One hypothesized explanation for water's anomalies imagines the existence of a liquid-liquid (LL) phase transition line separating two liquid phases and terminating at a LL critical point. We simulate the classic ST2 model of water for times up to 1000 ns and system size up to N = 729. We find that for state points near the LL transition line, the entire system flips rapidly between liquid states of high and low density. Our finite-size scaling analysis accurately locates both the LL transition line and its associated LL critical point. We test the stability of the two liquids with respect to the crystal and find that of the 350 systems simulated, only 3 of them crystallize and these 3 for the relatively small system size N = 343 while for all other simulations the incipient crystallites vanish on a time scales smaller than ≈ 100 ns.

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 1 μs. 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 1 μs. 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. © 2013 AIP Publishing LLC. [http://dx

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 outperforms previous ones in most cases. For the two benchmark cases tested, namely, the E. coli and the World Wide Web (WWW) networks, our results show that the improvement can be rather substantial, reaching up to 15% in the case of the WWW network.

We simulate the ST2 water model for time periods up to 1000 ns, and for four different system sizes, N = 6 3 , 7 3 , 8 3 , and 9 3. We locate the liquid-liquid phase transition line and its critical point in the supercooled region. Near the liquid-liquid phase transition line, we observe that the system continuously flips between the low-density and high-density liquid phases. We analyze the transition line further by calculating two thermodynamic response functions, the isobaric specific heat capacity C P and the isothermal compressibility K T. We use two different methods: (i) from fluctuations and (ii) with the relevant thermodynamic derivative. We find that the maxima of two different response functions coincide within the accuracy of our simulations. The lines of C P and K T maxima below the critical pressure approximate the Widom line which is continuous with the line of first-order transitions in the two-phase region where we observe the phase flipping.

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