Abstract
Recently, A.N. Gorban presented a rich family of universal Lyapunov functions for any linear or non-linear reaction network with detailed or complex balance. Two main elements of the construction algorithm are partial equilibria of reactions and convex envelopes of families of functions. These new functions aimed to resolve “the mystery” about the difference between the rich family of Lyapunov functions (f-divergences) for linear kinetics and a limited collection of Lyapunov functions for non-linear networks in thermodynamic conditions. The lack of examples did not allow to evaluate the difference between Gorban’s entropies and the classical Boltzmann–Gibbs–Shannon entropy despite obvious difference in their construction. In this paper, Gorban’s results are briefly reviewed, and these functions are analysed and compared for several mechanisms of chemical reactions. The level sets and dynamics along the kinetic trajectories are analysed. The most pronounced difference between the new and classical thermodynamic Lyapunov functions was found far from the partial equilibria, whereas when some fast elementary reactions became close to equilibrium then this difference decreased and vanished in partial equilibria.
Highlights
A convex function F ( N ) on U satisfies the partial equilibria criterion with a given thermodynamic Lyapunov function H and reversible reaction mechanism given by stoichiometric Equation (12) if argmin H ( N ) ⊆ argmin F ( N )
Ci + β i χ eq ci Equation (23) is very similar to the usual condition of detailed balance but we have to emphasise that it does not include any reaction rate constant, does not assume the reversibility of any reaction or microreversibility and just describes the minimisers of H in the given direction. It can be considered as the thermodynamic equilibrium condition for the elementary reaction with the stoichiometric vector γ and can differ from the kinetic equilibrium condition if the detailed balance is not assumed
The level sets for HΓ were found significantly different from the level sets of the classical thermodynamic Lyapunov function H ( N ) (2)
Summary
The classical example of the Lyapunov functional in kinetics was provided by Boltzmann in. The classical Lyapunov functions (1) and (2) have an important property, universality: they do not depend directly on the collision and reaction mechanisms and kinetic constants but on the equilibrium distributions (concentration and the detailed or complex balance condition in general form [8]). This universality can be considered as a manifestation of the universality of thermodynamics that does not depend on the microscopic details directly
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