Universal gorban's entropies: Geometric case study

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 constructionalgorithmarepartialequilibriaofreactionsandconvexenvelopesoffamiliesoffunctions. These new functions aimed to resolve "the mystery" about the difference between the rich family of Lyapunovfunctions(f-divergences)forlinearkineticsandalimitedcollectionofLyapunovfunctions fornon-linearnetworksinthermodynamicconditions. Thelackofexamplesdidnotallowtoevaluate 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 brie?y 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.