brusselator.qmd

---
title: "Brusselator Model"
subtitle: "Hexagonal Turing patterns with tight particle clusters"
---

## Field Dynamics

The concentration fields $(C_1, C_2)$ evolve according to the Brusselator reaction-diffusion system:

$$
\frac{\partial C_1}{\partial t} = D_1 \nabla^2 C_1 + \text{Da}_c \left( A - (B+1)C_1 + C_1^2 C_2 \right) + \chi \nabla^2 C_2
$$

$$
\frac{\partial C_2}{\partial t} = D_2 \nabla^2 C_2 + \text{Da}_c \left( B C_1 - C_1^2 C_2 \right)
$$

### Turing Analysis

The homogeneous steady state is $(C_1^*, C_2^*) = (A, B/A)$.

**Linear stability:** Perturbations $\delta C \propto e^{\lambda t + i\mathbf{k}\cdot\mathbf{x}}$ grow when:

$$
\lambda = \frac{1}{2}\left[ \text{tr}(\mathbf{J}_k) + \sqrt{\text{tr}(\mathbf{J}_k)^2 - 4\det(\mathbf{J}_k)} \right] > 0
$$

where $\mathbf{J}_k$ is the linearized Jacobian with diffusion:

$$
\mathbf{J}_k = \begin{pmatrix}
B - 1 - D_1 k^2 & A^2 \\
-B & -A^2 - D_2 k^2
\end{pmatrix}
$$

**Turing instability condition:**
$$
B > 1 + A^2 \quad \text{and} \quad D_2/D_1 > \frac{(A + \sqrt{B-1})^2}{B - 1 - A^2}
$$

### Parameters

| Symbol | Description | Optimal | Range |
|--------|-------------|---------|-------|
| $D_1$ | Activator diffusion | 0.015 | 0.01–0.5 |
| $D_2$ | Inhibitor diffusion | 0.15 | 0.1–1.0 |
| $\text{Da}_c$ | Damköhler number | 25 | 1–50 |
| $A$ | Feed concentration | 1.5 | 0.5–5.0 |
| $B$ | Reaction strength | 7.0 | 1.0–10.0 |
| $\chi$ | Cross-diffusion | 0 | -0.1–0.1 |

---

## Particle Dynamics: Diffusiophoresis

$$
\mathbf{v}_i = M_1 \nabla C_1\big|_{\mathbf{x}_i} + M_2 \nabla C_2\big|_{\mathbf{x}_i} + \sqrt{\frac{2}{\text{Pe}}} \boldsymbol{\eta}(t)
$$

**Particle → Field feedback:**

$$
\frac{\partial C_1}{\partial t}\bigg|_{\text{particle}} = -\gamma_c \sum_{j=1}^{N_p} w(\|\mathbf{x} - \mathbf{x}_j\|)
\quad , \quad
\frac{\partial C_2}{\partial t}\bigg|_{\text{particle}} = +\gamma_p \sum_{j=1}^{N_p} w(\|\mathbf{x} - \mathbf{x}_j\|)
$$

| Symbol | Description | Optimal |
|--------|-------------|---------|
| $M_1, M_2$ | Mobility coefficients | ±4 |
| $\gamma_c$ | Consumption rate | 180 |
| $\gamma_p$ | Production rate | -180 |

---

## Key Discoveries

::: {.callout-important}
## Mobility Sweet Spot
**M1=±4, M2=±4** is the optimal mobility magnitude. M1=±16 causes particle escape (0% retention). The ±4 range maintains strong field-particle coupling while preventing particle loss.
:::

::: {.callout-tip}
## Hexagonal Mode Selection
Extended simulation (6000 frames) reveals **ring→hexagonal symmetry breaking**. The concentric ring pattern evolves into hexagonal Turing spots.
:::

| Finding | Mechanism |
|---------|-----------|
| **Hexagonal is particle-type-independent** | 1, 2, or 3 particle types all achieve identical hexagonal mode |
| **~6 large spots** | Coarse spatial wavelength compared to FHN |
| **Tight clustering (0.6+)** | Particles form dense aggregates at field peaks |
| **Ring→hexagonal transition** | Occurs at 4000-6000 frames with extended simulation |

