Turkish Journal of Physics
On a lattice with equal spacing on the logarithmic scale, a momentum operator that respects the asymmetry of this nonlinear lattice yields a kinetics that can be understood in terms of diffusion on an underlying ultrametric space, if one also identifies the cannonical commutator with the time dilation operator. The motion to which this non-conventional kinetics corresponds is irreversible, with an explicit violation of time reversal symmetry resulting from the spreading with time of a probability distribution over a larger and larger volume of the phase space.
ERZAN, Ayşe and GORBON, Ayşe (1999) "q-calculus and irreversible dynamics on a hierarchical lattice," Turkish Journal of Physics: Vol. 23: No. 1, Article 2. Available at: https://journals.tubitak.gov.tr/physics/vol23/iss1/2