  Discrete formulations of (quantum) gravity in four spacetime dimensions build
space out of tetrahedra. We investigate a statistical mechanical system of
tetrahedra from a many-body point of view based on non-local, combinatorial
gluing constraints that are modelled as multi-particle interactions. We focus
on Gibbs equilibrium states, constructed using Jaynes' principle of constrained
maximisation of entropy, which has been shown recently to play an important
role in characterising equilibrium in background independent systems. We apply
this principle first to classical systems of many tetrahedra using different
examples of geometrically motivated constraints. Then for a system of quantum
tetrahedra, we show that the quantum statistical partition function of a Gibbs
state with respect to some constraint operator can be reinterpreted as a
partition function for a quantum field theory of tetrahedra, taking the form of
a group field theory.
