Tropical limit and the micro-macro correspondence

TL;DR

This paper explores how tropical mathematics provides a unified framework to connect microscopic and macroscopic objects in statistical models, emphasizing structure-preserving maps, order relations, and physical applications in thermodynamics and physics.

Contribution

It introduces a tropical algebra-based approach to relate microscopic and macroscopic systems, highlighting order relations, dualities, and physical interpretations in non-equilibrium physics.

Findings

Tropical algebra offers a consistent framework for macrosystems and their constituents.

The approach reveals underlying order relations via filters and ideals.

Applications include thermodynamics and non-equilibrium physics models.

Abstract

Tropical mathematics is used to establish a correspondence between certain microscopic and macroscopic objects in statistical models. Tropical algebra gives a common framework for macrosystems (subsets) and their elementary constituents (elements) that is well-behaved with respect to composition. This kind of connection is studied with maps that preserve a monoid structure. The approach highlights an underlying order relation that is explored through the concepts of filter and ideal. Particular attention is paid to asymmetry and duality between max- and min-criteria. Physical implementations are presented through simple examples in thermodynamics and non-equilibrium physics. The phenomenon of ultrametricity, the notion of tropical equilibrium and the role of ground energy in non-equilibrium models are discussed. Tropical symmetry, i.e. idempotence, is investigated.