E5: Active Brownian particles and the Vicsek model

Week 9. Due Wed 10 Mar 23:59, reviewed Thu 11 Mar

About 2–3 hours. Theory: Active matter notes. Code: src/vicsek/ (by Harish).

1. One active Brownian particle

Simulate 400 independent active Brownian particles with the finite-difference equations in the notes, using \(v = 1\), \(D_T = 0.1\), \(D_R = 0.5\), \(\Delta t = 0.01\) (arbitrary units) and \(2\times10^4\) time steps. Plot the MSD against time on log–log axes.

1. Identify the diffusive, ballistic and enhanced-diffusive regimes. At what times do the crossovers happen? Compare with \(D_T/v^2\) and \(\tau_R = 1/D_R\).
2. Compare the long-time slope with \(D_{\rm eff} = D_T + v^2\tau_R/2\).

2. The flocking transition

Use vicsek_model.generate_vicsek_data (see vicsek_model_exercise.py) with \(N = 400\), \(L = 14\) (density \(\approx 2\)), \(r = 1\), \(v = 0.1\) and 500 iterations.

1. For noise amplitudes \(\eta = 0.1, 0.2, \ldots, 1.0\), compute the order parameter \(\psi = \frac{1}{N}\left|\sum_i \mathbf v_i / v\right|\), averaged over the last 200 iterations. Plot \(\psi\) against \(\eta\).
2. Where is the transition? Run once more at the \(\eta\) closest to the transition. How much does \(\psi\) fluctuate in time there, compared with far from it?
3. Why is \(\psi \approx 1/\sqrt{N}\), not 0, in the disordered phase?