Infinite-dimensional genetic and evolution algebras generated by Gibbs measures

TL;DR

This paper introduces infinite-dimensional genetic and evolution algebras derived from Gibbs measures, extending finite-dimensional models and analyzing their properties, including infertility and algebraic decomposition.

Contribution

It extends finite-dimensional Gibbs algebra models to infinite dimensions using Gibbs measures, introducing infertility and algebraic decomposition.

Findings

Algebras are commutative, non-associative, with uncountable basis.

Infertility induces a decomposition into fertile ideals.

Properties of Gibbs measures facilitate algebraic analysis.

Abstract

Genetic and evolution algebras arise naturally from applied probability and stochastic processes. Gibbs measures describe interacting systems commonly studied in thermodynamics and statistical mechanics with applications in several fields. Here, we consider that the algebras are determined by configurations of finite spins on a countable set with their associated Gibbs distributions. The model preserves properties of the finite-dimensional Gibbs algebras found in the literature and extend their results. We introduce infertility in the genetic dynamics when the configurations differ macroscopically. It induces a decomposition of the algebra into a direct sum of fertile ideals with genetic realization. The proposed infinite-dimensional algebras are commutative, non-associative, with uncountable basis and zero divisors. The properties of Gibbs measures allow us to deal with the difficulties arising from the algebraic structure and obtain the results presented in this article.