Trae Active Transport Workflow

• theory simplification and nondimensionalization
• simulation parameter scans
• convergence and statistical validation
• figure generation for main Figures 1–4
• Trae IDE task orchestration and milestone tracking

2. Scientific hypotheses to test

H1. Productive-memory window

There exists an internal ridge in parameter space, rather than a boundary optimum, such that transport efficiency is maximized when

[ \Pi_m + \Pi_f \sim O(1), \qquad \Pi_p \gtrsim 1 ]

where (\Pi_m) is the memory-to-gate time ratio, (\Pi_f) is the feedback-delay-to-gate time ratio, and (\Pi_p) is the persistence-to-gate-length ratio.

H2. Speed-efficiency separation

The parameter set that minimizes mean first-passage time (MFPT) is generically different from the one that maximizes dissipation-normalized transport efficiency.

H3. Failure by trapping, not just slowing down

Outside the productive-memory window, transport degrades primarily because of prolonged wall trapping, revisit cycles, and stale steering, rather than a simple reduction in instantaneous speed.

H4. Weak-flow assistance and strong-flow breakdown

A weak co-flow can enlarge the productive window, while strong flow can collapse it by over-biasing trajectories toward wall hugging or wrong-gate overshoot.

H5. Geometry-robust collapse

The same nondimensional control law should approximately organize results across several geometry families.

3. Minimal model summary

State variables per particle:

• position: ((x,y))
• velocity: ((v_x,v_y))
• memory auxiliary variable: ((q_x,q_y))
• heading: (\theta)

Dynamics:

• active self-propulsion with speed scale (v_0)
• uniform background flow (\mathbf U=(U,0))
• wall soft-repulsion from signed distance field
• viscoelastic memory via single-exponential kernel implemented with Markovian embedding
• delayed alignment to the local negative navigation-field gradient
• translational and rotational noise

Primary observables:

• success probability (P_{\mathrm{succ}})
• MFPT and FPT quantiles
• total trap time and trap count
• wall-contact fraction
• drag-dissipation proxy (\Sigma_{\mathrm{drag}})
• transport proxy (J_{\mathrm{proxy}})
• dissipation-normalized efficiency (\eta_\sigma)
• alignment correlation and revisit statistics

4. Nondimensional control parameters

Use a reference geometry and baseline dynamics (no_memory, no_feedback, U=0) to define:

• gate-search length: (\ell_g)
• gate-crossing time: (\tau_g)
• persistence time: (\tau_p = D_r^{-1})
• persistence length: (\ell_p = v_0 \tau_p)
• effective memory time: (\tau_{\mathrm{mem}})

Recommended dimensionless groups:

[ \Pi_p = \frac{\ell_p}{\ell_g} = \frac{v_0 / D_r}{\ell_g} ]

[ \Pi_m = \frac{\tau_{\mathrm{mem}}}{\tau_g} ]

[ \Pi_f = \frac{\tau_f}{\tau_g} ]

[ \Pi_U = \frac{U}{v_0} ]

[ \Pi_W = \frac{w_{\mathrm{gate}}}{L} ]

[ \Pi_B = \frac{\delta_{\mathrm{wall}}}{L} ]

Optional secondary controls:

• memory strength ratio (\chi_m = \gamma_1 / (\gamma_0 + \gamma_1))
• translational noise ratio (\Theta_t = k_B T / (\gamma_0 v_0 \ell_g))
• rotational noise ratio (\Theta_r = D_r \tau_g)

5. Parameter table

5.1 Baseline dimensional defaults

5.2 Primary scan ranges in dimensionless form

5.3 Scan phases

6. Theory-to-simulation workflow

Stage T1. Baseline reference extraction

Compute (\ell_g) and (\tau_g) from the reference model:

• no memory
• no delayed feedback
• no external flow
• main geometry only

Output:

• reference_scales.json
• baseline_transition_stats.parquet

Acceptance:

• stable within 5% under seed and run-length doubling

Stage T2. Coarse-grained gate model

Fit a reduced-state description:

• bulk search
• wall sliding
• gate capture
• successful crossing / failure return

Target outputs:

• effective rates (k_{\mathrm{search}}, k_{\mathrm{slide}}, k_{\mathrm{cross}}, k_{\mathrm{return}})
• phenomenological productive-memory criterion

Acceptance:

• reduced model qualitatively reproduces ridge position and trapping growth

Stage T3. Small-delay / small-memory asymptotics

Derive and verify low-order corrections to crossing rate and alignment phase lag.

Output:

• note or appendix derivation
• symbolic expressions for local expansion
• comparison plots against simulations at small (\Pi_m, \Pi_f)

Stage T4. Large-delay mismatch regime

Characterize stale-control regime through revisit rate, trap-time growth, and alignment decorrelation.

Output:

• asymptotic narrative and scaling ansatz
• one summary panel for stale steering signature

7. Repository and file structure

Recommended repository layout:

project_root/
├─ README.md
├─ pyproject.toml
├─ configs/
│  ├─ geometries/
│  │  ├─ maze_main.yaml
│  │  ├─ channel_single_bottleneck.yaml
│  │  ├─ pore_array.yaml
│  │  └─ random_labyrinth.yaml
│  ├─ scans/
│  │  ├─ phaseA_validation.yaml
│  │  ├─ phaseB_ablations.yaml
│  │  ├─ phaseC_global_sobol.yaml
│  │  ├─ phaseD_refinement.yaml
│  │  ├─ phaseE_rare_events.yaml
│  │  └─ phaseF_geometry_transfer.yaml
│  └─ figures/
│     ├─ fig1_productive_memory.yaml
│     ├─ fig2_speed_efficiency_separation.yaml
│     ├─ fig3_trapping_mechanism.yaml
│     └─ fig4_geometry_collapse.yaml
├─ docs/
│  ├─ theory_notes.md
│  ├─ derivations/
│  ├─ manuscript_notes/
│  └─ trae_active_transport_workflow.md
├─ src/
│  ├─ core/
│  │  ├─ geometry.py
│  │  ├─ navigation.py
│  │  ├─ dynamics.py
│  │  ├─ kernels.py
│  │  ├─ observables.py
│  │  └─ rng.py
│  ├─ scans/
│  │  ├─ generate_design.py
│  │  ├─ run_point.py
│  │  ├─ run_batch.py
│  │  ├─ adaptive_refine.py
│  │  └─ rare_event.py
│  ├─ analysis/
│  │  ├─ aggregate.py
│  │  ├─ bootstrap.py
│  │  ├─ ridge_detection.py
│  │  ├─ collapse.py
│  │  └─ diagnostics.py
│  ├─ figures/
│  │  ├─ fig1.py
│  │  ├─ fig2.py
│  │  ├─ fig3.py
│  │  └─ fig4.py
│  └─ cli/
│     ├─ scan_cli.py
│     ├─ analysis_cli.py
│     └─ figure_cli.py
├─ jobs/
│  ├─ local/
│  ├─ slurm/
│  └─ manifests/
├─ data/
│  ├─ raw/
│  ├─ interim/
│  ├─ processed/
│  └─ reference/
├─ outputs/
│  ├─ logs/
│  ├─ tables/
│  ├─ figures/
│  └─ reports/
└─ tests/
   ├─ test_geometry.py
   ├─ test_dynamics.py
   ├─ test_observables.py
   ├─ test_reproducibility.py
   └─ test_figures.py

8. Parallel task decomposition

8.1 Unit of work

The atomic unit should be:

(geometry_id, model_variant, state_point_id, seed_chunk_id)