Cool iron below 1043 K and its spins stop asking. Each moment sits in the average field of all the others — a molecular field — and that self-reference has a switch: above a critical temperature the moments are a crowd, below it they are an army. The magnet then remembers its own history as a hysteresis loop. Down the center: temperature and field go in, the Weiss engine solves itself, the phase and the loop come out. The blue team builds it; the red team breaks it.
source P. Weiss, L’hypothèse du champ moléculaire et la propriété ferromagnétique, J. Phys. Théor. Appl. 6, 661 (1907) — journaldephysique.org. Historic, no DOI — cited by title/year. Rendered, not quoted.
Weiss put every moment in the mean field of the rest: an effective field λM proportional to the magnetization it is helping to create. Self-consistency gives the order-parameter equation
m = tanh( (Tc/T) · m ), m = M/Msat
The line m and the curve tanh cross only at the origin when T ≥ Tc; below Tc the tanh starts steeper than 1 and a second, nonzero crossing appears — spontaneous order, out of nothing but temperature. That fork is the phase transition.
Live, at the current temperature:
In the-curie-law the moments ignore each other: χ = C/T, a paramagnet that only ever diverges at absolute zero. Let them feel each other’s field and the very same algebra shifts the pole:
χ = C/(T − Tc) — the Curie–Weiss law.
Now the susceptibility blows up at a finite temperature Tc, and just below it, order appears with no field at all. One sign of cooperation turns a solitary law into a phase transition. Each sphere is the next one’s premise.
The blue team’s live check: re-solve the Weiss equation across temperature and confirm the transition — order below Tc, none above, and a Curie–Weiss pole at Tc. If red tampers with the law, this badge is where it shows.
Two knobs and two material constants:
| symbol | meaning | value |
|---|---|---|
| T | temperature | slider, as T/Tc |
| H | applied field | swept −Hmax…+Hmax |
| Tc | Curie point | 1043 K (α-iron) |
| C | Curie constant | 1 (reduced) |
Set the temperature; the engine solves the molecular-field equation and sweeps the field to trace the loop. Nothing is looked up.
Move the temperature — the order parameter, the susceptibility and the loop are all solved on the spot from m = tanh((Tc/T)m), never tabulated.
Proven, live: below Tc there is a spontaneous magnetization at zero field that vanishes continuously as T → Tc, the M–H curve is a hysteresis loop with remanence Mr and coercivity Hc that encloses area (energy lost per cycle), and above Tc the susceptibility follows χ = C/(T−Tc), diverging at the Curie point.
The blue witness (left) confirms the transition live; the red team (right) tries to erase it.
It is an AMBER idealization: the phase transition it discovers is real, the numbers at the edge are not. The honest exponents need the renormalization group (Wilson, 1971). Weiss saw the fork; he could not price it.
“A magnet loses all magnetization at its Curie point.” Cut. Only the spontaneous magnetization dies; above Tc a field still magnetizes it — as a paramagnet, χ = C/(T−Tc).
“Hysteresis is just internal friction.” Cut. The loop area is real dissipated energy, but its origin is domain-wall pinning and metastability, not friction — a memory, not a drag.
“Below Tc every spin points the same way.” Kept, corrected. Order is local: domains point every way and cancel until a field aligns them — which is why a fresh nail is not a magnet.
The red team’s move: flip one sign in the Curie–Weiss law — χ = C/(T + Tc) — so the pole moves to negative temperature. No divergence, no transition, no memory. The blue witness (window 7) is watching.
With the plus sign the susceptibility is finite everywhere and the spontaneous-M / divergence check fails — the witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.