Finite-size scaling investigation of the liquid-liquid critical point in ST2 water and its stability with respect to crystallization

The liquid-liquid critical point scenario of water hypothesizes the existence of two metastable liquid phases---low-density liquid (LDL) and high-density liquid (HDL)---deep within the supercooled region. The hypothesis originates from computer simulations of the ST2 water model, but the stability of the LDL phase with respect to the crystal is still being debated. We simulate supercooled ST2 water at constant pressure, constant temperature and constant number of molecules N for N<=729 and times up to 1000 ns. We observe clear differences between the two liquids, both structural and dynamical. Using several methods, including finite-size scaling, we confirm the presence of a liquid-liquid phase transition ending in a critical point. We find that the LDL is stable with respect to the crystal in 98% of our runs (we perform 372 runs for LDL or LDL-like states), and in 100% of our runs for the two largest system sizes (N=512 and 729, for which we perform 136 runs for LDL or LDL-like states). In all these runs tiny crystallites grow and then melt within 1000 ns. Only for N<=343 we observe six events (over 236 runs for LDL or LDL-like states) of spontaneous crystallization after crystallites reach an estimated critical size of about 70+/-10 molecules.

Optimal box-covering algorithm for fractal dimension of complex networks

The self-similarity of complex networks is typically investigated through computational algorithms the primary task of which is to cover the structure with a minimal number of boxes. Here we introduce a box-covering algorithm that not only outperforms previous ones, but also finds optimal solutions. For the two benchmark cases tested, namely, the E. Coli and the WWW networks, our results show that the improvement can be rather substantial, reaching up to 15% in the case of the WWW network.