README.md

Active particles confined 3D

The aim of this project is to build simulations describings the motion of active interacting particles under a cylindrical confinement. These simulations are based on Langevin equations and used the Euler-Mayurama algorithm. Active interactive particles evolve in different geometries, such as circular or squared. The dynamics is given by two Langevins equations, one for the position $\mathbf{\tilde{r}}(\tilde{x},\tilde{y},\tilde{z})$ of the particles and one for its orientation $\mathbf{e}$

$$ \begin{align} \frac{d}{d\tilde{t}}\mathbf{\tilde{r}} &= \tilde{v_s}\mathbf{e} - \tilde{\nabla}_{\tilde{R}}(\tilde{LP}) + \sqrt{2\tilde{D}_t}\tilde{\mathbf{\xi}_t},\\ \frac{d}{d\tilde{t}}\mathbf{e} &= \sqrt{2\tilde{D}_e}\mathbf{e}\times\tilde{\mathbf{\xi}_e}, \end{align} $$

where $\mathbf{e} = (e_x,e_y,e_z)^{T}$ is the orientational unit vector, $\tilde{v_s}$ is the self-propulsion, $\tilde{D_{t}}$ and $\tilde{D_{e}}$ are the translational and rotational diffusivities, respectively. Moreover, $\langle \tilde{\xi_{t_i}}(\tilde{t}')\tilde{\xi_{t_j}}(\tilde{t}) \rangle = \delta_{ij}\delta(\tilde{t}'-\tilde{t})$ and $\langle \tilde{\xi_{e_i}}(\tilde{t}')\tilde{\xi_{e_j}}(\tilde{t}) \rangle = \delta_{ij}\delta(\tilde{t}'-\tilde{t})$ are two Gaussian white noises. Moreover, the interactions between the particles is represented by using the Lennard-Jones potential

$$ \tilde{LP} = 4\tilde{\epsilon}[(\frac{\tilde{\sigma}}{\tilde{R}})^{12} - (\frac{\tilde{\sigma}}{\tilde{R}})^{6}], $$

where $\tilde{\epsilon}$ is the depth of the potential well, $\tilde{R}$ is the distance between two interacting particles. In this project, only the repulsive part of the potential is considered.

Visualizations

Video

video.mp4

Seventeen different particles

Particles trajectories