Author(s)

Y. A. Pykh & I. G. Malkina-Pykh

Abstract

In this report we summarize the results in the application of the Direct Lyapunov method to the generalized replicator systems with the weighted sum of nonlinear pairwise interactions. These complex systems define the properties of a system composed of objects that are coupled via nonlinear pairwise interactions. It is shown that there exist two types of thermodynamic Lyapunov functions: fitness-like and entropy-like. As an example it will be established that practically all known thermodynamic characteristics may be obtained from entropy-like Lyapunov functions for replicator systems. \“The positive time direction is associated with the increase of entropy. Let us emphasize the strong and very specific way in which the one-sidedness of time appears in the second law. According to its formulation it leads to the existence of a function having quite specific properties as expressed by the fact that for an isolated system it can only increase in time. Such functions play an important role in modern theory of stability as initiated by the classic work of Lyapunov. For this reason they are called Lyapunov functions (or functionals). The entropy S is a Lyapunov function for isolated systems. As shown in all textbooks thermodynamic potentials such as the Helmholtz or Gibbs free energy are also Lyapunov functions for other \“boundary conditions” (such as imposed values of temperature and volume). In all these cases the system evolves to an equilibrium state characterized by the existence of a thermodynamic potential. This equilibrium state is an \“attractor” for non-equilibrium states.” Ilya Prigogine Nobel Lecture, 8 December, 1977

Keywords