# flake8: noqa
import numpy as np
[docs]class DiffusionCoefficient:
def __init__(self, traj, timestep, atom_indices=None, molecule=False):
"""
This class calculates the Diffusion Coefficient for the given Trajectory using the Einstein Equation:
..math:: \\left \\langle \\left | r(t) - r(0) \\right | ^{2} \\right \\rangle = 2nDt
where r(t) is the position of atom at time t, n is the degrees of freedom and D is the Diffusion Coefficient
Solved herein by fitting with :math:`y = mx + c`, i.e. :math:`\\frac{1}{2n} \\left \\langle \\left | r(t) - r(0) \\right | ^{2} \\right \\rangle = Dt`, with m = D and c = 0
wiki : https://en.wikibooks.org/wiki/Molecular_Simulation/Diffusion_Coefficients
Parameters:
traj (Trajectory):
Trajectory of atoms objects (images)
timestep (Float):
Timestep between *each image in the trajectory*, in ASE timestep units
(For an MD simulation with timestep of N, and images written every M iterations, our timestep here is N * M)
atom_indices (List of Int):
The indices of atoms whose Diffusion Coefficient is to be calculated explicitly
molecule (Boolean)
Indicate if we are studying a molecule instead of atoms, therefore use centre of mass in calculations
"""
self.traj = traj
self.timestep = timestep
# Condition used if user wants to calculate diffusion coefficients for
# specific atoms or all atoms
self.atom_indices = atom_indices
if self.atom_indices is None:
self.atom_indices = [i for i in range(len(traj[0]))]
# Condition if we are working with the mobility of a molecule, need to
# manage arrays slightly differently
self.is_molecule = molecule
if self.is_molecule:
self.types_of_atoms = ["molecule"]
self.no_of_atoms = [1]
else:
self.types_of_atoms = sorted(
set(traj[0].symbols[self.atom_indices]))
self.no_of_atoms = [traj[0].get_chemical_symbols().count(
symbol) for symbol in self.types_of_atoms]
# Dummy initialisation for important results data object
self._slopes = []
@property
def no_of_types_of_atoms(self):
"""
Dynamically returns the number of different atoms in the system
"""
return len(self.types_of_atoms)
@property
def slopes(self):
"""
Method to return slopes fitted to datapoints. If undefined, calculate slopes
"""
if len(self._slopes) == 0:
self.calculate()
return self._slopes
@slopes.setter
def slopes(self, values):
"""
Method to set slopes as fitted to datapoints
"""
self._slopes = values
def _initialise_arrays(self, ignore_n_images, number_of_segments):
"""
Private function to initialise data storage objects. This includes objects to store the total timesteps
sampled, the average diffusivity for species in any given segment, and objects to store gradient and intercept from fitting.
Parameters:
ignore_n_images (Int):
Number of images you want to ignore from the start of the trajectory, e.g. during equilibration
number_of_segments (Int):
Divides the given trajectory in to segments to allow statistical analysis
"""
total_images = len(self.traj) - ignore_n_images
self.no_of_segments = number_of_segments
self.len_segments = total_images // self.no_of_segments
# These are the data objects we need when plotting information. First
# the x-axis, timesteps
self.timesteps = np.linspace(
0, total_images * self.timestep, total_images + 1)
# This holds all the data points for the diffusion coefficients,
# averaged over atoms
self.xyz_segment_ensemble_average = np.zeros(
(self.no_of_segments, self.no_of_types_of_atoms, 3, self.len_segments))
# This holds all the information on linear fits, from which we get the
# diffusion coefficients
self.slopes = np.zeros(
(self.no_of_types_of_atoms, self.no_of_segments, 3))
self.intercepts = np.zeros(
(self.no_of_types_of_atoms, self.no_of_segments, 3))
self.cont_xyz_segment_ensemble_average = 0
def calculate(self, ignore_n_images=0, number_of_segments=1):
"""
Calculate the diffusion coefficients, using the previously supplied data. The user can break the data into segments and
take the average over these trajectories, therefore allowing statistical analysis and derivation of standard deviations.
Option is also provided to ignore initial images if data is perhaps unequilibrated initially.
