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Mass-radius plot of the discharge: number of cluster sites inside radius r (log-log); the slope is the fractal dimension
Potential against distance from the centre (ring average, solid) and the bare-electrode solution (dashed)
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How to use
Drag to orbit, wheel or pinch to zoom, Space pauses, R restarts. The globe, the electrode and the driver were modelled in Blender (baked PBR textures); the discharge is the same growth model as the 2D page, here on a 3D grid, drawn as glowing filaments inside the glass. Add a finger from the sliders (direction azimuth and elevation) to see where the next streamers go.
The science
What a plasma ball does. A small high-voltage, high-frequency source drives a centre electrode inside a glass globe filled with a low-pressure noble gas mixture. The gas breaks down near the electrode and thin, branching filaments (streamers) reach out toward the glass, which is grounded by the surrounding air and, much more strongly, by a touching hand. Touching the glass gives the current an easier path, so the filaments bend toward the finger.
The model: dielectric breakdown. Niemeyer, Pietronero and Wiesmann (1984) described such discharges as a growth process. The electrostatic potential obeys Laplace's equation in the gas,
nabla^2 phi = 0, phi = 1 on the electrode and on the discharge channels, phi = 0 on the glass and the finger
The channels conduct, so they are equipotential with the electrode. The discharge extends by one lattice site at a time to a site next to it, chosen with probability
p(i) = (1 - phi_i)^eta / sum_j (1 - phi_j)^eta
(1 - phi is proportional to the voltage across the gap, so it is large near the tips of the discharge and small deep inside fjords where the field is screened). The exponent eta controls how strongly growth favours the tips: eta = 0 gives uniform growth (a compact Eden cluster), eta = 1 the classic branching fractal with dimension about 1.7 in 2D, and large eta gives a few long unbranched filaments. After every added site the potential is relaxed again with successive over-relaxation (SOR). When a filament touches the glass, the channel is complete: the page flashes, waits for the afterglow time and starts a new discharge with another random seed.
Fractal dimension. The mass-radius relation N(r) ~ r^D gives the dimension D from the slope of the log-log graph. The page measures it for the current discharge and for a sweep over eta; the checks compare it with 2 for eta = 0 and with the literature value near 1.7 for eta = 1.
Numerical checks. The solver is checked on cases with known answers: for a centre electrode in a disc the potential is ln(R / r) / ln(R / r0), so the ratio of ring-averaged potentials at two radii is fixed (and the flux through any ring is the same, Gauss law); for a sphere the potential is (1/r - 1/R) / (1/r0 - 1/R); the potential stays between 0 and 1 (maximum principle).
Honest simplifications. (1) Real plasma-globe filaments are driven by an RF field and ionisation waves; this is a static-field growth model with the Laplace field of a conductor, which reproduces the branching statistics but not the dynamics (the flicker here is only the restart of the process). (2) The lattice is square or cubic, so a faint grid anisotropy remains; the 3D drawing adds a small random offset to each site (visual only) to hide the grid. (3) A single filament from the electrode to the glass is complete after tens to a few hundred sites; real filaments carry current through plasma with its own conductivity. (4) The finger is a fixed grounded blob, not a real capacitive coupling. (5) The 3D model uses a coarse 41-cell grid; the dimension fit is rough. (6) Colours are artistic.
Checks run in the page. Disc Laplace solution (ratio and Gauss law), maximum principle, 3D sphere Laplace solution, fractal dimension for eta = 0 and eta = 1, and the monotonic fall of D with eta.