Order Module¶
Overview
Compute the cubatic order parameter [HAG15] for a system of particles using simulated annealing instead of Newton-Raphson root finding. |
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Compute the nematic order parameter for a system of particles. |
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Calculates the \(k\)-atic order parameter for 2D systems. |
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Compute the translational order parameter for each particle. |
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Compute the rotationally invariant Steinhardt order parameter \(q_l\) or \(w_l\) for a set of points [Ste83]. |
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Identifies solid-like clusters using dot products of \(q_{lm}\). |
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Calculates a measure of total rotational autocorrelation. |
Details
The freud.order
module contains functions which compute order
parameters for the whole system or individual particles. Order parameters take
bond order data and interpret it in some way to quantify the degree of order in
a system using a scalar value. This is often done through computing spherical
harmonics of the bond order diagram, which are the spherical analogue of
Fourier Transforms.
-
class
freud.order.
Cubatic
¶ Bases:
freud.util._Compute
Compute the cubatic order parameter [HAG15] for a system of particles using simulated annealing instead of Newton-Raphson root finding.
- Parameters
t_initial (float) – Starting temperature.
t_final (float) – Final temperature.
scale (float) – Scaling factor to reduce temperature.
n_replicates (unsigned int, optional) – Number of replicate simulated annealing runs. (Default value =
1
).seed (unsigned int, optional) – Random seed to use in calculations. If
None
, system time is used. (Default value =None
).
-
compute
(self, orientations)¶ Calculates the per-particle and global order parameter.
- Parameters
orientations ((\(N_{particles}\), 4)
numpy.ndarray
) – Orientations as angles to use in computation.
-
property
cubatic_tensor
¶ Rank 4 homogeneous tensor representing the optimal system-wide coordinates.
- Type
\(\left(3, 3, 3, 3 \right)\)
numpy.ndarray
-
property
global_tensor
¶ Rank 4 tensor corresponding to the global orientation. Computed from all orientations.
- Type
\(\left(3, 3, 3, 3 \right)\)
numpy.ndarray
-
n_replicates
¶ Number of replicate simulated annealing runs.
- Type
unsigned int
-
property
orientation
¶ The quaternion of global orientation.
- Type
\(\left(4 \right)\)
numpy.ndarray
-
property
particle_order
¶ Order parameter.
- Type
\(\left(N_{particles} \right)\)
numpy.ndarray
-
seed
¶ Random seed to use in calculations.
- Type
unsigned int
-
class
freud.order.
Hexatic
¶ Bases:
freud.locality._PairCompute
Calculates the \(k\)-atic order parameter for 2D systems.
The \(k\)-atic order parameter (called the hexatic order parameter for \(k = 6\)) is analogous to Steinhardt order parameters, and is used to measure order in the bonds of 2D systems.
The \(k\)-atic order parameter for a particle \(i\) and its \(n\) neighbors \(j\) is given by:
\(\psi_k \left( i \right) = \frac{1}{n} \sum_j^n e^{k i \phi_{ij}}\)
The parameter \(k\) governs the symmetry of the order parameter and typically matches the number of neighbors to be found for each particle. The quantity \(\phi_{ij}\) is the angle between the vector \(r_{ij}\) and \(\left(1, 0\right)\).
Note
2D:
freud.order.Hexatic
is only defined for 2D systems. The points must be passed in as[x, y, 0]
.- Parameters
k (unsigned int, optional) – Symmetry of order parameter. (Default value =
6
).weighted (bool, optional) – Determines whether to use neighbor weights in the computation of spherical harmonics over neighbors. If enabled and used with a Voronoi neighbor list, this results in the 2D Minkowski Structure Metrics \(\psi'_k\). (Default value =
False
)
-
compute
(self, system, neighbors=None)¶ Calculates the hexatic order parameter.
- Example::
>>> box, points = freud.data.make_random_system( ... box_size=10, num_points=100, is2D=True, seed=0) >>> # Compute the hexatic (6-fold) order for the 2D system >>> hex_order = freud.order.Hexatic(k=6) >>> hex_order.compute(system=(box, points)) freud.order.Hexatic(...) >>> print(hex_order.particle_order) [...]
