T-27 · Statistical mechanics

Counting becomes law.

Temperature is what counting looks like when the numbers get large. Two solids share energy in the most numerous way, counted here exactly. A gas forgets how it started: every disk begins at the same speed, and the speeds settle into the law of a hypotenuse with two Gaussian legs. A magnet orders below a temperature set by 1 + √2.

Counting, exactly

N oscillators can hold q quanta in C(q + N − 1, q) ways: stars and bars. Two solids side by side multiply their ways, and summed over every split they make the ways of one big solid, by Vandermonde's identity. These are whole numbers, computed exactly, however large.

all arrangements	4.5274 × 10^58 59 digits
Σ Ω_A Ω_B = Ω_(A+B)	Vandermonde, exactly
most likely split	q_A = 29 of 100, near 30
its chance	…

Equally cold

Energy flows until the split is the most numerous one, and there the two solids' entropies climb equally steeply: dS/dq, which is 1/T, is the same for both. That is what equal temperature means. Bigger solids make the peak sharper, until it is a certainty.

1/T of A, of B	…
difference	… shrinking as the solids grow

Why it is a law

The same count, read differently: for 300 and 700 oscillators the most likely split already has a chance of only a few per cent, yet the splits within a few per cent of it hold nearly all the probability.

the peak's own count	2.7812 × 10^57 58 digits
oscillators in all	100
relative width ~ 1/√N	…