Integrating Agents.jl with CellListMap.jl

This example illustrates how to integrate Agents.jl with CellListMap.jl, to accelerate the computation of short-ranged (within a cutoff) interactions in 2D and 3D continuous spaces. CellListMap.jl is a package that allows the computation of pairwise interactions using an efficient and parallel implementation of cell lists.

The system simulated

The example will illustrate how to simulate a set of particles in 2 dimensions, interacting through a simple repulsive potential of the form:

\[U(r) = k_i k_j\left[r^2 - (r_i+r_j)^2\right]^2~~~\textrm{for}~~~r \leq (r_i+r_j)\]
\[U(r) = 0.0~~~\textrm{for}~~~r \gt (r_i+r_j)\]

where $r_i$ and $r_j$ are the radii of the two particles involved, and $k_i$ and $k_j$ are constants associated to each particle. The potential energy function is a smoothly decaying potential with a maximum when the particles overlap.

Thus, if the maximum sum of radii between particles is much smaller than the size of the system, cell lists can greatly accelerate the computation of the pairwise forces.

Each particle will have different radii and different repulsion force constants and masses.

using Agents

Below we define the Particle type, which represents the agents of the simulation. The Particle type, for the ContinousAgent{2} space, will have additionally an id and pos (position) and vel (velocity) fields, which are automatically added by the @agent macro.

@agent Particle ContinuousAgent{2} begin
    r::Float64 # radius
    k::Float64 # repulsion force constant
Particle(; id, pos, vel, r, k, mass) = Particle(id, pos, vel, r, k, mass)

Required and data structures for CellListMap.jl

We will use the high-level interface provided by the PeriodicSystems module (requires version ≥0.7.22):

using CellListMap.PeriodicSystems
using StaticArrays

StaticArrays provides the SVector type, which is practical for the representation of various vector types (e.g., positions or velocities) in small amount of dimensions. Agents.jl uses NTuple{D, Float64} for that, which does not support vector operations out of the box. In the future, Agents.jl may also switch the pos type to a static vector.

Two auxiliary arrays will be created on model initialization, to be passed to the PeriodicSystem data structure:

  1. positions: CellListMap requires a vector of (preferentially) static vectors as the positions of the particles. To avoid creating this array on every call, a buffer to which the agent.pos positions will be copied is stored in this data structure.
  2. forces: In this example, the property to be computed using CellListMap.jl is the forces between particles, which are stored here in a Vector{<:SVector}, of the same type as the positions. These forces will be updated by the map_pairwise! function.

Additionally, the computation with CellListMap.jl requires the definition of a cutoff, which will be twice the maximum interacting radii of the particles, and the geometry of the the system, given by the unitcell of the periodic box.

More complex output data, variable system geometries and other options are supported, according to the CellListMap.PeriodicSystems user guide.

Model initialization

We create the model with a keyword-accepting function as is recommended in Agents.jl. The keywords here control number of particles and sizes.

function initialize_model(;
    sides=SVector(500.0, 500.0),
    # initial random positions
    positions = [sides .* rand(SVector{2,Float64}) for _ in 1:number_of_particles]

    # We will use CellListMap to compute forces, with similar structure as the positions
    forces = similar(positions)

    # Space and agents
    space2d = ContinuousSpace(Tuple(sides); periodic=true)

    # Initialize CellListMap periodic system
    system = PeriodicSystem(
        cutoff=2 * max_radius,
        output_name=:forces, # allows the system.forces alias for clarity

    # define the model properties
    # The clmap_system field contains the data required for CellListMap.jl
    properties = (
    model = ABM(Particle,

    # Create active agents
    for id in 1:number_of_particles
                r=(0.5 + 0.9 * rand()) * max_radius,
                k=(10 + 20 * rand()), # random force constants
                mass=10.0 + 100 * rand(), # random masses
                vel=(100 * randn(), 100 * randn()), # initial velocities

    return model
initialize_model (generic function with 1 method)

Computing the pairwise particle forces

To follow the CellListMap interface, we first need a function that computes the force between a single pair of particles. This function receives the positions of the two particles (already considering the periodic boundary conditions), x and y, their indices in the array of positions, i and j, the squared distance between them, d2, the forces array to be updated and the model properties.

Given these input parameters, the function obtains the properties of each particle from the model, and computes the force between the particles as (minus) the gradient of the potential energy function defined above.

The function must return the forces array, to follow the CellListMap API.

function calc_forces!(x, y, i, j, d2, forces, model)
    pᵢ = model[i]
    pⱼ = model[j]
    d = sqrt(d2)
    if d ≤ (pᵢ.r + pⱼ.r)
        dr = y - x
        fij = 2 * (pᵢ.k * pⱼ.k) * (d2 - (pᵢ.r + pⱼ.r)^2) * (dr / d)
        forces[i] += fij
        forces[j] -= fij
    return forces
calc_forces! (generic function with 1 method)

The model_step! function will use CellListMap to update the forces for all particles. The first argument of the call is the function to be computed for each pair of particles, which closes-over the model data to call the calc_forces! function defined above.

function model_step!(model::ABM)
    # Update the pairwise forces at this step
        (x, y, i, j, d2, forces) -> calc_forces!(x, y, i, j, d2, forces, model),
    return nothing
model_step! (generic function with 1 method)

Update agent positions and velocities

The agent_step! function will update the particle positions and velocities, given the forces, which are computed in the model_step! function. A simple Euler step is used here for simplicity. We need to convert the static vectors to tuples to conform the Agents API for the positions and velocities of the agents. Finally, the positions within the CellListMap.PeriodicSystem structure are updated.

function agent_step!(agent, model::ABM)
    id =
    dt =
    # Retrieve the forces on agent id
    f = model.system.forces[id]
    a = f / agent.mass
    # Update positions and velocities
    v = SVector(agent.vel) + a * dt
    x = SVector(agent.pos) + v * dt + (a / 2) * dt^2
    x = normalize_position(Tuple(x), model)
    agent.vel = Tuple(v)
    move_agent!(agent, x, model)
    # !!! IMPORTANT: Update positions in the CellListMap.PeriodicSystem
    model.system.positions[id] = SVector(agent.pos)
    return nothing
agent_step! (generic function with 1 method)

The simulation

Finally, the function below runs an example simulation, for 1000 steps.

function simulate(model=nothing; nsteps=1_000, number_of_particles=10_000)
    if isnothing(model)
        model = initialize_model(number_of_particles=number_of_particles)
        model, agent_step!, model_step!, nsteps, false,
simulate (generic function with 2 methods)

Which should be quite fast

model = initialize_model()
simulate(model) # compile
@time simulate(model)
 16.897631 seconds (22.00 k allocations: 2.029 MiB)

and let's make a nice video with less particles, to see them bouncing around. The marker size is set by the radius of each particle, and the marker color by the corresponding repulsion constant.

using InteractiveDynamics
using CairoMakie
model = initialize_model(number_of_particles=1000)
    "celllistmap.mp4", model, agent_step!, model_step!;
    framerate=20, frames=200, spf=5,
    title="Bouncing particles with CellListMap.jl acceleration",
    as=p -> p.r, # marker size
    ac=p -> p.k # marker color