Entropy fingerprint

The entropy parameter was introduced by Piaggi et al for identification of defects and distinction between solid and liquid. The entropy paramater \(s_s^i\) is defined as,

\[s_s^i = -2\pi\rho k_B \int_0^{r_m} [g_m^i(r)\ln g_m^i(r) - g_m^i(r) + 1] r^2 dr\]

where \(r_m\) is the upper bound of integration and \(g_m^i\) is radial distribution function centered on atom \(i\),

\[g_m^i(r) = \frac{1}{4\pi\rho r^2} \sum_j \frac{1}{\sqrt{2\pi\sigma^2}} \exp{-(r-r_{ij})^2/(2\sigma^2)}\]

\(r_{ij}\) is the interatomic distance between atom \(i\) and its neighbors \(j\) and \(\sigma\) is a broadening parameter.

The averaged version of entropy parameters \(\bar{s}_s^i\) can be calculated in two ways, either using a simple averaging over the neighbors given by,

\[\bar{s}_s^i = \frac{\sum_j s_s^j + s_s^i}{N + 1}\]

or using a switching function as described below,

\[\bar{s}_s^i = \frac{\sum_j s_s^i f(r_{ij}) + s_s^i}{\sum_j f(r_{ij}) + 1}\]

\(f(r_{ij})\) is a switching parameter which depends on \(r_a\) which is the cutoff distance. The switching function shows a value of 1 for \(r_{ij} << r_a\) and 0 for \(r_{ij} >> r_a\). The switching function is given by,

\[f(r_{ij}) = \frac{1-(r_{ij}/r_a)^N}{1-(r_{ij}/r_a)^M}\]

Entropy parameters can be calculated in pyscal using the following code,

import pyscal.core as pc
sys = pc.System()
sys.find_neighbors(method="cutoff", cutoff=0)
sys.calculate_entropy(1.4*lattice_constant, averaged=True)
atoms = sys.atoms
entropy = [atom.entropy for atom in atoms]
average_entropy = [atom.avg_entropy for atom in atoms]

The value of \(r_m\) is provided in units of lattice constant. Further parameters shown above, such as \(\sigma\) can be specified using the various keyword arguments. The above code does a simple averaging over neighbors. The switching function can be used by,

sys.calculate_entropy(1.4*lattice_constant, ra=0.9*lattice_constant, switching_function=True, averaged=True)

In pyscal, a slightly different version of \(s_s^i\) is calculated. This is given by,

\[s_s^i = -\rho \int_0^{r_m} [g_m^i(r)\ln g_m^i(r) - g_m^i(r) + 1] r^2 dr\]

The prefactor \(2\pi k_B\) is dropped in the entropy values calculated in pyscal.