Statistical Mechanics and Categorical Entropy

TL;DR

This paper explores the connection between categorical entropy in A-infinity-categories and von Neumann entropy in quantum lattices, proposing a unified framework that reveals deep structural links between algebraic and statistical mechanical concepts.

Contribution

It introduces a lattice model for categorical entropy, unifies it with von Neumann entropy, and proposes a condition linking algebraicity properties to statistical mechanics.

Findings

Von Neumann entropy per site converges to the logarithm of an algebraic integer.

A lattice model for categorical entropy is constructed from endofunctors.

A gauged lattice framework unifies the two entropy notions.

Abstract

This paper investigates the relationship between categorical entropy and von Neumann entropy of quantum lattices. We begin by studying the von Neumann entropy, proving that the average von Neumann entropy per site converges to the logarithm of an algebraic integer in the low-temperature and thermodynamic limits. Next, we turn to categorical entropy. Given an endofunctor of a saturated A-infinity-category, we construct a corresponding lattice model, through which the categorical entropy can be understood in terms of the information encoded in the model. Finally, by introducing a gauged lattice framework, we unify these two notions of entropy. This unification leads naturally to a sufficient condition for a conjectural algebraicity property of categorical entropy, suggesting a deeper structural connection between A-infinity-categories and statistical mechanics.