# Density Module¶

Overview

 freud.density.FloatCF Computes the real pairwise correlation function. freud.density.ComplexCF Computes the complex pairwise correlation function. freud.density.GaussianDensity Computes the density of a system on a grid. freud.density.LocalDensity Computes the local density around a particle. freud.density.RDF Computes RDF for supplied data.

Details

The freud.density module contains various classes relating to the density of the system. These functions allow evaluation of particle distributions with respect to other particles.

## Correlation Functions¶

class freud.density.FloatCF(rmax, dr)

Computes the real pairwise correlation function.

The correlation function is given by $$C(r) = \left\langle s_1(0) \cdot s_2(r) \right\rangle$$ between two sets of points $$p_1$$ (ref_points) and $$p_2$$ (points) with associated values $$s_1$$ (ref_values) and $$s_2$$ (values). Computing the correlation function results in an array of the expected (average) product of all values at a given radial distance $$r$$.

The values of $$r$$ where the correlation function is computed are controlled by the rmax and dr parameters to the constructor. rmax determines the maximum distance at which to compute the correlation function and dr is the step size for each bin.

Note

2D: freud.density.FloatCF properly handles 2D boxes. The points must be passed in as [x, y, 0]. Failing to set z=0 will lead to undefined behavior.

Note

Self-correlation: It is often the case that we wish to compute the correlation function of a set of points with itself. If points is the same as ref_points, not provided, or None, we omit accumulating the self-correlation value in the first bin.

Module author: Matthew Spellings <mspells@umich.edu>

Parameters
• rmax (float) – Maximum pointwise distance to include in the calculation.

• dr (float) – Bin size.

Variables
accumulate

Calculates the correlation function and adds to the current histogram.

Parameters
compute

Calculates the correlation function for the given points. Will overwrite the current histogram.

Parameters
plot

Plot correlation function.

Parameters

ax (matplotlib.axes.Axes) – Axis to plot on. If None, make a new figure and axis. (Default value = None)

Returns

Axis with the plot.

Return type
reset

Resets the values of the correlation function histogram in memory.

class freud.density.ComplexCF(rmax, dr)

Computes the complex pairwise correlation function.

The correlation function is given by $$C(r) = \left\langle s_1(0) \cdot s_2(r) \right\rangle$$ between two sets of points $$p_1$$ (ref_points) and $$p_2$$ (points) with associated values $$s_1$$ (ref_values) and $$s_2$$ (values). Computing the correlation function results in an array of the expected (average) product of all values at a given radial distance $$r$$.

The values of $$r$$ where the correlation function is computed are controlled by the rmax and dr parameters to the constructor. rmax determines the maximum distance at which to compute the correlation function and dr is the step size for each bin.

Note

2D: freud.density.ComplexCF properly handles 2D boxes. The points must be passed in as [x, y, 0]. Failing to set z=0 will lead to undefined behavior.

Note

Self-correlation: It is often the case that we wish to compute the correlation function of a set of points with itself. If points is the same as ref_points, not provided, or None, we omit accumulating the self-correlation value in the first bin.

Module author: Matthew Spellings <mspells@umich.edu>

Parameters
• rmax (float) – Maximum pointwise distance to include in the calculation.

• dr (float) – Bin size.

Variables
accumulate

Calculates the correlation function and adds to the current histogram.

Parameters
compute

Calculates the correlation function for the given points. Will overwrite the current histogram.

Parameters
plot

Plot complex correlation function.

Parameters

ax (matplotlib.axes.Axes) – Axis to plot on. If None, make a new figure and axis. (Default value = None)

Returns

Axis with the plot.

Return type
reset

Resets the values of the correlation function histogram in memory.

## Gaussian Density¶

class freud.density.GaussianDensity(*args)

Computes the density of a system on a grid.

Replaces particle positions with a Gaussian blur and calculates the contribution from each to the proscribed grid based upon the distance of the grid cell from the center of the Gaussian. The resulting data is a regular grid of particle densities that can be used in standard algorithms requiring evenly spaced point, such as Fast Fourier Transforms. The dimensions of the image (grid) are set in the constructor, and can either be set equally for all dimensions or for each dimension independently.

