Tropical limit and the micro-macro correspondence

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.

arXiv comment33 pages. Typos have been corrected, some statements have been clarified. Latest version

References in corpus (15)

• The Entropy Production Fluctuation Theorem and the Nonequilibrium Work Relation for Free Energy Differences
• Aspects of Holographic Entanglement Entropy
• Inconsistent thermostatistics and negative absolute temperatures
• Tropical Geometry of Statistical Models
• Gibbs, Boltzmann, and negative temperatures
• The metastate approach to thermodynamic chaos
• On the dispute between Boltzmann and Gibbs entropy
• Cellular Automata and Ultra-Discrete Painlevé Equations
• Principles of classical statistical mechanics: A perspective from the notion of complementarity
• Comment on "Consistent thermostatistics forbids negative absolute temperatures"
• Thermodynamics and the moment map
• Tropical Cramer Determinants Revisited
• Tropical Limit in Statistical Physics
• Geometry of the basic statistical physics mapping
• Zeros and amoebas of partition functions

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• Universal Constants as Manifestations of Relativity