Monte Carlo Simulations. In this section, using copula-GARCH based Monte Carlo simulation, I examine the size and power of the entropy-based test and show that the entropy test has reliable sizes, and has higher power in finite samples than the Hong, Tu, and Xxxx (2007) test.
Monte Carlo Simulations. In this work the phase boundaries were estimated by means of Metropolis Monte Carlo (MMC) sampling [65] within the isobaric semi-grand canonical ensemble (N, P, T, ) for defect free binary alloys and the isobaric grand canonical ensemble (N, P, T) for ternary alloys and binary alloys containing defects. Both MMC methods include three types of trials of which two are common to both methods: (i) a random displacement of all atoms from their current positions (by this trial lattice relaxation and vibrational entropy are accounted for); (ii) the overall volume change of the simulation box (this trial allows the desired pressure to be maintained, even if a structural transition were to occur). In the semi-grand canonical ensemble the third trial consists of the change of species of a randomly picked atom (by this trial the equilibrium composition is sampled), while in the grand canonical ensemble the latter consists of the random exchange of two atoms of different species (by this trial the equilibrium configurations are sampled). The decision on the acceptance of the new configuration is based on the standard Metropolis algorithm [65] and one set of these trials is termed an "MC step". The phase boundaries are obtained by scanning the chemical potential difference, versus composition. A plot of such a curve is obtained at every desired temperature and any discontinuity in the latter is interpreted as a phase transition (see [66] and references therein for more details). In order to identify the solubility limit at a given temperature, the average composition of the last three points (which show statistical scatter) is taken before the phase transition occurs, the maximum spread in the latter serving as error bar. By definition the semi-grand canonical ensemble works in a single phase region. Therefore small simulation boxes can be used without loss in accuracy. For our simulations we used boxes containing as little as 1024 atoms and 1.0E+5 MC steps proved enough to reach full convergence of enthalpy and composition. The phase boundaries from MMC simulations in grand canonical ensemble are obtained by scanning the temperature (50 K temperature grid) for different compositions (0.25-2% Ni and 0-10% Cr). In this way the phase boundary is determined as the limit below which precipitation is observed and above which solubility is observed. To determine precipitation the solute cluster size distribution averaged over the last five million MC steps is visualized...