• Constructor Calls:

Initialize with all dimensions identical:

freud.density.GaussianDensity(width, r_cut, sigma)


Initialize with each dimension specified:

freud.density.GaussianDensity(width_x, width_y, width_z, r_cut, sigma)


Module author: Joshua Anderson <joaander@umich.edu>

Parameters
• width (unsigned int) – Number of pixels to make the image.

• width_x (unsigned int) – Number of pixels to make the image in x.

• width_y (unsigned int) – Number of pixels to make the image in y.

• width_z (unsigned int) – Number of pixels to make the image in z.

• r_cut (float) – Distance over which to blur.

• sigma (float) – Sigma parameter for Gaussian.

Variables
compute

Calculates the Gaussian blur for the specified points. Does not accumulate (will overwrite current image).

Parameters
plot

Plot Gaussian Density.

Parameters

ax (matplotlib.axes.Axes) – Axis to plot on. If None, make a new figure and axis. (Default value = None)

Returns

Axis with the plot.

Return type

## Local Density¶

class freud.density.LocalDensity(r_cut, volume, diameter)

Computes the local density around a particle.

The density of the local environment is computed and averaged for a given set of reference points in a sea of data points. Providing the same points calculates them against themselves. Computing the local density results in an array listing the value of the local density around each reference point. Also available is the number of neighbors for each reference point, giving the user the ability to count the number of particles in that region.

The values to compute the local density are set in the constructor. r_cut sets the maximum distance at which data points are included relative to a given reference point. volume is the volume of a single data points, and diameter is the diameter of the circumsphere of an individual data point. Note that the volume and diameter do not affect the reference point; whether or not data points are counted as neighbors of a given reference point is entirely determined by the distance between reference point and data point center relative to r_cut and the diameter of the data point.

In order to provide sufficiently smooth data, data points can be fractionally counted towards the density. Rather than perform compute-intensive area (volume) overlap calculations to determine the exact amount of overlap area (volume), the LocalDensity class performs a simple linear interpolation relative to the centers of the data points. Specifically, a point is counted as one neighbor of a given reference point if it is entirely contained within the r_cut, half of a neighbor if the distance to its center is exactly r_cut, and zero if its center is a distance greater than or equal to r_cut + diameter from the reference point’s center. Graphically, this looks like:

Note

2D: freud.density.LocalDensity properly handles 2D boxes. The points must be passed in as [x, y, 0]. Failing to set z=0 will lead to undefined behavior.

Module author: Joshua Anderson <joaander@umich.edu>

Parameters
• r_cut (float) – Maximum distance over which to calculate the density.

• volume (float) – Volume of a single particle.

• diameter (float) – Diameter of particle circumsphere.

Variables
compute

Calculates the local density for the specified points. Does not accumulate (will overwrite current data).

Parameters

class freud.density.RDF(rmax, dr, rmin=0)

Computes RDF for supplied data.

The RDF ($$g \left( r \right)$$) is computed and averaged for a given set of reference points in a sea of data points. Providing the same points calculates them against themselves. Computing the RDF results in an RDF array listing the value of the RDF at each given $$r$$, listed in the R array.

The values of $$r$$ to compute the RDF are set by the values of rmin, rmax, dr in the constructor. rmax sets the maximum distance at which to calculate the $$g \left( r \right)$$, rmin sets the minimum distance at which to calculate the $$g \left( r \right)$$, and dr determines the step size for each bin.

Module author: Eric Harper <harperic@umich.edu>

Note

2D: freud.density.RDF properly handles 2D boxes. The points must be passed in as [x, y, 0]. Failing to set z=0 will lead to undefined behavior.

Parameters
• rmax (float) – Maximum interparticle distance to include in the calculation.

• dr (float) – Distance between histogram bins.

• rmin (float, optional) – Minimum interparticle distance to include in the calculation (Default value = 0).

Variables

Changed in version 0.7.0: Added optional rmin argument.

accumulate

Calculates the RDF and adds to the current RDF histogram.

Parameters
compute

Calculates the RDF for the specified points. Will overwrite the current histogram.

Parameters
plot

ax (matplotlib.axes.Axes) – Axis to plot on. If None, make a new figure and axis. (Default value = None)
reset