  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.
