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    "# Entropy - Enthalpy parameters"
   ]
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    "## Entropy fingerprint\n",
    "\n",
    "The entropy parameter was introduced by Piaggi et\n",
    "al {cite}`Piaggi2017` for identification of defects and\n",
    "distinction between solid and liquid. The entropy paramater $s_s^i$ is\n",
    "defined as,\n",
    "\n",
    "> $$s_s^i = -2\\pi\\rho k_B \\int_0^{r_m} [g_m^i(r)\\ln g_m^i(r) - g_m^i(r) + 1] r^2 dr$$\n",
    "\n",
    "where $r_m$ is the upper bound of integration and $g_m^i$ is radial\n",
    "distribution function centered on atom $i$,\n",
    "\n",
    "> $$g_m^i(r) = \\frac{1}{4\\pi\\rho r^2} \\sum_j \\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\exp{-(r-r_{ij})^2/(2\\sigma^2)}$$\n",
    "\n",
    "$r_{ij}$ is the interatomic distance between atom $i$ and its neighbors\n",
    "$j$ and $\\sigma$ is a broadening parameter.\n",
    "\n",
    "The averaged version of entropy parameters $\\bar{s}_s^i$ can be\n",
    "calculated in two ways, either using a simple averaging over the\n",
    "neighbors given by,\n",
    "\n",
    "> $$\\bar{s}_s^i = \\frac{\\sum_j s_s^j + s_s^i}{N + 1}$$\n",
    "\n",
    "or using a switching function as described below,\n",
    "\n",
    "> $$\\bar{s}_s^i = \\frac{\\sum_j s_s^i f(r_{ij}) + s_s^i}{\\sum_j f(r_{ij}) + 1}$$\n",
    "\n",
    "$f(r_{ij})$ is a switching parameter which depends on $r_a$ which is the\n",
    "cutoff distance. The switching function shows a value of 1 for\n",
    "$r_{ij} << r_a$ and 0 for $r_{ij} >> r_a$. The switching function is\n",
    "given by,\n",
    "\n",
    "> $$f(r_{ij}) = \\frac{1-(r_{ij}/r_a)^N}{1-(r_{ij}/r_a)^M}$$\n",
    "\n",
    "Entropy parameters can be calculated in pyscal using the following code,\n",
    "\n",
    "``` python\n",
    "import pyscal.core as pc\n",
    "sys = pc.System()\n",
    "sys.read_inputfile('conf.dump')\n",
    "sys.find_neighbors(method=\"cutoff\", cutoff=0)\n",
    "lattice_constant=4.00\n",
    "sys.calculate_entropy(1.4*lattice_constant, averaged=True)\n",
    "atoms = sys.atoms\n",
    "entropy = [atom.entropy for atom in atoms]\n",
    "average_entropy = [atom.avg_entropy for atom in atoms]\n",
    "```\n",
    "\n",
    "The value of $r_m$ is provided in units of lattice constant. Further\n",
    "parameters shown above, such as $\\sigma$ can be specified using the\n",
    "various keyword arguments. The above code does a simple averaging over\n",
    "neighbors. The switching function can be used by,\n",
    "\n",
    "``` python\n",
    "sys.calculate_entropy(1.4*lattice_constant, ra=0.9*lattice_constant, switching_function=True, averaged=True)\n",
    "```\n",
    "\n",
    "In pyscal, a slightly different version of $s_s^i$ is calculated. This\n",
    "is given by,\n",
    "\n",
    "> $$s_s^i = -\\rho \\int_0^{r_m} [g_m^i(r)\\ln g_m^i(r) - g_m^i(r) + 1] r^2 dr$$\n",
    "\n",
    "The prefactor $2\\pi k_B$ is dropped in the entropy values calculated in\n",
    "pyscal.\n",
    "\n",
    "## References\n",
    "\n",
    "```{bibliography} ../references.bib\n",
    ":filter: docname in docnames\n",
    ":style: unsrt\n",
    "```"
   ]
  }
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