This post provides an interactive introduction to a simple description of self-propelled particles, the “active Brownian particle” model. Use the sliders to explore how the system’s behavior changes with its parameters.

Active Brownian particles are self-propelled disks that “swim” along an orientation \(\theta\) while subject to translational and rotational noise. In the simulation below, each particle obeys overdamped Langevin dynamics in a periodic box of side \(L\), in which their positions \(\mathbf{r}_i\) and orientations \(\theta_i\) are subject to the following equations:

\[\dot{\mathbf{r}}_i = v_0\,\hat{\mathbf{e}}(\theta_i) + \mu \sum_{j \neq i} \mathbf{F}_{ij} + \sqrt{2 D_T}\,\boldsymbol{\eta}_i(t),\] \[\dot{\theta}_i = \sqrt{2 D_R}\,\xi_i(t),\]

where \(v_0\) is the propulsion speed, \(D_T\) and \(D_R\) are translational and rotational diffusion coefficients, \(\boldsymbol{\eta}_i(t)\) and \(\xi_i(t)\) are delta-correlated Gaussian white noises with \(\langle \eta_{i\alpha}(t)\,\eta_{j\beta}(t')\rangle = \delta_{ij}\,\delta_{\alpha\beta}\,\delta(t-t')\) for the translational components (\(\alpha,\beta \in \{x,y\}\)) and \(\langle \xi_i(t)\,\xi_j(t')\rangle = \delta_{ij}\,\delta(t-t')\) for the scalar rotational noise, and \(\mathbf{F}_{ij}\) is a pairwise harmonic repulsion (\(V = \tfrac{k}{2}(1 - r/\sigma)^2\) for \(r < \sigma\), zero otherwise). The simulation uses Euler–Maruyama integration with \(\mu = 1\), \(k = 100\), \(\sigma = 1\), and \(\Delta t = 0.0005\).

At high \(v_0\), low \(D_R\), and moderate density, the system undergoes motility-induced phase separation (MIPS) — particles pile up into dense clusters even though the interactions are purely repulsive. The default parameters are roughly within this regime. Try it out!

You can optionally color the particles by their orientation \(\theta\), local crystalline order (\(\lvert\psi_4\rvert\) or \(\lvert\psi_6\rvert\)), or local particle area fraction \(\phi\).

box size 40
area fraction 0.70 (N=1425)
propulsion speed 4.00
rotational diffusivity 0.20
translational diffusivity 0.01
timesteps per frame 60

I recommend playing around with the parameters! For example, set the self-propulsion speed v₀ and diffusion coefficients DR and DT to zero, crank up the area fraction, and watch as the hexatic order evolves and grain boundaries form. Load this preset. What happens as you slowly increase the self-propulsion speed?

Useful references:

  1. P. Romanczuk et al., Active Brownian particles, Eur. Phys. J. Special Topics 202, 1 (2012)

  2. M.C. Marchetti et al., Hydrodynamics of active matter, Rev. Mod. Phys. 85, 1143 (2013)

  3. C. Bechinger et al., Active particles in complex and crowded environments, Rev. Mod. Phys. 88, 045006 (2016)