Boids Model - Basic model.

• - Set random initial position/speed to \(N\) particles.

• - Set attraction/repulsion force depending on pairwise distances

• • \(\displaystyle F(p_i)=\sum_j f(\|p_j-p_i\|) \frac{p_j-p_i}{\|p_j-p_i\|} \)

• • Example
  • • - Inverse of distance \(\displaystyle f(x) = \frac{\alpha_1}{x^2}-\frac{\alpha_2}{x^4}\)
    • - Exponential/Gaussian \(\displaystyle f(x) = \alpha_1 \exp\left(-k\left(\frac{x-x_0}{x_0}\right)^2\right)-\alpha_2 \exp\left(\frac{x}{x_0}\right) \)

• - Integrate position and speed through time

• • \(v^{t+\Delta t}(p_i)=v^{t}(p_i)+\Delta t\;F(p_i^t)\)
  • \(p^{t+\Delta t}(p_i)=p^{t}(p_i)+\Delta t\;v^{t+\Delta t}(p_i)\)