---

## Stripe/Labyrinth Mode (Block 4 Discovery)

::: {.callout-important}
## Breakthrough: Turing Boundary Controls Mode Selection
At the Turing instability boundary $B/(1+A^2) = 1$, the system transitions from hexagonal spots to **labyrinth/stripe patterns**. This is controlled by the Brusselator parameters $A$ and $B$.
:::

### Turing Bifurcation Map

| Regime | Condition | Pattern |
|--------|-----------|---------|
| Sub-Turing | $B/(1+A^2) < 1$ | No patterns (disordered) |
| Turing boundary | $B/(1+A^2) \approx 1$, low $A$ ($\leq 3$) | **Stripes/Labyrinth** |
| Deep Turing | $B/(1+A^2) > 1$ | Hexagonal spots |
| Extreme Turing | $B \geq 15$ | Pixel-scale noise (unusable) |

### Key Constraints

- **Stripe mode requires 1-type uniform particles**: Multi-type configurations (even with identical parameters) destroy stripe organization
- **Low A is critical**: $B/(1+A^2) = 1$ at $B=13, A=3.46$ gives spots, NOT stripes. Low $A$ ($\leq 3$) is required alongside the ratio
- **Practical limit**: $B \leq 13$ for 100x100 mesh; $B=15$ produces pixel-scale patterns too fine for particle organization

### Stripe vs Hexagonal Parameters

| Parameter | Hex Mode | Stripe Mode |
|-----------|----------|-------------|
| $A$ | 4.5 | 3.0 |
| $B$ | 6.5 | 10.0 |
| $B/(1+A^2)$ | 0.29 (deep) | 1.0 (boundary) |
| n_particle_types | 1-3 | 1 only |

### Robustness Hierarchy

Hexagonal (survives multi-type, reversed feedback) > Stripe (1-type uniform only)

---

## Nonlinear Diffusion: Labyrinthine and Vermiform Modes (Block 11)

::: {.callout-important}
## Breakthrough: PDE Modification Unlocks New Morphologies
After exhausting all 8 particle-level code features (80 iterations), the LLM modified the Brusselator PDE itself: **concentration-dependent diffusion** breaks single-wavelength lock and produces qualitatively new pattern types inaccessible to any parameter tuning.
:::

### Nonlinear Diffusion (NLD)

The activator diffusion becomes concentration-dependent (Gambino et al. 2013):

$$
D_1(C_1) = D_1^0 \left[ 1 + \delta \frac{(C_1 - A)^2}{A^2} \right]
$$

where $\delta$ controls the nonlinearity strength. At concentration peaks ($C_1 \gg A$), diffusion increases, spreading the activator and breaking the fixed Turing wavelength.

### NLD Pattern Selection

| B/A Ratio | NLD $\delta$ | Pattern | Best Example |
|:---------:|:----------:|---------|:------------:|
| 1.36 ($A$=5.5, $B$=7.5) | 2.0 | Hexagonal (slightly larger spots) | Iter 82 (7/10) |
| **1.83** ($A$=3.0, $B$=5.5) | **2.0** | **Labyrinthine** ridges | **Iter 83** (7/10) |
| **2.50** ($A$=2.0, $B$=5.0) | **2.0** | **Vermiform** filamentary chains | **Iter 87** (7/10) |
| 1.36 ($A$=5.5, $B$=7.5) | 3.0 | Over-damped hexagonal | Iter 86 (6/10) |

### Key Constraints

- **NLD requires moderate coupling**: |$\chi$| $\leq$ 8 and consumption $\leq$ 80. Stronger coupling ($\chi=-16$) with NLD $\delta \geq 1.0$ causes blowup (Iter 81)
- **High B/A ratio required**: NLD alone at $A=5.5/B=7.5$ does not trigger labyrinthine transition --- the B/A ratio must be $\geq 1.8$
- **$\delta=2.0$ is optimal**: $\delta=3.0$ over-damps fields (Iter 86). $\delta=1.0$ is insufficient for labyrinthine at moderate coupling
- **Vermiform = strongest fields ever**: Iter 87 achieved pattern\_growth=294, nearly $2\times$ the previous maximum