- Parameters
system – Any object that is a valid argument to
freud.locality.NeighborQuery.from_system
.neighbors (
freud.locality.NeighborList
or dict, optional) – Either aNeighborList
of neighbor pairs to use in the calculation, or a dictionary of query arguments (Default value: None).
-
default_query_args
¶ The default query arguments are
{'mode': 'nearest', 'num_neighbors': self.k}
.
-
k
¶ Symmetry of the order parameter.
- Type
unsigned int
-
property
particle_order
¶ Order parameter.
- Type
\(\left(N_{particles} \right)\)
numpy.ndarray
-
plot
(self, ax=None)¶ Plot order parameter distribution.
- Parameters
ax (
matplotlib.axes.Axes
, optional) – Axis to plot on. IfNone
, make a new figure and axis. (Default value =None
)- Returns
Axis with the plot.
- Return type
-
class
freud.order.
Nematic
¶ Bases:
freud.util._Compute
Compute the nematic order parameter for a system of particles.
- Parameters
u (\(\left(3 \right)\)
numpy.ndarray
) – The nematic director of a single particle in the reference state (without any rotation applied).
-
compute
(self, orientations)¶ Calculates the per-particle and global order parameter.
- Example::
>>> orientations = np.array([[1, 0, 0, 0]] * 100) >>> director = np.array([1, 1, 0]) >>> nematic = freud.order.Nematic(director) >>> nematic.compute(orientations) freud.order.Nematic(u=[...])
- Parameters
orientations (\(\left(N_{particles}, 4 \right)\)
numpy.ndarray
) – Orientations to calculate the order parameter.
-
property
director
¶ The average nematic director.
- Type
\(\left(3 \right)\)
numpy.ndarray
-
property
nematic_tensor
¶ 3x3 matrix corresponding to the average particle orientation.
- Type
\(\left(3, 3 \right)\)
numpy.ndarray
-
property
particle_tensor
¶ \(\left(N_{particles}, 3, 3 \right)\)
numpy.ndarray
: One 3x3 matrix per-particle corresponding to each individual particle orientation.
-
u
¶ The normalized reference director (the normalized vector provided on construction).
- Type
\(\left(3 \right)\)
numpy.ndarray
-
class
freud.order.
RotationalAutocorrelation
¶ Bases:
freud.util._Compute
Calculates a measure of total rotational autocorrelation.
For any calculation of rotational correlations of extended (i.e. non-point) particles, encoding the symmetries of these particles is crucial to appropriately determining correlations. For systems of anisotropic particles in three dimensions, representing such equivalence can be quite mathematically challenging. This calculation is based on the hyperspherical harmonics as laid out in [KDvAG19]. Generalizations of spherical harmonics to four dimensions, hyperspherical harmonics provide a natural basis for periodic functions in 4 dimensions just as harmonic functions (sines and cosines) or spherical harmonics do in lower dimensions. The idea behind this calculation is to embed orientation quaternions into a 4-dimensional space and then use hyperspherical harmonics to find correlations in a symmetry-aware fashion.
The choice of the hyperspherical harmonic parameter \(l\) determines the symmetry of the functions. The output is not a correlation function, but rather a scalar value that measures total system orientational correlation with an initial state. As such, the output can be treated as an order parameter measuring degrees of rotational (de)correlation. For analysis of a trajectory, the compute call needs to be done at each trajectory frame.
- Parameters
l (int) – Order of the hyperspherical harmonic. Must be a positive, even integer.
-
compute
(self, ref_orientations, orientations)¶ Calculates the rotational autocorrelation function for a single frame.
- Parameters
ref_orientations ((\(N_{orientations}\), 4)
numpy.ndarray
) – Orientations for the initial frame.orientations ((\(N_{orientations}\), 4)
numpy.ndarray
) – Orientations for the frame of interest.
-
property
particle_order
¶ Rotational autocorrelation values calculated for each orientation.
