  Dust grains are an important component of the interstellar medium (ISM) of
galaxies. We present the first direct measurement of the residence times of
interstellar dust in the different ISM phases, and of the transition rates
between these phases, in realistic hydrodynamical simulations of the
multi-phase ISM. Our simulations include a time-dependent chemical network that
follows the abundances of H^+, H, H_2, C^+ and CO and take into account
self-shielding by gas and dust using a tree-based radiation transfer method.
Supernova explosions are injected either at random locations, at density peaks,
or as a mixture of the two. For each simulation, we investigate how matter
circulates between the ISM phases and find more sizeable transitions than
considered in simple mass exchange schemes in the literature. The derived
residence times in the ISM phases are characterised by broad distributions, in
particular for the molecular, warm and hot medium. The most realistic
simulations with random and mixed driving have median residence times in the
molecular, cold, warm and hot phase around 17, 7, 44 and 1 Myr, respectively.
The transition rates measured in the random driving run are in good agreement
with observations of Ti gas-phase depletion in the warm and cold phases in a
simple depletion model, although the depletion in the molecular phase is
under-predicted. ISM phase definitions based on chemical abundance rather than
temperature cuts are physically more meaningful, but lead to significantly
different transition rates and residence times because there is no direct
correspondence between the two definitions.
