  Water modeling is a challenging problem. Its anomalies are difficult to
reproduce, promoting the proliferation of a large number of computational
models, among which researchers select the most appropriate for the property
they study. In this chapter, we introduce a coarse-grained model introduced by
Franzese and Stanley (FS) that accounts for the many-body interactions of
water. We review mean-field calculations and Monte Carlo simulations on water
monolayers for a wide range of pressures and temperatures, including extreme
conditions. The results show the presence of two dynamic crossovers and explain
the origin of diffusion anomalies. Moreover, the model shows that all the
different scenarios, proposed in the last decades as alternative explanations
of the experimental anomalies of water, can be related by the fine-tuning of
the many-body (cooperative) interaction. Once this parameter is set from the
experiments, the FS model predicts a phase transition between two liquids with
different densities and energies in the supercooled water region, ending in a
liquid-liquid critical point. From this critical point stems a liquid-liquid
Widom line, i.e., the locus of maxima of the water correlation length, that in
the FS model can be directly calculated. The results are consistent with the
extrapolations from experiments. Furthermore, they agree with those from
atomistic models but make predictions over a much wider thermodynamic region,
allowing for a better interpretation of the available experimental data. All
these findings provide a coherent picture of the properties of water and
confirm the validity of the FS model that has proved to be useful for
large-scale simulations of biological systems.