- Type
(\(N_{orientations}\))
numpy.ndarray
-
class
freud.order.
SolidLiquid
¶ Bases:
freud.locality._PairCompute
Identifies solid-like clusters using dot products of \(q_{lm}\).
The solid-liquid order parameter [tW95][FHND10] uses a Steinhardt-like approach to identify solid-like particles. First, a bond parameter \(q_l(i, j)\) is computed for each neighbor bond.
If
normalize_q
is true (default), the bond parameter is given by \(q_l(i, j) = \frac{\sum_{m=-l}^{l} \text{Re}~q_{lm}(i) q_{lm}^*(j)} {\sqrt{\sum_{m=-l}^{l} \lvert q_{lm}(i) \rvert^2} \sqrt{\sum_{m=-l}^{l} \lvert q_{lm}(j) \rvert^2}}\)If
normalize_q
is false, then the denominator of the above expression is left out.Next, the bonds are filtered to keep only “solid-like” bonds with \(q_l(i, j)\) above a cutoff value \(q_{threshold}\).
If a particle has more than \(S_{threshold}\) solid-like bonds, then the particle is considered solid-like. Finally, solid-like particles are clustered.
- Parameters
l (unsigned int) – Spherical harmonic quantum number l.
q_threshold (float) – Value of dot product threshold when evaluating \(q_l(i, j)\) to determine if a bond is solid-like. For \(l=6\), 0.7 is generally good for FCC or BCC structures [FHND10].
solid_threshold (unsigned int) – Minimum required number of adjacent solid-like bonds for a particle to be considered solid-like for clustering. For \(l=6\), 6-8 is generally good for FCC or BCC structures.
normalize_q (bool) – Whether to normalize the dot product (Default value =
True
).
-
property
cluster_idx
¶ \(\left(N_{particles}\right)\)
numpy.ndarray
: Solid-like cluster indices for each particle.
-
property
cluster_sizes
¶ The sizes of all clusters.
- Type
\((N_{clusters}, )\)
np.ndarray
-
compute
(self, system, neighbors=None)¶ Compute the order parameter.
- Parameters
system – Any object that is a valid argument to
freud.locality.NeighborQuery.from_system
.neighbors (
freud.locality.NeighborList
or dict, optional) –Either a
NeighborList
of neighbor pairs to use in the calculation, or a dictionary of query arguments (Default value: None).
-
default_query_args
¶ No default query arguments.
-
l
¶ Spherical harmonic quantum number l.
- Type
unsigned int
-
property
largest_cluster_size
¶ The largest cluster size.
- Type
unsigned int
-
property
nlist
¶ Neighbor list of solid-like bonds.
-
property
num_connections
¶ The number of solid-like bonds for each particle.
- Type
\(\left(N_{particles}\right)\)
numpy.ndarray
-
plot
(self, ax=None)¶ Plot solid-like cluster distribution.
- Parameters
ax (
matplotlib.axes.Axes
, optional) – Axis to plot on. IfNone
, make a new figure and axis. (Default value =None
)- Returns
Axis with the plot.
- Return type
-
property
ql_ij
¶ Bond dot products \(q_l(i, j)\). Indexed by the elements of
self.nlist
.- Type
\(\left(N_{bonds}\right)\)
numpy.ndarray
-
class
freud.order.
Steinhardt
¶ Bases:
freud.locality._PairCompute
Compute the rotationally invariant Steinhardt order parameter \(q_l\) or \(w_l\) for a set of points [Ste83].
Implements the local rotationally invariant \(q_l\) or \(w_l\) order parameter described by Steinhardt. For a particle \(i\), we calculate the average order parameter by summing the spherical harmonics between particle \(i\) and its neighbors \(j\) in a local region: \(\overline{q}_{lm}(i) = \frac{1}{N_b} \displaystyle\sum_{j=1}^{N_b} Y_{lm}(\theta(\vec{r}_{ij}), \phi(\vec{r}_{ij}))\).
