Give a few thousand point particles a short-ranged force law, sort them into a handful of species, and let the strength of the interaction depend on which species is looking at which. Then do one slightly illegal thing: let the matrix of interaction strengths be asymmetric, so that species 1 can chase species 2 while species 2 runs from species 1.

That single change is the whole model. There is no chemistry in it, no bonding rule, no template, nothing that encodes “make a cell membrane.” What comes out anyway are cells with skins, crawling worms, rotating mills, and clusters that divide when they get too large.

Drag on the viewport to stir; hold shift to pull instead of push. The matrix on the right is live — drag any cell up or down and watch the structures reorganise within a second or two.

The model

Each particle carries a position, a velocity, and a species label $s_i \in \{1,\dots,K\}$. Every pair closer than a cutoff $r_\max$ interacts through a radial force that depends only on the reduced separation $r = |\mathbf{r}_{ij}| / r_\max$ and on the matrix entry $A_{s_i s_j}$:

$$ f(r) = \begin{cases} \dfrac{r}{\beta} - 1, & 0 \le r < \beta \\[6pt] A_{s_i s_j}\left(1 - \dfrac{\left|2r - 1 - \beta\right|}{1 - \beta}\right), & \beta \le r < 1 \\[6pt] 0, & r \ge 1 \end{cases} $$

The first branch is a species-independent core repulsion: it rises linearly from $-1$ at contact to $0$ at $r = \beta$, and it is the only thing stopping the system from collapsing to a point. The second is a symmetric tent peaking midway between $\beta$ and $1$, scaled by the matrix entry — positive entries attract, negative entries repel. Beyond the cutoff, nothing.

Dynamics are overdamped, which is the right limit for anything moving through a viscous medium at this scale. Momentum is not conserved and is not meant to be:

$$ \mathbf{v}_i \leftarrow \mathbf{v}_i\, 2^{-\Delta t / \tau} + \mathbf{F}_i\, \Delta t, \qquad \mathbf{x}_i \leftarrow \mathbf{x}_i + \mathbf{v}_i\, \Delta t \pmod{\mathbf{L}} $$

with $\tau$ the velocity half-life (the Velocity half-life slider) and periodic boundaries in both directions, so the domain is a torus with no walls and no corners for structures to hide in.

Why the asymmetry matters

If $A$ is symmetric, the force law derives from a pair potential. The system has a well-defined energy, it relaxes downhill into that energy’s minima, and what you get is condensed-matter-flavoured: droplets, phase separation, occasionally a lattice. Everything stops moving eventually.

If $A$ is asymmetric, $f_{ij} \neq -f_{ji}$, no potential exists, and there is no energy to minimise. Each pair is a tiny motor injecting momentum into the system, and structures persist because they are continuously driven, not because they are stable. This is the defining feature of active matter — the same reason a school of fish and a flock of birds have dynamics no equilibrium statistical mechanics will reproduce.

The Matrix symmetry slider interpolates between the two. Set it to 1 and watch the system go quiet.

Things worth trying

  • Chase loops. Set a 3-species cycle by hand: 1 attracted to 2, 2 attracted to 3, 3 attracted to 1, with the reverse entries negative. You get travelling packets that never settle.
  • Self-enclosing cells. Make one species self-attracting and a second species attracted to the first but self-repelling. The second forms a shell around the first — a membrane, from two numbers.
  • Cutoff sweep. Fix a matrix you like, then walk $r_\max$ from 0.02 upward. Structures have a preferred size set by $r_\max$, and there is a fairly sharp point where local clusters percolate into a single network.
  • Core radius. Small $\beta$ gives dense, liquid-like blobs; large $\beta$ gives open, chain-like structures, because the attractive tent gets squeezed into a narrow shell.
  • Turn the force off. Set the gain to 0 and the system is a collisionless gas coasting to a halt under drag — a useful sanity check that nothing but the matrix is doing the work.

Every control is captured in the URL, matrix included. Copy link gives you a permalink that reproduces the exact configuration, quantised matrix and random seed and all.

Implementation notes

The simulation is plain JavaScript with no dependencies and no build step — static/sims/particle-life/particle-life.js is the whole thing. Three details make it fast enough to run at a few thousand particles inside a blog post:

Uniform hash grid. Naive pair enumeration is $O(N^2)$; at $N = 4000$ that is 16 million distance checks per step, far too slow for 60 fps. Instead particles are binned into a grid whose cell size is at least $r_\max$, so every interacting pair lies within the $3\times3$ block around a particle’s own cell. Binning uses a counting sort into preallocated typed arrays, which costs one pass to count, one prefix sum, and one pass to place — no allocation and no per-frame garbage. Cost per step drops to $O(N)$ at fixed density.

Grid-order traversal. The force loop iterates particles in the sorted grid order rather than by index. Neighbours in space end up adjacent in memory, which turns most of the inner loop’s reads into cache hits.

Batched drawing. Species are stored in contiguous blocks, so rendering emits one canvas path per species — $K$ fill calls per frame instead of $N$.

Everything lives in Float32Arrays, the integrator is semi-implicit Euler (positions updated only after every velocity is known), and the random number generator is a seeded mulberry32, so a given seed reproduces a run exactly. The engine object is DOM-free and exported on window.ParticleLife, which makes it straightforward to drive headlessly if you want to measure something rather than watch it.

One deliberate compromise: there is a speed cap. Push the force gain and time step high enough and semi-implicit Euler will go unstable, so velocities are clamped and non-finite values reset. It is a guard rail, not physics — if you are near it, shorten the half-life or lower the gain rather than trusting what you see.

References

The model was popularised by Jeffrey Ventrella as Clusters and by Tom Mohr’s Particle Life implementation; the force law above follows the formulation in Tom Mohr’s write-up. The physics it gestures at is the literature on active matter and non-reciprocal interactions — a good entry point is Fruchart, Hanai, Littlewood and Vitelli, Non-reciprocal phase transitions, Nature 592, 363 (2021), which works out what non-reciprocity does to phase behaviour far more carefully than a canvas demo can.