## Section: Scientific Foundations

### From particle dynamics to continuum mechanics

DPD is well adapted to describe biological cells. However, it is a
very time consuming method which becomes difficult to use if the
number of particles exceeds the order of 10^{5} -10^{6} (unless
distributed computing is used). On the other hand, PDEs of
continuum mechanics are essentially more efficient for numerical
simulations. Moreover, they can be studied by analytical methods
which have a crucial importance for the understanding of
relatively simple test cases. Thus we need to address the question
about the relation between DPD and PDE. The difficulty follows
already from the fact that molecular dynamics with the
Lennard-Jones potential can describe very different media,
including fluids (compressible, incompressible, non-Newtonian, and
so on) and solids (elastic, elasto-plastic, and so on).
Introduction of dissipative terms in the DPD models can help to
justify the transition to a continuous medium because each medium
has a specific to it law of dissipation. Our first results
[26] show the correspondence between a DPD model and
Darcy's law describing fluid motion in a porous medium. However,
we cannot expect a rigorous justification in the general case and
we will have to carry out numerical comparison of the two
approaches.

An interesting approach is related to hybrid models where PDEs of continuum mechanics are considered in the most part of the domain, where we do not need a microscopical description, while DPD in some particular regions are required to consider individual cells.