  Drawing inspiration from transportation theory, in this work we introduce the
notions of "well-structured" and "stable" Gibbs states and we investigate their
implications for quantum thermodynamics and its resource theory approach via
thermal operations. It turns out that, in the quasi-classical realm, global
cyclic state transfers are impossible if and only if the Gibbs state is stable.
Moreover, using a geometric approach by studying the so-called
thermomajorization polytope we prove that any subspace in equilibrium can be
brought out of equilibrium via thermal operations. Interestingly, the case of
some subsystem being in equilibrium can be witnessed via degenerate extreme
points of the thermomajorization polytope, assuming the Gibbs state of the
system is well structured. These physical considerations are complemented by
simple new constructions for the polytope's extreme points as well as for an
important class of extremal Gibbs-stochastic matrices.
