Non-commutative geometry and irreversibility

A kinetics built upon $q$-calculus, the calculus of discrete dilatations, is shown to describe diffusion on a hierarchical lattice. The only observable on this ultrametric space is the "quasi-position" whose eigenvalues are the levels of the hierarchy, corresponding to the volume ofphase space available to the system at any given time. Motion along the lattice of quasi-positions is irreversible.