Source
Vinjanampathy & Anders (2015; Contemporary Physics 57, 2016), LIT-010, §3.2 and §3.6, read in full in NOTE-011. The review derives the fluctuation relations in full; the primary sources it cites are Jarzynski, Crooks and Tasaki, its refs [3]–[5]. The result is standard, and the record files it on the review’s derivation.
What was actually shown
Conditions. The system starts in the thermal state at inverse temperature β for the initial Hamiltonian. Its evolution between the two energy measurements is a closed unitary V. Work is the two-point random variable W_{m,n} = E_m^(τ) − E_n^(0) (Eq. 24), with distribution Eq. 25.
Result. The quantum Jarzynski equality ⟨e^{−βW}⟩ = e^{−βΔF} and the Tasaki–Crooks relation follow (Eqs. 26–30). The proof uses only the normalisation Σ_n p_{m|n} = 1 of the transition probabilities (Eq. 29). Jensen’s inequality then gives ⟨W⟩ ≥ ΔF. The review notes that these relations are identical in form to the classical ones (Eqs. 18–21), where the closed-system proof rests on Liouville’s theorem (Eq. 20).
Why the form is classical. Each run starts and ends in an energy eigenstate, so the coherences of the state in the energy basis play no role in the statistics. The review records that this is exactly why the two-point definition is contested (§3.6, citing Kammerlander & Anders). NOTE-011 grades the formal equivalence strong and its physical reading dependent on that choice (C3).
What could have come out otherwise. A quantum correction term could have appeared in the exponential average. Under these conditions it cannot: the proof’s only use of V is the normalisation of the transition probabilities it induces.
What this does not say
- That quantum fluctuations of work obey the classical relations in general. The equality holds for this definition of work. For the ensemble work of Eq. 2, or for work extracted from coherences (Eq. 12), the review gives other results, and it lists the definitions as unreconciled (THEORY-416).
- That the relations hold for open evolution. The derivation assumes a closed unitary between the measurements. The Crooks form needs detailed balance with the bath (§3.2).
- That quantum thermodynamics has no new effects. The review’s own §7 says the field had made few experimentally checkable predictions of new effects; this identity is one reason, not a demonstration that none exist.
- That the review is the authority. It reproduces the derivation; the record’s LIT entry advises citing the primary papers for formulas.
Connections
- Carries further THEORY-431, the classical trajectory-level fluctuation theorem with Jarzynski’s equality as a corollary. This is the same identity widened to closed quantum systems under the two-point definition of work.
- The four notions of work, and why naming one matters, are THEORY-416.