ActiveTrail-2D

ActiveTrail-2D is a 2D simulation program written in modern Fortran that models the collective dynamics of self-driven particles (e.g., ants or active agents) moving on a discrete grid and interacting indirectly through pheromone-like fields. The model combines stochastic motion, local interactions, and probabilistic behavioral rules, leading to emergent collective organization and trail formation.

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🧠 Physical and Mathematical Model

The system represents a population of $N_f$ active agents moving on a rectangular grid $(n \times m)$ of spacing $(\Delta x, \Delta y)$, confined between reflective boundaries.

1. Particle Motion

Each particle $i$ has position $(x_i, y_i)$ and velocity $(u_i, v_i)$ evolving according to a stochastic dynamics:

$$ m \frac{du_i}{dt} = -\gamma u_i + F_i^{\text{rand}} + F_i^{\text{rep}}, $$

where:

• $m$ is the particle mass;
• $\gamma$ is the friction coefficient;
• $F_i^{\text{rand}}$ is a stochastic force (Gaussian white noise in the Euler–Maruyama version);
• $F_i^{\text{rep}}$ is a short-range repulsive force preventing particle overlap.

In the v0.1 base version, random forces are uniform in $[-\Delta x, \Delta x]$, while velocities and positions are updated via a 4th-order Runge–Kutta scheme.
In the extended versions (v0.2+), the update follows the Euler–Maruyama stochastic integrator with Gaussian noise.

2. Repulsive Interaction

Particles interact only when they are within a cutoff distance $d_0$. The deterministic repulsive force follows a linear penalty model:

$$ \mathbf{F}_{ij}^{\text{rep}} = k_{\text{rep}}(d_0 - r_{ij})\hat{\mathbf{r}}_{ij}, \quad r_{ij} < d_0, $$

where $k_{\text{rep}}$ is a stiffness coefficient.
The total deterministic force on particle $i$ is $\mathbf{F}i^{\text{rep}} = \sum_j \mathbf{F}{ij}^{\text{rep}}$.

3. Pheromone (Trace) Field

A binary field $F(x,y,t)$ marks the grid cells visited by particles:

• When a particle occupies a cell, that cell’s pheromone value becomes 1 (deposition rule).
• When a cell is empty, its pheromone value evaporates with probability $p$ per time step (evaporation rule).

This dynamic field provides indirect communication among agents — creating a memory of past trajectories that guides future motion statistically.

4. Behavioral Rule: Stay or Leave

When a particle enters a cell without pheromone, it must decide whether to stay or move on:

• With probability q, it stays (remains in that cell for the next step);
• With probability 1–q, it receives a small random “kick” and moves away.

If the cell already contains pheromone, the particle always stays (reinforcement of trail-following). Here, we set $q < 0.5$ so the particle has a higher probability to stay in a cell that has a pheromone trace.

5. Boundary Conditions

Reflective conditions ensure confinement: $$ x_i, y_i \in [x_{\min}, x_{\max}] \times [y_{\min}, y_{\max}]. $$

Whenever a particle attempts to exit the domain, its position is slightly displaced inward.

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⚙️ Simulation Algorithm

1. Random initialization of particle positions.
2. Mapping of particles into discrete grid cells.
3. At each time step:
   • Generate stochastic forces (random or Gaussian).
   • Compute short-range repulsion (if enabled).
   • Update velocities and positions (RK4 or EM).
   • Handle collisions (one particle per cell).
   • Apply pheromone-based behavioral rules (stay/leave).
   • Evaporate pheromone field.
   • Record particle and field states to output.

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📂 Output Files

File	Description	Format
trajectories.csv	Particle positions and velocities per time step	CSV (t, id, x, y, u, v)
traces.csv	Coordinates of marked cells (pheromone traces)	CSV (t, x, y)
active_trail_trajectories.gif	Animation of particle trajectories	GIF (via Gnuplot)
active_trail_traces.gif	Animation of trace points over time	GIF (via Gnuplot)

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🧩 Key Parameters

Parameter	Meaning	Default
n, m	Grid resolution	50 × 50
Nf	Number of particles	10,000
dt	Time step	0.01
p	Evaporation probability	0.6
q	Stay probability (if no trace)	0.8
time	Total simulated time	20.0
out_stride	Output frequency (steps between frames)	10

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🧮 Numerical and Algorithmic Details

• Integrator: 4th-order Runge–Kutta (v0.1); Euler–Maruyama (v0.2+).
• Random Forces: Uniform ([-dx, dx]) or Gaussian (σ√dt).
• One-Particle-per-Cell Rule: Prevents overlap by local displacement.
• RNG: Seeded once for consistent random sequences.
• Boundary Handling: Reflective displacement inward.
• Output: CSV files for postprocessing and animation via Gnuplot.

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🎞️ Visualization

The included postprocess.gnu script can generate:

• Animated trajectories (active_trail_trajectories.gif)
• Pheromone trace evolution (active_trail_traces.gif)

Usage:

gnuplot postprocess.gnu

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🧑‍💻 Authors

Developed and maintained at LCEC–UNB
(Laboratório de Computação Científica em Escoamentos Complexos).

Developer: Prof. Rafael Gabler Gontijo

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© 2025 LCEC–UNB. Open research and educational use permitted with citation.