Entropy points the arrow of time. Bring a hot body and a cold body into contact and heat only ever flows one way — from hot to cold — and the total entropy of the isolated pair only ever climbs, reaching its ceiling exactly when the two temperatures meet. Rendered from the closed forms, not quoted.
source R. Clausius, The Mechanical Theory of Heat (1865) — archive.org/mechanicaltheor04claugoog amber 1865 collection; scan is the 1867 Hirst translation — year cited, not a stable per-edition DOI.
Two bodies, heat capacity C each, at temperatures Th > Tc, sealed from the world. Entropy of a body warmed from T to T′ is ΔS = C·ln(T′/T) (from dS = dQ/T, dQ = C·dT).
S = k ln Ω = -k Σ p·ln p (k = 1) dS = dQ / T (reversible heat)Move heat dQ from hot to cold: the hot body loses dQ/Th, the cold gains dQ/Tc, and the pair’s entropy shifts by
dS = dQ·( 1/Tc − 1/Th ) > 0 for Th > TcLet them run to a common Tf. Total: ΔS = C·ln(Tf² / ThTc) ≥ 0, since Tf = (Th+Tc)/2 ≥ √(ThTc) (AM ≥ GM).
This is the sphere that gives time a direction. the-entropy supplies the state function S; Clausius’s second law tells it which way to point — never down in isolation. Downstream it sets the hard ceiling on the-carnot engine: no cycle beats η = 1 − Tc/Th, because beating it would demand dS < 0.
Boltzmann’s S = k ln Ω (next door) explains why the arrow exists: more microstates below, none above.
Re-runs every closed form live at load and after any button press: heat-flow sign, mixing positivity, reversible equality, the equilibrium maximum, and the wider stat-mech family (Z, F, detailed balance, equipartition, Fermi–Dirac). It also arms the planted void in window 6 and confirms the second law catches it.
Set the two starting temperatures and the shared heat capacity. Nothing is quoted — every number below is computed from these three.
total entropy vs. the split of energy (Ta from cold→hot); the green peak is equilibrium:
Proven, live: heat flows hot→cold, ΔS > 0 for every unequal pair, ΔS = 0 iff the temperatures already match, and S is maximal exactly at Ta = Tf. The arrow only points one way.
“Entropy is just disorder, and things get orderly all the time — crystals, life, my tidy desk.” True locally, and no violation: those systems are open. A freezing puddle dumps latent heat into the room; the puddle’s S falls but the room’s rises more. The law is about the isolated total, dSuniverse ≥ 0 — and that is what the panel isolates and checks.
“Heat can flow cold→hot on its own if you just wait.”
→ No isolated process does this; dS would be negative. It needs work paid in (a fridge) — Clausius’s own statement of the law.
“A reversible process has dS > 0.”
→ Reversible means dS = 0 exactly (Clausius equality); strict increase is the signature of irreversibility.
“Maxwell’s demon breaks it for free.”
→ Landauer: erasing the demon’s memory costs ≥ kT·ln2 of entropy, restoring the ≥ 0 balance.
The planted void, disclosed: force heat the wrong way — cold→hot spontaneously — which flips dQ’s sign so dS < 0. The second law is now violated on purpose. The Witness (7) re-runs the dS ≥ 0 gate and must go red.