Circulation against how fast the bucket turns

At its defaults it draws circulation against how fast the bucket turns. The circulation round the rim of a bucket of radius 1 mm, against the angular velocity it is spun at. An ordinary liquid ends up rotating with the bucket, and its circulation is 2Ω times the area — the straight dashed line, continuous in Ω and with no special value anywhere on it. A superfluid's velocity is the gradient of a phase, so it can carry circulation only in whole units of h/m = 9.969e-8 m²/s. Below 0.256 radians per second it carries none at all: the bucket turns and the liquid does not, which is what Hess and Fairbank measured. Above it the circulation is a staircase of 13 steps, each exactly one quantum high and each 1.59e-2 radians per second wide. The staircase runs below the classical line by the ln(R/a) quanta the threshold costs, a fixed lag: at 102 radians per second the two agree to 0.24 per cent, which is why a rotating superfluid looks like a rotating liquid at any speed a bucket is normally spun at.

vortex-lattice is one function in lib/figures/fluids.js — matter that will not hold a shape, and the forces in it. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The circulation round the rim of a bucket of radius 1 mm, against the angular velocity it is spun at. An ordinary liquid ends up rotating with the bucket, and its circulation is 2Ω times the area — the straight dashed line, continuous in Ω and with no special value anywhere on it. A superfluid's velocity is the gradient of a phase, so it can carry circulation only in whole units of h/m = 9.969e-8 m²/s. Below 0.256 radians per second it carries none at all: the bucket turns and the liquid does not, which is what Hess and Fairbank measured. Above it the circulation is a staircase of 13 steps, each exactly one quantum high and each 1.59e-2 radians per second wide. The staircase runs below the classical line by the ln(R/a) quanta the threshold costs, a fixed lag: at 102 radians per second the two agree to 0.24 per cent, which is why a rotating superfluid looks like a rotating liquid at any speed a bucket is normally spun at.

A slice through the shadow a released cloud of a million atoms casts after it has expanded, the measurement by which condensates are seen, at 1.1, 0.8 and 0.5 times the transition temperature: the thermal atoms, whose spread of speeds sets the cloud's width, give a broad rounded profile; the condensed atoms, all in one state with only the small spread of momenta their trap allowed, give a narrow peak at the centre, drawn here with the shape repulsion between the atoms gives it. Above Tc there is only the broad cloud. At 0.8 Tc a sharp peak stands up from its middle, holding 47 per cent of the atoms; at 0.5 Tc the peak holds 87 per cent and the thermal wings have shrunk. The two-component shape — a narrow peak on a broad base — was the signature that announced the first gaseous condensates in 1995.

The fraction of a gas of bosons in the ground state of a harmonic trap, against temperature in units of the large-N transition temperature kTc = 0.94 ħω N^⅓, computed by summing the occupation of every level of the trap, for 1 × 10^3, 1 × 10^5, 1 × 10^7 atoms (solid); dashed, 1 − (T/Tc)³, the limit of infinitely many; dotted, 1 − (T/Tc)^1.5, the same gas in a box. Below Tc the excited levels are full and the remaining atoms are forced into the lowest. A thousand atoms condense later, the transition rounded and moved down to about 0.94 Tc — a gas that small has too few levels for the large-N count to be exact — while ten million follow the cube to within a per cent. The cube is the trap's: a box's excited states fill as T^1.5 and its condensate grows more slowly below the transition.

A trap holding 10⁵ atoms at 0.7 Tc: the number of atoms in each level, n ħω above the ground state, on a logarithmic axis, computed with the exact chemical potential, which sits 15.80 millionths of kT below the ground level. The excited levels hold 36701 atoms between them — the most they can hold at this temperature, set by the trap's number of states, which grows as the square of the energy — and the ground state holds the other 63299, 63.3 per cent of the gas, in a single quantum state. Each excited level holds a few hundred atoms spread over hundreds of states; the ground level holds tens of thousands in one.

The transition temperature kTc = 0.94 ħω N^⅓ against the number of atoms, for trap frequencies of 20, 100 and 500 hertz, on logarithmic axes. A million atoms in a 100 hertz trap condense below 451 nanokelvin; a thousand, below 45 nK. Tighter traps raise the temperature in proportion to their frequency, and more atoms raise it as the cube root of their number — a thousandfold more atoms only tenfold. The temperatures are set by the trap and the count, not by the atoms' species: rubidium and sodium in the same trap condense at the same temperature, though at different densities.

The heat capacity per atom, in units of Boltzmann's constant, of 10⁵ ideal bosons in a harmonic trap, against temperature over Tc, found by differentiating the energy summed over every level (solid); dashed, the large-N law below Tc, 12ζ(4)/ζ(3) (T/Tc)³ = 10.80 at Tc; dotted, the classical 3 per atom that a trapped gas reaches when hot. The heat capacity climbs as the cube of the temperature while the excited levels fill, reaches 9.9 near Tc — over three times the classical value — and falls steeply above it towards 3. The sharp fall is the transition: below Tc heat goes into exciting atoms out of the condensate, above it only into spreading a gas that has no condensate left to draw on. When the energy of trapped rubidium was measured against its temperature in 1996, the change of slope at Tc that this peak implies was there — thermodynamic evidence that the condensate is a phase transition and not merely a dense centre.

physicscheck asserts something about vortex-lattice that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.