Inequalities for non-equilibrium fluctuations of work

Five previously unknown inequalities relating equilibrium free energy differences and non-equilibrium work fluctuations are derived, and lucid path to derivation of many similar inequalities is presented. These results are based upon combined exploitation of the Jarzynski equality and the generalization of the scheme for producing uncertainty-type inequalities due to H. Weyl. The inequalities may possibly lead to better understanding of behavior of the equilibrium free-energy estimators from non-equilibrium experimental data in many important applications concerning biological, chemical, and physical molecular processes.

💡 Research Summary

The paper presents a systematic derivation of five previously unknown inequalities that link equilibrium free‑energy differences (ΔF) with the statistical fluctuations of non‑equilibrium work (W). The starting point is the Jarzynski equality ⟨e^{−βW}⟩ = e^{−βΔF}, which allows one to compute ΔF from an ensemble of work measurements performed far from equilibrium. While the equality is exact, practical applications suffer from large statistical errors because the exponential average is dominated by rare low‑work events, especially when the number of experimental repetitions is limited. To tighten the relationship between ΔF and observable work statistics, the authors combine the Jarzynski framework with a generalized uncertainty‑type inequality originally introduced by H. Weyl. Weyl’s inequality is a Cauchy–Schwarz‑type bound that holds for any real‑valued function f(W): (⟨f⟩)^2 ≤ ⟨f^2⟩·⟨1⟩, where ⟨·⟩ denotes an average over the work distribution. By choosing the weighting function f(W)=e^{−αβW} with α∈