Mathematics

Imagine considering a material aggregate as either a molecular system or a continuous medium, and wishing to relate the two representations. Assume that the fields entering the continuum description, such as deformation and stress, are adequately sampled on an array of positions, whose typical spacing H is enormously larger than the average intermolecular distance d. Associate with each of these macroscopic sampling positions an Andersen-Parrinello-Rahman (APR) cell, whose reference size h is large enough with respect to d in order to allow for a decent sampling of the microscopic molecular states, and still much smaller than H: H >> h >> d. Now, let the molecules in each cell interact directly with each other (and with their h-neighboring images), while being indirectly affected by those in the H-neighboring cells via the collective degrees of freedom of the deforming APR cell, governed by the force balance and compatibility equations of the continuous medium (sampled at the H scale). In turn, the stress-deformation relation characterizing the response of the continuous medium arises as an emergent property of molecular dynamics on the h scale. Based on joint ongoing work with Manuela Minozzi and Matteo Paoluzzi, I will present and discuss a method for playing this sort of multi-scale game.