Parameters:
ignore_n_images (Int):
Number of images you want to ignore from the start of the trajectory, e.g. during equilibration
number_of_segments (Int):
Divides the given trajectory in to segments to allow statistical analysis
"""
# Setup all the arrays we need to store information
self._initialise_arrays(ignore_n_images, number_of_segments)
for segment_no in range(self.no_of_segments):
start = segment_no * self.len_segments
end = start + self.len_segments
seg = self.traj[ignore_n_images + start:ignore_n_images + end]
# If we are considering a molecular system, work out the COM for the
# starting structure
if self.is_molecule:
com_orig = np.zeros(3)
for atom_no in self.atom_indices:
com_orig += seg[0].positions[atom_no] / \
len(self.atom_indices)
# For each image, calculate displacement.
# I spent some time deciding if this should run from 0 or 1, as the displacement will be zero for
# t = 0, but this is a data point that needs fitting too and so
# should be included
for image_no in range(0, len(seg)):
# This object collects the xyz displacements for all atom
# species in the image
xyz_disp = np.zeros((self.no_of_types_of_atoms, 3))
# Calculating for each atom individually, grouping by species
# type (e.g. solid state)
if not self.is_molecule:
# For each atom, work out displacement from start coordinate
# and collect information with like atoms
for atom_no in self.atom_indices:
sym_index = self.types_of_atoms.index(
seg[image_no].symbols[atom_no])
xyz_disp[sym_index] += np.square(
seg[image_no].positions[atom_no] - seg[0].positions[atom_no])
# Calculating for group of atoms (molecule) and work out squared
# displacement
else:
com_disp = np.zeros(3)
for atom_no in self.atom_indices:
com_disp += seg[image_no].positions[atom_no] / \
len(self.atom_indices)
xyz_disp[0] += np.square(com_disp - com_orig)
# For each atom species or molecule, use xyz_disp to calculate
# the average data
for sym_index in range(self.no_of_types_of_atoms):
# Normalise by degrees of freedom and average overall atoms
# for each axes over entire segment
denominator = (2 * self.no_of_atoms[sym_index])
for xyz in range(3):
self.xyz_segment_ensemble_average[segment_no][sym_index][xyz][image_no] = (
xyz_disp[sym_index][xyz] / denominator)
# We've collected all the data for this entire segment, so now to
# fit the data.
for sym_index in range(self.no_of_types_of_atoms):
self.slopes[sym_index][segment_no], self.intercepts[sym_index][segment_no] = self._fit_data(self.timesteps[start:end],
self.xyz_segment_ensemble_average[segment_no][sym_index])
def _fit_data(self, x, y):
"""
Private function that returns slope and intercept for linear fit to mean square diffusion data
Parameters:
x (Array of floats):
Linear list of timesteps in the calculation
y (Array of floats):
Mean square displacement as a function of time.
"""
# Simpler implementation but disabled as fails Conda tests.
# from scipy.stats import linregress
# slope, intercept, r_value, p_value, std_err = linregress(x,y)
# Initialise objects
slopes = np.zeros(3)
intercepts = np.zeros(3)
# Convert into suitable format for lstsq
x_edited = np.vstack([np.array(x), np.ones(len(x))]).T
# Calculate slopes for x, y and z-axes
for xyz in range(3):
slopes[xyz], intercepts[xyz] = np.linalg.lstsq(
x_edited, y[xyz], rcond=-1)[0]
return slopes, intercepts
def get_diffusion_coefficients(self):
"""
Returns diffusion coefficients for atoms (in alphabetical order) along with standard deviation.
All data is currently passed out in units of Å^2/<ASE time units>
To convert into Å^2/fs => multiply by ase.units.fs
To convert from Å^2/fs to cm^2/s => multiply by (10^-8)^2 / 10^-15 = 10^-1
"""
slopes = [np.mean(self.slopes[sym_index])
for sym_index in range(self.no_of_types_of_atoms)]
std = [np.std(self.slopes[sym_index])
for sym_index in range(self.no_of_types_of_atoms)]
return slopes, std
def plot(self, ax=None, show=False):
"""
Auto-plot of Diffusion Coefficient data. Provides basic framework for visualising analysis.
