Overview
SelfPropelledVoronoi.jl is a Julia package designed for simulating two-dimensional (2D) systems of self-propelled particles with periodic boundary conditions. The central idea is that particle interactions and spatial organization are determined by a dynamic Voronoi tessellation of the simulation domain. Each particle corresponds to a Voronoi cell, and the properties of these cells (like area and perimeter) influence particle behavior.
Core Mechanics
The behavior of particles in SelfPropelledVoronoi.jl is governed by a few key mechanical and kinetic principles:
Voronoi Tessellation: The simulation space is continuously partitioned into Voronoi cells, with each cell uniquely associated with a particle. The geometry of these cells (area, perimeter, number of neighbors) determines the forces acting on the cells.
Area and Perimeter Elasticity:
- Each particle (cell) has a preferred or "target" area (
A0) and a target perimeter (p0). - Deviations from these target values result in an elastic energy penalty. For example, if a cell's actual area
Ais different fromA0, there is an energy contribution proportional toKA * (A - A0)^2, whereKAis an area stiffness constant. A similar term applies for perimeter deviations with a perimeter stiffnessKP. - These energy penalties translate into forces that drive the cells to relax towards their target geometries, influencing particle movement and rearrangement.
- Each particle (cell) has a preferred or "target" area (
Active Particle Motion (Self-Propulsion):
- Particles are "active," meaning they generate their own motion. Each particle
ihas an intrinsic self-propulsion force of magnitudef0_i. - This force is directed along an orientation vector
(cos(θ_i), sin(θ_i)), whereθ_iis the particle's current orientation angle.
- Particles are "active," meaning they generate their own motion. Each particle
Rotational Diffusion:
- The orientation
θ_iof each particle is not fixed but evolves stochastically over time due to rotational diffusion. - This is modeled by adding a random angular displacement at each time step, with the rate of change controlled by a rotational diffusion constant
Dr_i.
- The orientation
Periodic Boundary Conditions:
- The simulation is performed in a rectangular domain with periodic boundary conditions. This means that particles exiting one side of the box re-enter from the opposite side, allowing for the simulation of bulk system behavior without edge effects.
By combining these elements, SelfPropelledVoronoi.jl provides a tool to explore how local interactions and active forces at the particle level give rise to complex, large-scale behaviors in 2D active systems. The reference to Bi et al. (Physical Review X 6.2 (2016): 021011) provides an introduction to this model.
Specifically, it solves the following equations of motion for every particle:
\[d\mathbf{r} = \left(\frac{1}{\gamma} \mathbf{F} + v_0 \hat{\mathbf{n}}\right)dt,\]
in which $\mathbf{r}$ is the particle position, $\gamma$ is the friction constant, $\mathbf{F}$ is the potential force $\mathbf{F} = - \nabla H$, $v_0$ is the active velocity, and $\hat{\mathbf{n}}$ is the direction of the active force that is governed by the angle $\theta$ which satisfies a stochastic differential equation:
\[d\theta = \sqrt{2D_r}dW,\]
in which $D_r$ is the rotational diffusion constant and $dW$ is a Wiener process (standard white noise). The potential force is generated by a Hamiltonian $H$, which is given by
\[H = \sum_i \left(K_{A,i}(A_i-A_{0,i})^2+K_{P,i}(P_i-P_{0,i})^2\right)\]
of which the parameters are explained above.