A partial ordering of sets, making mean entropy monotone

TL;DR

This paper introduces a partial order on sets based on entropy measures, demonstrating that mean entropy decreases with this order, with applications to lattice systems and energy behavior in statistical mechanics.

Contribution

It defines a new order relation for subsystems based on entropy and volume, proving mean entropy's monotonicity and applying it to lattice systems in statistical mechanics.

Findings

Mean entropy decreases with the defined order relation.

Application to lattice systems shows energy decreases in blow-up sequences.

Establishes inequalities for subsystem entropies under certain conditions.

Abstract

Consider a state of a system with several subsystems. The entropies of the reduced state on different subsystems obey certain inequalities, provided there is an equivalence relation, and a function measuring volumes or weights of subsystems. The entropy per unit volume or unit weight, the mean entropy, is then decreasing with respect to an order relation of the subsystems, defined in this paper. In the context of statistical mechanics a lattice system is studied in detail, and a decrease of mean energy is deduced for blow-up sequences of regular and irregular octogons.