Pairwise interactions origin of entropy functions

TL;DR

This paper develops a unified theory linking pairwise interactions, dynamical systems, and entropy functions, showing that nonlinear pairwise interactions underpin all known entropy measures through generalized replicator dynamics and Lyapunov functions.

Contribution

It introduces a novel theoretical framework connecting pairwise interactions with entropy functions, generalizing Fisher's theorem and deriving new distance measures between probability distributions.

Findings

Generalized replicator equations derived from interaction hypotheses.

Energy-like Lyapunov functions generalize Fisher's theorem.

Negative relative entropy acts as a convex distance measure.

Abstract

In this paper we combine the three universalisms: pairwise interactions concept, dynamical systems theory and relative entropy analysis to develop a theory of entropy issues. We introduce two hypotheses concerning the structure and types properties of the system's entities interactions and derive generalized replicator dynamic equations. Then we construct energy-like and entropy-like Lyapunov-Meyer functions (LMF) for these equations. We show that energy-like LMF contains no information about the equilibrium of the system and is a substantial generalization of the Fisher's fundamental theorem of natural selection. If there are exists nontrivial equilibrium point for generalized replicator system then we construct entropy-like LMF for this system and prove that it is a relative entropy function or the function of information divergence. We prove that negative relative entropy is a convex function for a probability space and receive new distance measure between two probability distributions. Also we use Legendre-Donkin-Fenchel transformation for dual coordinates. As result we establish the set of next important links: nonlinear pairwise interaction - generalized Fisher (replicator) equations - Lyapunov-Meyer functions - relative entropy - distance measure - LDF-transformation - duality. In particular it follows from these links that nonlinear pairwise interaction is the origin of all known entropy functions.