Source
Barato & Seifert (2015), LIT-1885, letter eqs. (1)–(24) and Supplemental Material §§I–IV, read in full in NOTE-1480 from arXiv 1502.05944v2.
What was actually shown
The quantity. For a current X_α (a count of transitions with antisymmetric increments dᵢⱼ) in a steady state with entropy production rate σ, Q_α ≡ σtε²_α = 2σD_α/J_α², with J the mean rate and D the diffusion coefficient of X_α (eqs. 6–10). Both σt and 1/ε² grow linearly in t, so Q is time-independent.
Three results.
- Biased walk: Q = A coth(A/2) ≥ 2, increasing in the affinity A (eq. 4).
- Linear response, any network: Q_α = 2[1 + (positive semidefinite form)/(J_α)²] ≥ 2, equality when only A_α ≠ 0 (eq. 13, supplement §I).
- Unicyclic network with N states and affinity A, any distance from equilibrium: Q ≥ (A/N) coth(A/2N) ≥ 2, the minimum at uniform rates (eq. 20, supplement §III, with a uniqueness step the authors say they could not prove).
For multicyclic networks far from equilibrium: numerical minimisation and random sampling on seven networks, all ≥ 2, minima reached in linear response; “we conjecture that Q_α ≥ 2 also for general multicyclic networks” (p. 4).
My check. Computing J and D from the top eigenvalue of the tilted generator: Q matches A coth(A/2) and (A/N) coth(A/2N) at N = 3, 5 to six digits; over 2,998 random complete four-state graphs with rates 10^{U(−3,3)}, min Q = 2.08.
What could have failed. A strongly driven, many-cycle network might have bought precision cheaply through cycle interference. None of the sampled networks did.
What this does not say
- Anything about occupation observables. The time spent in a state has finite uncertainty even at equilibrium (p. 1). The bound does not say that keeping a system in a maintained state reliably costs dissipation. It says that a reliable flux, such as a production rate counted over time, does. How long a maintained state lasts is the escape-rate question (THEORY-1264).
- That the general case is proved here. Proofs for general networks came later and are not held.
- That systems sit at the bound. Q = 2 is reached only near equilibrium with a single affinity, where output is slow.
- That dissipation selects anything. The bound constrains fluctuations at a given state; it is not an extremum principle (contrast THEORY-671).
- For irreversible models. With one-way reactions the affinity is infinite and the bound is trivially met. A model like CASE-189’s must be made reversible before the bound bites.