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Markov state model of the two-state behaviour of water
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Calculating the minimized RMSD of all 120 permutations does not yet require to calculate the transformation matrix from the SVD, only the corresponding singular values,45 which is most efficiently done via the roots of the cubic characteristic polynomial (see http://www.rohitab.com/discuss/topic/36251-c-svd-of-3x3-matrix/). While this approach is potentially numerically unstable, it has been verified that the round-off errors are small for this particular problem. Only for the permutation that reveals the smallest RMSD, the full SVD is then calculated. See W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, 1992).
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One expects cr(r) to go to zero for large enough r. However, due to the limited size of the simualation box and the fact that the MD simulation has been performed in the NVT ensemble, a small anti-correlation (i.e., negative cr(r)) is observed for large r. Since LDL and HDL have different densities, a particular domain at one position of the simulation box implies a larger likelihood of the opposite domain far away from it since the total volume is constant. The fitting of the correlation function therefore included a small negative pedestial, which has been subtracted in the presentation of Fig. 5(b).
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With the help of a Markov State Model (MSM), two-state behaviour is resolved for two computer models of water in a temperature range from 255 K to room temperature (295 K). The method is first validated for ST2 water, for which the so far strongest evidence for a liquid-liquid phase transition exists. In that case, the results from the MSM can be cross-checked against the radial distribution function
5(r) of the 5th-closest water molecule around a given reference water molecule. The latter is a commonly used local order parameter, which exhibits a bimodal distribution just above the liquid-liquid critical point that represents the low-density form of the liquid (LDL) and the high density liquid. The correlation times and correlation lengths of the corresponding spatial domains are calculated and it is shown that they are connected via a simple diffusion model. Once the approach is established, TIP4P/2005 will be considered, which is the much more realistic representation of real water. The MSM can resolve two-state behavior also in that case, albeit with significantly smaller correlation times and lengths. The population of LDL-like water increases with decreasing temperature, thereby explaining the density maximum at 4 °C along the lines of the two-state model of water.
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