Source
Crooks (1999), LIT-522 (Phys. Rev. E 60, 2721), read in full in NOTE-413 from arXiv v4. Jarzynski (1997), LIT-517 (Phys. Rev. Lett. 78, 2690), read in full in NOTE-417 from the arXiv preprint. Seifert (2012), LIT-523 (Rep. Prog. Phys. 75, 126001), read in NOTE-420: §§1–4, 6, 7, 8.1–8.2, 9.1, 9.3–9.4, 10 and 12 closely, the case studies and heat engines skimmed. No version of record was seen for any of the three.
What was actually shown
The work equality (LIT-517). Switch a classical system that starts in canonical equilibrium from H_0 to H_1 along a fixed path in a fixed time. Then ⟨e^{−βW}⟩ = e^{−βΔF} (Eq. 2), whatever the path and the switching time. The proof is exact for an isolated Hamiltonian system, by Liouville’s theorem (Eqs. 6–8, checked in NOTE-417), and for a Nosé–Hoover thermostat with no further assumption (p. 9). For a physical reservoir it needs weak coupling (Eqs. 9–10), which the author states. Thermodynamic integration and the Zwanzig formula are its slow and fast limits. ⟨W⟩ ≥ ΔF follows by Jensen’s inequality, “rather than by invoking the increase of entropy” (p. 6). If W is Gaussian, W_diss = βσ²/2 (Eq. 12).
The fluctuation theorem (LIT-522). Assume Markovian dynamics that are microscopically reversible: a path’s probability over its reverse’s is e^{−βQ} (Eq. 5). Define the entropy production of a path as ω = ln ρ(x_{−τ}) − ln ρ(x_{+τ}) − βQ (Eq. 6). If ω is odd under time reversal, then P_F(+ω)/P_R(−ω) = e^{ω} exactly, at any finite time (Eq. 2), and ⟨e^{−ω}⟩ = 1 (Eq. 4), so ⟨ω⟩ ≥ 0. ω is odd in two classes (§III):
- processes that start in equilibrium and relax to equilibrium again, where ω = β(W − ΔF) and the work equality is one line (Eqs. 9–10);
- time-symmetric driving into a reversal-invariant steady state, where P(+ω)/P(−ω) = e^{ω} for whole cycles (Eq. 12).
If ω is Gaussian, its variance is twice its mean, which gives the Green–Kubo relations “without the standard assumption that the system is close to equilibrium” (§IV). NOTE-413 checked the derivations.
The unification (LIT-523). Choosing a conjugate dynamics (reversed, dual or dual-reversed) gives a master theorem (§4, Eqs. 78–79) from which the integral, detailed and Crooks-type theorems all follow. The integral theorem for total entropy production, ⟨e^{−Δs_tot}⟩ = 1, holds for any initial state, driving and duration (Eq. 64). Apparent violations of the second law are of order 1 in the entropy produced, so exponentially rare in system size (pp. 19–20). Dissipated work bounds the distinguishability of forward and reverse processes, ⟨w⟩ − ΔF ≥ T·D[p(x(t))‖p_eq(x(t))] (Eq. 150, read with T as NOTE-420 corrects it).
What could have come out otherwise. These are theorems, so what can fail is the premises. The Crooks paper shows one way: its long-time heat theorem is “wholly inaccurate” for Δt ≤ 256 steps in its own model (Fig. 4). The review rests the thermodynamic reading of every theorem on noise that is “not affected by the presence of a time-dependent force” (p. 11), and says this “could be violated in experiments” (§5.1).
What this does not say
- That the second law has been derived. The integral theorems are not a proof of it, because a Langevin or master-equation dynamics builds irreversibility in (LIT-523, §3.3.1, §12).
- Which steady state a system adopts. The results are exact symmetries of fluctuations, not extremum principles for the mean. They neither support nor refute minimum or maximum entropy production (NOTE-413, NOTE-420). The framework that succeeded the Brussels school for small systems uses no variational selection principle for steady states.
- That the work equality is a practical method for large systems. The average is dominated by rare runs with W far below ⟨W⟩, which “pretty much rules out macroscopic systems” (LIT-517, p. 7). The steady-state theorem is formally exact but cannot be measured directly, since “we have no independent method for calculating the probability of a state in a nonequilibrium ensemble” (LIT-522, p. 5).
- That the theorem holds for any endpoint. It needs ω odd under reversal. A run that ends in a nonequilibrium distribution that is not reversal-invariant is outside it, so it does not carry over to the setting of LIT-327 as it stands (NOTE-413).
- That “dissipated work” has one meaning. Here W_diss = W − ΔF, with ΔF between equilibrium states. LIT-327’s dissipation subtracts the nonequilibrium free energy instead. The two agree only when the system ends in equilibrium (NOTE-417).
- Anything quantum, non-Markovian or strongly coupled. All three works are classical. The physical-reservoir case of the work equality needs weak coupling, and what it becomes under strong coupling is left open.
- The distribution of work. The equality fixes one exponential moment of p(W). The relation between forward and reverse distributions is Crooks’s result.
Connections
- The record’s thermodynamics of computation. THEORY-026 rests on an average-level identity for a discrete Markov chain and uses no fluctuation theorem; these works are its background, not its source. THEORY-030’s subject is framed by LIT-523 §7.3, which restores the second law under feedback through Landauer’s erasure cost and reports an erasure experiment approaching the Landauer bound for slow cycles. The quantum review LIT-010 re-derives both classical relations.
- England’s macrostate bound (THEORY-679). A coarse-grained second law of the same family, derived under detailed balance and extended to time-symmetric driving by citing Crooks.
- Dissipative adaptation (THEORY-412) takes the Crooks relation and its macrostate form as premises.
- Maximum entropy production (THEORY-671). Dewar’s path-entropy argument (LIT-521) produces a fluctuation theorem of the same form, p(σ) = e^{τσ/k_B} p(−σ), from an inferred path measure, not from a dynamics. It explains the theorem if real path statistics are maximum-entropy, and does not show that they are (NOTE-426).