# Example M6: Calculate local entropy

This Python modifier function computes the local entropy fingerprint via numerical integration of each particle’s local pair separation function \(g(r)\). The pair separation function is computed by visiting the neighbor particles within a spherical cutoff region around each particle. Please refer to the documentation page for this modifier in the OVITO user manual to learn more about this analysis function.

The code makes use of the following Python APIs:

A `CutoffNeighborFinder` is created to visit the neighbors of each particle within a spherical cutoff region and compute their distances to the central particle.

The modifier function stores the computed per-particle entropy values in a new output property, which is created by by calling the `create_property()` method.

To average the computed entropy values over a neighborhood of particles in a post-processing step, the modifier function employs the built-in `ComputePropertyModifier` of OVITO. It is invoked as a sub-routine with the `DataCollection.apply()` method.

    from ovito.data import CutoffNeighborFinder, DataCollection
    from ovito.modifiers import ComputePropertyModifier
    import numpy as np

    def modify(frame: int, data: DataCollection, cutoff = 5.0, sigma = 0.2, use_local_density = False, compute_average = False, average_cutoff = 5.0):
        # Validate input parameters:
        assert(cutoff > 0.0)
        assert(sigma > 0.0 and sigma < cutoff)
        assert(average_cutoff > 0)

        # Show message in OVITO's status bar:
        yield 'Calculating local entropy'

        # Overall particle density:
        global_rho = data.particles.count / data.cell.volume

        # Initialize neighbor finder:
        finder = CutoffNeighborFinder(cutoff, data)

        # Create output array for local entropy values
        local_entropy = np.empty(data.particles.count)

        # Number of bins used for integration:
        nbins = int(cutoff / sigma) + 1

        # Table of r values at which the integrand will be computed:
        r = np.linspace(0.0, cutoff, num=nbins)
        rsq = r**2

        # Precompute normalization factor of g_m(r) function:
        prefactor = rsq * (4 * np.pi * global_rho * np.sqrt(2 * np.pi * sigma**2))
        prefactor[0] = prefactor[1] # Avoid division by zero at r=0.

        # Iterate over input particles:
        for particle_index in range(data.particles.count):
            yield particle_index / data.particles.count

            # Get distances r_ij of neighbors within the cutoff range.
            r_ij = finder.neighbor_distances(particle_index)

            # Compute differences (r - r_ji) for all {r} and all {r_ij} as a matrix.
            r_diff = np.expand_dims(r, 0) - np.expand_dims(r_ij, 1)

            # Compute g_m(r):
            g_m = np.sum(np.exp(-r_diff**2 / (2.0*sigma**2)), axis=0) / prefactor

            # Estimate local atomic density by counting the number of neighbors within the
            # spherical cutoff region:
            if use_local_density:
                local_volume = 4/3 * np.pi * cutoff**3
                rho = len(r_ij) / local_volume
                g_m *= global_rho / rho
            else:
                rho = global_rho

            # Compute integrand:
            integrand = np.where(g_m >= 1e-10, (g_m * np.log(g_m) - g_m + 1.0) * rsq, rsq)

            # Integrate from 0 to cutoff distance:
            local_entropy[particle_index] = -2.0 * np.pi * rho * np.trapz(integrand, r)

        # Output the computed per-particle entropy values to the data pipeline.
        data.particles_.create_property('Entropy', data=local_entropy)

        # If requested by the user, perform the spatial averaging of the local entropy value using
        # the built-in ComputePropertyModifier. The math expression computes the sum of all entropy values
        # of the particles within the local neighborhood, divided by the number of particles in the neighborhood.
        if compute_average:
            data.apply(ComputePropertyModifier(
                output_property = 'Entropy',
                operate_on = 'particles',
                cutoff_radius = average_cutoff,
                expressions = ['Entropy / (NumNeighbors + 1)'],
                neighbor_expressions = ['Entropy / (NumNeighbors + 1)']))