Give a system a temperature and a ladder of energy levels, and it does not choose freely: the chance of finding it in a state falls off exponentially with that state's energy. The rule is P(state) = e−E/kT / Z — one weight per state, divided by the sum of all of them. Down the center, data flows: the levels and the temperature go in, the engine computes the populations, the proven distribution comes out. The blue team builds and defends it; the red team tries to break it.
source Boltzmann, L. (1877), Über die Beziehung zwischen dem zweiten Hauptsatze… und der Wahrscheinlichkeitsrechnung, Sitzungsber. Kais. Akad. Wiss. Wien, math.-naturwiss. Classe, 76, 373–435 — no stable canonical URL; cited by author/title/year. Rendered in natural units (k=1), not quoted.
Every state i of energy Ei gets a raw weight — its Boltzmann factor e−Ei/kT. Normalise by the sum Z = Σ e−Ei/kT (the partition function) and the weights become probabilities that sum to 1:
Pi = e−Ei/kT / Z
Four things fall straight out, and the engine holds all four: the Pi sum to 1; a higher level is strictly less likely; the ratio of any two is Pi/Pj = e−(Ei−Ej)/kT — depending on the gap alone; and the limits pin down the ends (below).
Live populations for the current ladder:
| level | E | P |
|---|
The factor e−E/kT is the seed of a whole family. Sum it over states and you have the partition function Z — from whose logarithm every thermodynamic quantity is a derivative.
Swap energy E for a negative score and temperature for a scale, and the very same normalised exponential is the softmax a transformer uses to pick a token. Boltzmann's 1877 weight is the shape both borrow. Each sphere is the next one's premise.
The blue team's live check: recompute the distribution from scratch and confirm it still normalises, still falls with energy, and still obeys the ratio law and both temperature limits. If red flips the sign, this badge is where it shows.
Two things go in. A ladder of energy levels — here En = n·Δ, the ground level at 0 — and a temperature T (in units where Boltzmann's constant k = 1, so kT is just T). Temperature is the only knob that says how much energy the system can afford to spend climbing the ladder.
Cold (small T): the system hugs the ground. Hot (large T): it spreads out toward equal occupation. That trade is the whole instrument — and it is what you feed the panel below.
Populations are computed on the spot from P = e−E/kT/Z — never looked up.
| n | En | factor e−E/kT | Pn | population |
|---|
Change any control — Z, the factors and the populations are recomputed from the formula, exactly.
What the machine produces, proven: a set of populations that sum to 1, fall strictly with energy, obey Pi/Pj = e−(Ei−Ej)/kT, and hit the right ends — uniform as T→∞, ground-only as T→0. The current numbers are above; the guarantees are the output.
The blue team's witness (left) reconfirms these live; the red team (right) tries to make them wrong.
It also treats levels as distinguishable and non-degenerate. Real spectra have degeneracy gi (a factor gie−Ei/kT), and identical quantum particles obey Bose–Einstein or Fermi–Dirac statistics, of which Boltzmann is only the dilute, high-temperature limit. This panel is the clean classical core, not the whole of statistical mechanics.
"Higher energy just means lower probability, linearly." Cut. The fall is exponential in E/kT, not linear — the ratio law is computed in the machine and depends on the gap alone.
"At any temperature the ground state is the most likely." Cut. Most probable single state, yes — but as T→∞ every state approaches equal probability, and a degenerate band of excited states can hold more total population than the ground.
"Z is just a normalising nuisance." Kept, corrected. Z is the generating function: −∂lnZ/∂β gives the mean energy — the engine checks that identity to 1e-6.
The red team's move: flip one sign — use e+E/kT instead of e−E/kT. Now high-energy states look more likely, and as T→0 the system piles into the top level instead of the ground. The blue team's witness (window 7) is watching.
Flip the exponent's sign and the distribution inverts — the fall becomes a climb, the T→0 limit lands on the wrong state. The witness recomputes, the monotone and ground-state checks fail, and it turns red. Nothing is faked; the attack is real and it is caught.