The Thermodynamic Characteristics Of
Systems Governing By The Weighted Sum Of
Nonlinear Pairwise Interactions
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
Related Book
Other papers in this volume
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