Condensation

As the mean interaction is raised, when does a Particle Life gas condense into bound structure?

▶ Run the simulationSee the measured result

Units: E[isolated fraction] of the t = 0 bond graph — exact: (1−p)^(N−1), p = V_ball(2)/V_box = 1.1635528e-3, mean degree c = 1.3951

How the lab tests it

Add a uniform shift μ to the seed's interaction matrix, sweep μ from repulsive to attractive, and at each value measure the bound fraction (particles in any cluster) once the field re-settles.

What it looks for

the condensation point μ_c where the bound fraction crosses ½ — an OPEN question, no closed-form answer

Random geometric graph calculator — isolated fraction, degree and bonds

Scatter N points at random in a periodic box and join every pair closer than R: that is a random geometric graph, and it is the one configuration of the simulation above that can be solved exactly. The world's question — where a Particle Life gas condenses — has no closed-form answer, but the instrument that measures the condensation does, because at t = 0 the module's 1200 particles are iid uniform on a 40×18×40 torus and its bond graph is precisely Gilbert's. Nothing here is typed in. One primitive is declared — a point falls inside another point's ball with probability V_ball/V_box — and the isolated fraction, the degree distribution, both bond-count moments and the dimer fraction are all counted out of it; fed this lab's own box the forward direction returns 0.2476070871763943, which is the number this world's oracle scores against, computed here rather than copied. The finite population is the whole subtlety, so it is a knob: θ = 1 is the exact binomial law and θ → 0 is the Poisson approximation textbooks usually reach for, which runs 0.0813% high at these settings and 0.61% high at R = 2.8 — small, but the leading term −c·p/2 is predictable before either is computed, and the page prints it. Two limits are real rather than decorative: past R = half the shortest box side the ball wraps around the torus and overlaps itself, so V_ball/V_box counts the same region twice and is a probability of nothing; and past t = 0 the dynamics has carved a hard core into the point field with its universal short-range repulsion, the positions stop being independent, and every law here stops applying. One thing the page refuses outright: the order parameter the simulation puts on screen thresholds at component size ≥ 4, and the expectation of a k ≥ 4 cluster-integral series has no closed form. This lab measures that floor at 0.46445 ± 0.00088 over 800 random configurations and says so; the trimer share printed in the census direction is what that measurement leaves over, which is arithmetic on a measured number and not a prediction of it. Nor does anything here decide whether the graph percolates.