Universal Gorban’s Entropies: Geometric Case Study

Recently, A.N. Gorban presented a rich family of universal Lyapunov functions for anylinear or non-linear reaction network with detailed or complex balance. Two main elements of theconstruction 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 ofLyapunov functions (f -divergences) for linear kinetics and a limited collection of Lyapunov functionsfor non-linear networks in thermodynamic conditions. The lack of examples did not allow to evaluatethe difference between Gorban’s entropies and the classical Boltzmann–Gibbs–Shannon entropydespite 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. Thelevel sets and dynamics along the kinetic trajectories are analysed. The most pronounced differencebetween the new and classical thermodynamic Lyapunov functions was found far from the partialequilibria, whereas when some fast elementary reactions became close to equilibrium then thisdifference decreased and vanished in partial equilibria.