Entropy fingerprint¶

EntropyFingerprint gives the pair entropy fingerprint of each atom, a measure of local order that needs no reference structure: low (more negative) in crystals, high in liquids and gases.

import numpy as np
import mdapy as mp
from mdapy.analysis import EntropyFingerprint
from mdapy.build import build_crystal

# fcc copper of 4 x 4 x 4 cubic cells, slightly disordered
crystal = build_crystal("fcc", 3.615, elements="Cu", repeat=(4, 4, 4))
shake = np.random.default_rng(0).normal(0, 0.05, (len(crystal), 3))
system = crystal.with_columns(positions=crystal["positions"] + shake)

result = EntropyFingerprint(cutoff=5.0, sigma=0.2, average_cutoff=3.0).compute(system)
result.entropy  # in units of the Boltzmann constant
system = system.with_columns(result)  # the column entropy_average

In a polycrystal, cut in half:

from mdapy.build import build_crystal, build_polycrystal
from mdapy.visualization import render

grains = build_polycrystal(
    build_crystal("fcc", 3.615, elements="Cu"), mp.Box(50.0), 5, seed=4
)
fingerprint = EntropyFingerprint(cutoff=5.0, sigma=0.2, average_cutoff=3.0).compute(
    grains
)
render(grains, "images/entropy.png", atoms=grains["positions"][:, 0] < 25,
       color=fingerprint.entropy, width=800, height=600, legend="entropy")  # fmt: skip

Entropy fingerprint of a polycrystal

The averaged entropy is lowest in the ordered grains (dark) and highest at the boundaries and their junctions (green and yellow).

Parameters¶

Name	Type	Default	Unit	Meaning
`cutoff`	float64		Å	neighbors are closer than it
`sigma`	float64	0.2	Å	width of the Gaussian of each neighbor, below `cutoff`
`local_density`	bool	`False`		use the density of each atom's neighbors instead of that of the system
`average_cutoff`	float64	`None`	Å	average each value with those of the neighbors closer than it

compute(system, *, neighbors=None, progress=None, cancel=None) takes the arguments of every analysis. A given neighbor list must reach the larger of cutoff and average_cutoff; without one, the neighbors within cutoff are found and used at once, never stored.

Results¶

An EntropyResult:

Attribute	Type	Meaning
`entropy`	(N,) float64	the fingerprint of each atom, in units of the Boltzmann constant; its column `entropy`, with the options in the name: `entropy_average`, `entropy_local_density`, `entropy_average_local_density`
`average`	bool	whether the values are averaged

Errors¶

The constructor raises InvalidParameter for a cutoff that is not positive, a sigma not between 0 and the cutoff, or an average_cutoff that is not positive; compute raises it for a neighbor list that does not fit.

Definition¶

For each atom \(i\) a local radial distribution is built from its neighbors \(j\) closer than the cutoff \(r_c\), each smeared by a Gaussian of width \(\sigma\):

\[ g_i(r) = \frac{1}{4\pi\rho r^2} \sum_j \frac{1}{\sqrt{2\pi}\,\sigma} e^{-(r - r_{ij})^2 / 2\sigma^2}, \qquad s_i = -2\pi\rho \int_0^{r_c} \big[g_i \ln g_i - g_i + 1\big]\, r^2 \,\mathrm dr , \]

where \(\rho\) is the number density of the system or, with local_density, the number of neighbors over the volume of the cutoff sphere. \(g_i\) is evaluated at \(r = k\sigma\) up to the cutoff and the integral is the trapezoid rule over those points, as in LAMMPS; where \(g_i < 10^{-10}\) the integrand is \(r^2\). With average_cutoff, \(s_i\) is averaged with the values of the neighbors closer than it.

LAMMPS compute entropy/atom spreads each neighbor over seven points only; MDAPY uses the whole Gaussian, so that equivalent atoms get equal values. The two agree to about 0.5 %.