Emergence and Breaking of Duality Symmetry in Generalized Fundamental Thermodynamic Relations

Abstract

Thermodynamic laws are limiting behavior of the statistics of repeated measurements of an arbitrary system with a priori probability distribution. A duality symmetry arises, between Massieu-Guggenheim entropy and Gibbs entropy, in the limit of large number of measurements. This yields the fundamental thermodynamic relation and Hill-Gibbs-Duhem (HGD) equation as a dual pair. We show if the system itself has a second macroscopic limit that satisfies Callen's postulate that entropy being an Eulerian homogeneous function of all extensive variables, the symmetry is lost: the HGD equation reduces to the Gibbs-Duhem equation. This theory provides better logic to textbook thermodynamics, a clarification on nanothermodynamics, as well as novel ideas for a thermodynamic-like framework for single-cell biology.

Duality Symmetry, Two Entropy Functions, and an Eigenvalue Problem in Gibbs' Theory

“…In recent years, through activities in stochastic thermodynamics and applying the mathematical theory of large deviations to the asymptotic behavior of the LLN, "the essence of a thermodynamic description is not found in its connection to conservation laws, microscopic reversibility, or the equilibrium state relation they entail, despite the central role those play" in the current teaching; see [1] and the references cited within. In addition to the derivation of fluctuation relations in connection to various nonequilibrium entropy productions, more recent results on the unification of stochastic chemical kinetics and Gibbsian thermochemistry [2], as well as the probabilistic elucidation of T. L. Hill's thermodynamics of small systems (1963) [3,4], have all further substantiated the above claim.…”

Thermodynamic Behavior of Statistical Event Counting in Time: Independent and Correlated Measurements

“…Eq. 42b has the form of the Hill-Gibbs-Duhem equation [1]. The Ψ/t ∼ o(1) as t, n → ∞, yields a singular LFT of ϕ in Eq.…”

“…Recent studies [1,2] on (i) the probabilistic foundation of Hill's nanothermodynamics (1963) [3,4], (ii) the duality symmetry hidden in Gibbs' statistical mechanics and thermochemistry [5,6], and (iii) the revisitation of the Gibbs and the Shannon entropies, all have shown central importance of the large deviations theory [7][8][9][10] for the asymptotic, entropic description of statistical observables, such as empirical sample mean values and counting frequencies, as well as the key role played by conjugate variables in establishing the concept of an equation of state (EoS). It was revealed that the Gibbs entropy as a function of mean internal energy is the Legendre-Fenchel dual to the Massieu potential as a function of inverse temperature, and the Helmhotz free energy as a function of a set of energies of states in k B T units, {u i }, is the Legendre-Fenchel dual to the Shannon-Sanov relative entropy as a function of empirical counting frequencies {ν i }.…”

“…Statistical thermodynamics is a theory about emergent phenomenon [11]. The recent mathematical development [1,2] in fact had been captured presciently by P. W. Anderson in 1972 [12],…”

“…Following the logic of classical thermodynamics as presented by H. B. Callen in [13] and further developed recently in [1,2], a macroscopic thermodynamics like analysis for the Φ(t, n) in (1) can be carried out. This yields an equation of state (EoS) among two conjugate intensive variables corresponding to time (t) and statistical counting (n).…”