Towards quantum equilibrium

The beryllium atom in a superposition of twelve of its states with equal real coefficients, its four electrons moved by the de Broglie–Bohm guidance equation. Each point of light is one electron: amber spin up, blue spin down. On the left, 4000 atoms started from the distribution of the ground state alone, out of equilibrium with the wave; on the right, 2000 started from |Ψ|² at t = 0, in equilibrium. Beneath, for both, the coarse-grained H-function on the state-amplitude marginal S1: zero in equilibrium, up to the noise of a finite sample, and larger the further from it. It is a divergence of one distribution from another rather than a distance in the geometric sense; “distance from equilibrium” is its plain-language name here.

Computed with CASINO v3.1.851 (pilotdrive): adaptive Dormand–Prince steps at relative tolerance 10⁻¹⁰. The full ensembles are drawn at their computed times, every 2.5 a.u., each held until the next. 300 trajectories of each were computed again with an output every 0.05 a.u.: the moving lights and their trails, each shown at the computed time at or before the clock. The two computations agree closely for most; by 200 a.u. 31 of the 300 on the left and 6 on the right differ by more than 0.01 bohr. Nothing is interpolated: here the trails are straight segments through every second computed position, 0.1 a.u. apart (in the film, every one), and H is joined by straight lines between its computed times. Positions are rounded to 0.01 bohr. Trajectories the integrator could not continue (49 left, 29 right) leave the picture at the time they stop. Each point of the full clouds is twice as bright on the right, which has half as many atoms.

At 1× the clock runs at 3 atomic units of time a second (one a.u. is 0.0242 femtoseconds). Drag to turn the atoms; pinch or scroll to zoom; space plays and pauses.