### Mode Hierarchy (Complete)

Hexagonal (deep Turing, most robust) > Labyrinthine ($B/A \approx 1.8$ + NLD) > Vermiform ($B/A \approx 2.5$ + NLD) > Stripe ($B/(1+A^2) = 1$, 1-type only)

---

## Exploration Gallery

### Iter 3 (5/10)

![Strong radial Turing patterns develop in C1/C2 fields with clear ring/spot structures. Two particle types show spatial segregation: orange particles cluster at concentration maxima while blue particles spread to interstitial regions. Late frames show increasing complexity with multiple nested ring structures.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_003.png){.lightbox group="bruss"}

### Iter 12 (7/10)

![2-type with same-sign mobilities (both attracted to C1 peaks). Fields develop vigorous Turing patterns with ring structures breaking into regular spot arrays. Both types cluster at C1 hotspots but with differential strength: orange forms tight inner cores, blue forms looser outer halos. Late frames show star-like co-localized core-shell micro-clusters at each Turing spot.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_012.png){.lightbox group="bruss"}

### Iter 14 (8/10)

![C1/C2 fields develop elaborate multi-lobed flower/yin-yang patterns with asymmetric branching. Three particle types create tissue-like architecture: blue forms innermost core clusters, orange occupies intermediate zones forming branch arms, green creates outermost boundary layer. Cross-type adhesion sharpens inter-type boundaries into discrete layered domains. Late frames show a distinctive flower/mandala morphology with 3--5 branching arms and clear type stratification within each arm.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_014.png){.lightbox group="bruss"}

### Iter 31 (6/10)

![Fields develop clean concentric ring patterns (bullseye). Two particle types form remarkably clean annular bands --- blue inner core, orange outer ring --- that remain stable throughout. No hexagonal breakup or satellite spots emerge. Weber-Fechner sensing compresses the dynamic range enough to suppress hexagonal instability while maintaining radial structure.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_031.png){.lightbox group="bruss"}

### Iter 36 (5/10)

![Fields develop the strongest Turing patterns in this batch with multiple spots across the domain. Both types cluster at the same spot centers --- blue surrounded by orange --- forming co-localized core-shell micro-clusters. Same-sign mobilities produce co-localization rather than segregation.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_036.png){.lightbox group="bruss"}

### Iter 45 (8/10)

![Hexagonal Turing spots develop in both fields at 150x150 mesh resolution. Particles self-organize into elaborate flower/mandala with radial arms and type-specific petal sorting. 3-type segregation creates concentric core-ring evolving into multi-armed rosette with radial spokes. Higher resolution enables more elaborate petal structure.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_045.png){.lightbox group="bruss"}

### Iter 83 (7/10)

![Fields show labyrinthine/vermiform Turing patterns (not hexagonal spots). The hex-to-labyrinth transition produces very strong field development with interconnected ridges rather than isolated spots. Particles form scattered clusters tracking field maxima along labyrinthine ridges.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_083.png){.lightbox group="bruss"}

### Iter 85 (7/10)

![Fields develop from noise through concentric rings into labyrinthine multi-lobed flower/mandala with 3-type segregation. Particle rows show 3-layer stratified flower (blue core, orange intermediate, green outer) with elaborate branching lobes that trace labyrinthine field topology. More dispersed branching than compact flower morphologies --- the labyrinthine field creates wider-spaced branches.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_085.png){.lightbox group="bruss"}

### Iter 87 (7/10)

![Dramatically strong patterns with fragmented labyrinthine topology and many small-scale features. Particles form scattered elongated vermiform/worm-like clusters that trace the field topology --- a qualitatively new morphology of filamentary particle chains weaving through the domain.](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_087.png){.lightbox group="bruss"}