A partial ordering of sets, making mean entropy monotone

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.