Source
Perunov, Marsland and England (Phys. Rev. X 6:021036, 2016), LIT-518, Eq. (8), read in full in NOTE-416 from arXiv v1. The macrostate relation it starts from is England (2013), LIT-515, read in NOTE-419.
What was actually shown
The identity. In the paper’s notation, with Ψ = β⟨Q⟩ and Φ = ln⟨e^{−βQ}⟩ + Ψ,
ln[π(I→II)/π(I→III)] = ln(Ω_II/Ω_III) + ln[π̃(II→I)/π̃(III→I)] + ΔΨ − ΔΦ,
where π̃ is the time-reversed protocol and the averages are over forward paths from I ending in each outcome. Ψ − Φ = −ln⟨e
Re-derivation (promotion audit, batch 077). Microscopic reversibility gives, path by path, P[path]/P̃[reversed path] = e^{βQ[path]}. Summing over the paths from microstate x to y, π̃(y→x) = π(x→y)⟨e^{−βQ}⟩{x→y}. Start the forward process uniformly on I and each reverse process uniformly on its own macrostate J. Then π(I→J) = Ω_I⁻¹ Σ π(x→y) and π̃(J→I) = Ω_J⁻¹ Σ π(x→y)⟨e^{−βQ}⟩{x→y}, so π̃(J→I)/π(I→J) = (Ω_I/Ω_J)⟨e^{−βQ}⟩_{I→J}. Writing this for J = II and J = III and dividing gives the identity. Uniform starting densities are what make the state-count term exactly ln(Ω_II/Ω_III); the paper reaches it under near-uniformity and no-correlation assumptions (its Eq. 5), so this entry states the exact version with its condition.
The corollary. By Jensen’s inequality Φ ≥ 0, with equality only when Q does not vary across the paths. So if the two outcomes have equal state counts and equal reversal probabilities, the likelier is the one with larger −ln⟨e^{−βQ}⟩: more heat released on average, less spread across paths. Heat released in futile cycles cancels and drives no drift.
What this does not say
- That driven matter tends toward adapted states. That is THEORY-412, which extends this entry and stays Proposed: it needs real systems in which the other two terms can be held equal.
- That driven systems maximize dissipation. Ψ alone does not fix the odds; Φ and the other two terms count too (THEORY-671 is the maximum-entropy-production contrast).
- Anything for feedback-controlled drives, or for starting densities far from uniform on the macrostates.