Uniqueness of Lieb–Yngvason entropy-meter entropy for non-extensive and non-equilibrium systems: conditions for comparability, global calibration, and validity domain

Lieb and Yngvason showed that an 'entropy meter' — a normal, scalable system with an established entropy — can, in principle, define an entropy for systems that cannot be scaled (gravitating bodies, surface-dominated systems) and for non-equilibrium states, via two variational bounds $S_-$ and $S_+$ on the meter's entropy change. Their Theorem 1 yields a unique, additive entropy only under the comparability assumption (B2), which they emphasize is highly nontrivial and cannot be expected to hold generally. The authors' 2022 review states that comparability in both extensions is postulated rather than derived. The open problem is to characterize the classes of systems and dynamical regimes for which comparability (or weaker verifiable conditions) guarantees $S_- = S_+$ ; establish realistic conditions for global consistency of chart calibrations; delineate the mesoscopic/quantum validity domain of the adiabatic-accessibility ordering; and develop computation of $S_\pm$ in concrete models.

Background

In the Lieb–Yngvason axiomatization of the second law, the basic primitive is the relation of adiabatic accessibility: for states $X$ , $Y$ of a thermodynamic system one writes $X \prec Y$ if $X$ can be transformed into $Y$ by an adiabatic process, understood operationally as a 'work process' that leaves no mark on the universe except the raising or lowering of a weight — no adiabatic enclosure, slowness, or notions of heat and temperature are presupposed. Two states are comparable if $X \prec Y$ or $Y \prec X$ . States can be composed (products $\Gamma_1 \times \Gamma_2$ of state spaces) and, for 'normal' systems, scaled ( $\lambda X$ multiplies extensive variables while leaving intensive properties unchanged). For normal systems the entropy is fixed by a double variational principle: picking reference states $X_0 \prec\prec X_1$ (meaning $X_0 \prec X_1$ but not $X_1 \prec X_0$ ), one defines $S_-(X)$ as a supremum and $S_+(X)$ as an infimum over scaled compositions of the references, and the common value, when it exists, is an essentially unique, additive, extensive entropy characterizing $\prec$ (i.e., $X \prec Y$ iff $S(X) \le S(Y)$ ). The pivotal ingredient is comparability: for simple systems in equilibrium it was derived from structural axioms (convexity of the state space, nestedness of forward sectors — the sets of states adiabatically accessible from a given state — and the thermal join and mixing axioms, which govern thermal contact between compound systems and the comparison of differently scaled copies), which is one of the deepest parts of the 1999 Lieb–Yngvason construction. Two extensions push the framework beyond its original domain. First, for non-equilibrium states of a normal system, where scaling is unavailable, two entropy functions $S_- \le S_+$ are defined by sandwiching a non-equilibrium state between equilibrium states; uniqueness is equivalent to every non-equilibrium state being adiabatically equivalent to some equilibrium state. Lieb and Yngvason argued this is implausible in general: it plausibly holds near equilibrium where classical irreversible thermodynamics (local equilibrium, entropy density integrated over cells) applies, but fails when fluxes must be treated as independent variables, as in extended irreversible thermodynamics with Cattaneo-type heat transport; their glued-copper toy model exhibits a finite gap $S_+ - S_-$ when equilibration via a Carnot machine competes with irreversible leakage through the connecting layer. Second, for 'non-extensive' systems — meaning macroscopic systems that cannot be scaled without changing intrinsic properties, such as gravitating bodies or systems with sizeable surface contributions (not Tsallis-type generalized entropies, which fail additivity and are thereby excluded from this framework) — the 2014 paper introduces an entropy meter: a normal system $\Gamma_0$ with known entropy $S$ , coupled adiabatically to the system $\Gamma$ of interest. Under the order-theoretic axioms A1–A3 and A6 (reflexivity, transitivity, consistency, stability) plus a calibration condition B1, two functions on $\Gamma$ are defined variationally from the meter's entropy change: $S_-(X) = \sup\{S(Z') : (X_1, Z') \prec (X, Z_0)\}$ and $S_+(X) = \inf\{S(Z'') : (X, Z_0) \prec (X_1, Z'')\}$ . Under the additional comparability assumption B2 — every pair of states in any multiple product of the spaces involved is comparable — Theorem 1 shows $S_- = S_+$ , and the common function is an entropy on $\Gamma$ : essentially unique up to an additive constant and additive on products. Without B2, $S_\pm$ remain monotone bounds bracketing every monotone state function (Proposition 1), but they depend nontrivially on the reference points and need not be additive. Because B1 can only be expected when $X$ and the reference $X_1$ are close (a small meter cannot move a large system far), the state space must be covered by overlapping 'charts', raising the question whether the additive constants on overlapping charts can be chosen globally consistently; the authors give only an in-principle remedy — a sufficiently large scalable copy of $\Gamma_0$ , 'which might not be practical physically'. They further note that near atomic dimensions the relation $\prec$ itself becomes 'fuzzy' due to quantum entanglement with surroundings, so the framework's domain of validity in size, coherence, and correlation-strength parameters is undetermined, and that the variational formulas may be inconvenient for practical determination of entropy.

Problem Statement

Convert the Lieb–Yngvason entropy-meter construction — in which a normal, scalable system with an established entropy is coupled adiabatically to the system of interest, and an entropy for the latter is defined through the variational bounds $S_-$ and $S_+$ computed from the meter's entropy change — from an in-principle existence result into a well-posed and applicable entropy definition for non-extensive (non-scalable) and non-equilibrium systems. The components, as identified by the sources, jointly serve this single objective — the uniqueness conditions are the central missing piece, and global calibration, validity domain, and computability are the steps required for the construction to be usable: (1) Core: characterize the classes of physically relevant systems and dynamical regimes — non-equilibrium states of normal systems and non-scalable systems coupled to an entropy meter — for which the comparability assumption (B2: every pair of states in any multiple product of the state spaces involved, including the meter space, is comparable under the adiabatic-accessibility relation $\prec$ ) holds, or identify weaker, physically motivated conditions that guarantee $S_- = S_+$ , i.e., essential uniqueness (up to an additive constant) and additivity of the entropy on products; in particular, establish comparability from dynamical laws in concrete models, as the 2013 conclusions explicitly request. (2) Give explicit, physically realizable conditions under which local chart calibrations of the meter (the local form of the calibration condition B1, which requires that small changes of the meter state suffice to relate a system state to the reference point) are globally consistent (no additive-constant mismatch along chains of overlapping charts), replacing the in-principle remedy of a sufficiently large scalable copy of the meter. (3) Delineate the domain of validity of the adiabatic-accessibility ordering (the relation $X \prec Y$ , read as ' $X$ can be transformed into $Y$ by an adiabatic process') and the entropy-meter construction in system size, quantum coherence, and correlation-strength parameters, given that entanglement with surroundings is expected to render the ordering non-classical or 'fuzzy'. (4) Develop methods to compute or approximate the entropy-meter bounds $S_\pm$ in concrete models beyond the existence proofs, and assess their robustness to experimental imperfections. An answer may resolve the core (1) alone if it delivers verifiable conditions covering a class of non-equilibrium or non-scalable systems not already covered by the equilibrium simple-system comparability result; a result for a single concrete model does not by itself resolve (1). A complete answer to the full objective addresses (1)–(4) within the classical adiabatic-accessibility framework.

References

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