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Nano-science and nano-technology as well as the fine description of the structure and mechanics of materials from the nanometric to the micrometric scales need descriptions ranging from the quantum mechanics to the kinetic theory descriptions characteristic of statistical mechanics. This paper explores the modelling at these scales and points out the main challenges related to the numerical solution of such models that sometimes are discrete but involve an extremely large number of particles (as in the case of molecular dynamics simulations or coarse-grained molecular dynamics) and other times are continuous but they are defined in highly multidimensional spaces leading to the well known curse of dimensionality issues.

Atoms, Molecules and Flows: Recent Advances and New Challenges in their Multi-Scale Numerical Modeling at the Beginning of the Third Millenium