waves, with global polar order negligible and the wave number selected ⟶ $k \sim \alpha^{0.6}$ for periodic boundaries (Rana et al., 13 Oct 2025).
• At a critical $\alpha_c$, defects become unstable (by an Eckhaus instability), and ordered traveling waves with finite global polar order emerge. Their frequency and velocity often scale linearly with $\alpha$.
• For Dirichlet/Neumann boundary conditions, traveling wave states are prohibited; instead, intermittent polar domains or oppositely polarized partitions arise for large nonreciprocity.

These behaviors trace to the underlying bifurcation structure: exceptional points at which linearized eigenvalues coalesce mark transitions between static and oscillatory (traveling) states (Saha et al., 2020, Suchanek et al., 2023, Johnsrud et al., 10 Mar 2025). The propagation velocity is typically

$v = \pm \sqrt{\delta^2 - \delta_c^2},$

with $\delta$ the nonreciprocal coupling strength, and $\delta_c$ the instability threshold (Suchanek et al., 2023).

3. Nonreciprocal Coupling: Microscopic Origins and Conservation Laws

The NRCH model arises robustly from microscopic models where nonreciprocal coupling is present, such as coupled Ising lattices (with different interlattice coupling strengths) and spin-exchange (Kawasaki) dynamics (Blom et al., 1 Jul 2025). Spin-exchange updates conserve each species’ magnetization, enforcing the divergence structure and generating NRCH-type equations in the macroscopic limit: $$\tau \partial_t m^a = \nabla \cdot ([1 - (m^a)^2]/2\, \nabla(\delta {\mathcal F}/\delta m^a - (K_a-K_b)/2\, m^b))$$ with analogous equations for $m^b$. Genuine nonreciprocal couplings ($K_a \neq K_b$) break detailed balance and cause dynamical phenomena such as persistent currents and oscillatory instabilities (generalized Hopf bifurcations).

Spin-flip (Glauber) dynamics, by contrast, yield nonreciprocal Allen–Cahn models without conservation. The conserved NRCH model's linear instability spectrum exhibits zero growth at $k=0$ (mass conservation) and bands of unstable $k$ set by the nonreciprocal parameter.

4. Amplitude Equation, Instabilities, and Universal Classification

Recent theoretical work situates the NRCH equation as the universal amplitude equation for conserved–Hopf instabilities in isotropic systems (Frohoff-Hülsmann et al., 2023). The generic expansion near an oscillatory bifurcation yields

$$\partial_t A = \nabla^2 \big( \alpha_1 A + \alpha_2 B - D_A \nabla^2 A + N_A(A,B)\big),\quad \partial_t B = \nabla^2 \big( \beta_1 A + \beta_2 B - D_B \nabla^2 B + N_B(A,B)\big)$$

for slow envelopes $A,B$, including both linear and cubic nonlinearities.

This fills a previously missing case in the hierarchy of pattern-forming equations: systems can destabilize via (i) stationary/oscillatory; (ii) large/small scale; (iii) with/without conservation laws. The NRCH model universally encodes out-of-equilibrium pattern formation due to conserved large-scale oscillatory modes, classified as "conserved–Hopf" amplitude equations. In many regimes, localized patches, slanted homoclinic snaking, and coexistence between uniform and oscillatory/crystalline states occur, sometimes predicted via "spurious" Maxwell constructions (Frohoff-Hülsmann et al., 2020, Greve et al., 13 Feb 2024).

5. Sharp Interface Reduction and Coherent Structure Dynamics

In the sharp-interface regime, matched asymptotic methods allow reduction of diffuse NRCH models to modified Mullins–Sekerka type equations (Gomez et al., 26 Sep 2025). For the BM model (a conservative, spatially extended FitzHugh–Nagumo analog), the interface evolution is coupled to the auxiliary field with jump and curvature conditions: $$[w] = [v] = 0,\quad w - \theta v = \gamma \kappa,\quad [\nabla w] \cdot n = - [\nabla v] \cdot n = -2 c_n$$ with $\theta$ measuring nonreciprocity. For $\theta < -1$, traveling wave trains emerge with speed $c_0$ determined by a transcendental equation $\xi + \theta \tanh \xi = 0$ and explicit stability thresholds. For $\theta \ll -1$, further instabilities produce undulated interfaces or spatiotemporal chaos. The classical result that interface motion minimizes length fails; NRCH systems may support periodic, propagating, or chaotic interfaces, providing access to analytically tractable coherent structure regimes.

