In this article, we define the transport dimension of probability measures on R m using ramified optimal transportation theory. We show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called " the dimensional distance, " on… (More)
We prove Lieb-Robinson bounds and the existence of the thermo-dynamic limit for a general class of irreversible dynamics for quantum lattice systems with time-dependent generators that satisfy a suitable decay condition in space.
We consider a beam of two-level randomly excited atoms that pass one-by-one through a one-mode cavity. We show that in the case of an ideal cavity, i.e. no leaking of photons from the cavity, the pumping by the beam leads to an unlimited increase in the photon number in the cavity. We derive an expression for the mean photon number for all times. Taking… (More)