For \(q_l\), this is then combined in a rotationally invariant fashion to remove local orientational order as follows: \(q_l(i)=\sqrt{\frac{4\pi}{2l+1} \displaystyle\sum_{m=-l}^{l} |\overline{q}_{lm}|^2 }\).
For \(w_l\), it is then defined as a weighted average over the \(\overline{q}_{lm}(i)\) values using Wigner 3j symbols (Clebsch-Gordan coefficients). The resulting combination is rotationally (i.e. frame) invariant.
The average argument in the constructor provides access to a variant of this parameter that performs a average over the first and second shell combined [LD08]. To compute this parameter, we perform a second averaging over the first neighbor shell of the particle to implicitly include information about the second neighbor shell. This averaging is performed by replacing the value \(\overline{q}_{lm}(i)\) in the original definition by the average value of \(\overline{q}_{lm}(k)\) over all the \(k\) neighbors of particle \(i\) as well as itself.
The
norm
attribute argument provides normalized versions of the order parameter, where the normalization is performed by averaging the \(q_{lm}\) values over all particles before computing the order parameter of choice.- Parameters
l (unsigned int) – Spherical harmonic quantum number l.
average (bool, optional) – Determines whether to calculate the averaged Steinhardt order parameter. (Default value =
False
)wl (bool, optional) – Determines whether to use the \(w_l\) version of the Steinhardt order parameter. (Default value =
False
)weighted (bool, optional) – Determines whether to use neighbor weights in the computation of spherical harmonics over neighbors. If enabled and used with a Voronoi neighbor list, this results in the 3D Minkowski Structure Metrics \(q'_l\). (Default value =
False
)wl_normalize (bool, optional) – Determines whether to normalize the \(w_l\) version of the Steinhardt order parameter. (Default value =
False
)
-
compute
(self, system, neighbors=None)¶ Compute the order parameter.
- Example::
>>> box, points = freud.data.make_random_system(10, 100, seed=0) >>> ql = freud.order.Steinhardt(l=6) >>> ql.compute((box, points), {'r_max':3}) freud.order.Steinhardt(l=6, ...)
- Parameters
system – Any object that is a valid argument to
freud.locality.NeighborQuery.from_system
.neighbors (
freud.locality.NeighborList
or dict, optional) –Either a
NeighborList
of neighbor pairs to use in the calculation, or a dictionary of query arguments (Default value: None).
-
default_query_args
¶ No default query arguments.
-
l
¶ Spherical harmonic quantum number l.
- Type
unsigned int
-
property
particle_order
¶ Variant of the Steinhardt order parameter for each particle (filled with
nan
for particles with no neighbors).- Type
\(\left(N_{particles}\right)\)
numpy.ndarray
-
plot
(self, ax=None)¶ Plot order parameter distribution.
- Parameters
ax (
matplotlib.axes.Axes
, optional) – Axis to plot on. IfNone
, make a new figure and axis. (Default value =None
)- Returns
Axis with the plot.
- Return type
-
property
ql
¶ \(\left(N_{particles}\right)\)
numpy.ndarray
: \(q_l\) Steinhardt order parameter for each particle (filled withnan
for particles with no neighbors). This is always available, no matter which options are selected.
-
class
freud.order.
Translational
¶ Bases:
freud.locality._PairCompute
Compute the translational order parameter for each particle.
Note
2D:
freud.order.Translational
is only defined for 2D systems. The points must be passed in as[x, y, 0]
.- Parameters
k (float, optional) – Symmetry of order parameter. (Default value =
6.0
).
-
compute
(self, system, neighbors=None)¶ Calculates the local descriptors.
- Parameters
system – Any object that is a valid argument to
freud.locality.NeighborQuery.from_system
.neighbors (
freud.locality.NeighborList
or dict, optional) –Either a
NeighborList
of neighbor pairs to use in the calculation, or a dictionary of query arguments (Default value: None).
-
default_query_args
¶ The default query arguments are
{'mode': 'nearest', 'num_neighbors': int(self.k)}
.
-
property
particle_order
¶ Order parameter.
- Type
\(\left(N_{particles} \right)\)
numpy.ndarray