6. Entropy Production, Fluctuation–Dissipation, and Thermodynamic Signatures

The NRCH model generically violates equilibrium fluctuation–dissipation relations (Johnsrud et al., 4 Feb 2025). The entropy production rate $S$ is quantified mode-by-mode via Fourier decomposition: $$S = \sum_k \sigma^k,\quad \sigma^k = \frac{4L q_k^2\delta}{\epsilon}\Big[ \beta\, \mathrm{Re} \left\langle\phi_A^k \phi_B^{-k}\right\rangle + (\kappa+\delta)\langle |\phi_A^k|^2 \rangle \Big]$$ where $\delta$ is the nonreciprocal parameter and $\epsilon$ the noise strength (Suchanek et al., 2023). Surging entropy production near static–dynamic transitions is associated with fluctuations of long-wavelength Goldstone modes and the emergence of traveling waves.

Generalized fluctuation–dissipation identities relate the deviation from equilibrium in two-point correlation and response functions directly to a renormalized entropy production vertex: $$(i\omega/D)\left[G_{R+} - G_{R+}^\dagger - (i\omega/D) C_{R-} \right] = G_{R-}^\dagger \Sigma_{R+} G_{R-}$$ clarifying how nonequilibrium activity modifies linear response and statistical fluctuations (Johnsrud et al., 4 Feb 2025).

7. Generalizations, Spinodal Dynamics, and Mobility Effects

Nonreciprocal Model B variants explore the role of density-dependent mobilities and interfacial nonreciprocity (Sahoo et al., 4 Aug 2025). Here, the kinetic coefficients $L_{ij}$ crucially affect stability: for distinct mobility forms (e.g., ideal gas, interdiffusion) the same free energy landscape may host stability islands or generically avoid exceptional point transitions via first order 'jumps' in the spinodal lengthscale. Multiple competing modes, triple points, and complex spinodal dynamics arise. Interfacial nonreciprocity (parameter $\delta$) further expands the regime of oscillatory (travelling) instabilities, though bounded by global stability conditions, as determined analytically by conditions such as $q < \sqrt{-\mathrm{tr}\tilde{H}/\mathrm{tr}L K}$.

References to Representative Models and Key Works

• Scalar Active Mixtures and Minimal NRCH: (Saha et al., 2020, Frohoff-Hülsmann et al., 2023, Johnsrud et al., 10 Mar 2025)
• Sharp Interface Dynamics of BM Model: (Gomez et al., 26 Sep 2025)
• Disorder–Order Transition and Defect Dynamics: (Rana et al., 13 Oct 2025)
• Nonlocal and Nonreciprocal CH Models on Graphs/Random Walk Spaces: (Mazón et al., 2022)
• Entropy Production and Generalized FDR: (Suchanek et al., 2023, Johnsrud et al., 4 Feb 2025)
• Model B/Emergent Mobility Effects: (Sahoo et al., 4 Aug 2025)
• Universal Amplitude Equation: (Frohoff-Hülsmann et al., 2023, Frohoff-Hülsmann et al., 2022)
• Bifurcation Structures and Localized States: (Frohoff-Hülsmann et al., 2020, Greve et al., 13 Feb 2024)

Summary

The nonreciprocal Cahn–Hilliard model constitutes a fundamental extension of classical phase-separation dynamics, built from asymmetric coupling that drives persistent currents, traveling waves, oscillatory patches, and defect-laden microstructures, with sharply defined disorder–order and coarsening transitions. These behaviors are grounded in both rigorous macroscopic derivations from nonreciprocal microscopic models and systematic amplitude equation analyses. NRCH models illuminate how conservation laws, broken reciprocal symmetry, and active matter principles combine to produce rich spatiotemporal organization beyond the reach of equilibrium paradigms, with impact on the design and interpretation of active materials, phase field theories, and biological patterning.