Entropy parameter#

The entropy parameter was introduced by Piaggi et al [1] 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 by using a simple averaging over the neighbors given by,

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

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

from pyscal3 import System
sys = System('conf.dump')
sys.find.neighbors(method="cutoff", cutoff=0)
avg_entropy = sys.calculate.entropy(1.4*lattice_constant, averaged=True)

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.

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.


  1. Piaggi, P. M. & Parrinello, M. Entropy based fingerprint for local crystalline order. Journal of Chemical Physics 147, (2017).