Parameters:
ax (Matplotlib.axes.Axes)
Axes object on to which plot can be created
show (Boolean)
Whether or not to show the created plot. Default: False
"""
# Necessary if user hasn't supplied an axis.
import matplotlib.pyplot as plt
# Convert from ASE time units to fs (aesthetic)
from ase.units import fs as fs_conversion
if ax is None:
ax = plt.gca()
# Define some aesthetic variables
color_list = plt.cm.Set3(np.linspace(0, 1, self.no_of_types_of_atoms))
xyz_labels = ['X', 'Y', 'Z']
xyz_markers = ['o', 's', '^']
# Create an x-axis that is in a more intuitive format for the view
graph_timesteps = self.timesteps / fs_conversion
for segment_no in range(self.no_of_segments):
start = segment_no * self.len_segments
end = start + self.len_segments
label = None
for sym_index in range(self.no_of_types_of_atoms):
for xyz in range(3):
if segment_no == 0:
label = 'Species: {} ({})'.format(
self.types_of_atoms[sym_index], xyz_labels[xyz])
# Add scatter graph for the mean square displacement data
# in this segment
ax.scatter(graph_timesteps[start:end], self.xyz_segment_ensemble_average[segment_no][sym_index][xyz],
color=color_list[sym_index], marker=xyz_markers[xyz], label=label, linewidth=1, edgecolor='grey')
# Print the line of best fit for segment
line = np.mean(self.slopes[sym_index][segment_no]) * fs_conversion * \
graph_timesteps[start:end] + \
np.mean(self.intercepts[sym_index][segment_no])
if segment_no == 0:
label = 'Segment Mean : %s' % (
self.types_of_atoms[sym_index])
ax.plot(graph_timesteps[start:end], line, color='C%d' % (
sym_index), label=label, linestyle='--')
# Plot separator at end of segment
x_coord = graph_timesteps[end - 1]
ax.plot([x_coord,
x_coord],
[-0.001,
1.05 * np.amax(self.xyz_segment_ensemble_average)],
color='grey',
linestyle=":")
# Plot the overall mean (average of slopes) for each atom species
# This only makes sense if the data is all plotted on the same x-axis timeframe, which currently we are not - everything is plotted sequentially
# for sym_index in range(self.no_of_types_of_atoms):
# line = np.mean(self.slopes[sym_index])*graph_timesteps+np.mean(self.intercepts[sym_index])
# label ='Mean, Total : %s'%(self.types_of_atoms[sym_index])
# ax.plot(graph_timesteps, line, color='C%d'%(sym_index), label=label, linestyle="-")
# Aesthetic parts of the plot
ax.set_ylim(-0.001, 1.05 * np.amax(self.xyz_segment_ensemble_average))
ax.legend(loc='best')
ax.set_xlabel('Time (fs)')
ax.set_ylabel(r'Mean Square Displacement ($\AA^2$)')
if show:
plt.show()
def print_data(self):
"""
Output of statistical analysis for Diffusion Coefficient data. Provides basic framework for understanding calculation.
"""
from ase.units import fs as fs_conversion
# Collect statistical data for diffusion coefficient over all segments
slopes, std = self.get_diffusion_coefficients()
# Useful notes for any consideration of conversion.
# Converting gradient from Å^2/fs to more common units of cm^2/s => multiplying by (10^-8)^2 / 10^-15 = 10^-1
# Converting intercept from Å^2 to more common units of cm^2 => multiply by (10^-8)^2 = 10^-16
#
# Note currently in ASE internal time units
# Converting into fs => divide by 1/(fs_conversion) => multiply by
# (fs_conversion)
# Print data for each atom, in each segment.
for sym_index in range(self.no_of_types_of_atoms):
print('---')
print(r'Species: %4s' % self.types_of_atoms[sym_index])
print('---')
for segment_no in range(self.no_of_segments):
print(r'Segment %3d: Diffusion Coefficient = %.10f Å^2/fs; Intercept = %.10f Å^2;' %
(segment_no, np.mean(self.slopes[sym_index][segment_no]) * fs_conversion, np.mean(self.intercepts[sym_index][segment_no])))
# Print average overall data.
print('---')
for sym_index in range(self.no_of_types_of_atoms):
print('Mean Diffusion Coefficient (X, Y and Z) : %s = %.10f Å^2/fs; Std. Dev. = %.10f Å^2/fs' %
(self.types_of_atoms[sym_index], slopes[sym_index] * fs_conversion, std[sym_index] * fs_conversion))
print('---')