multipoles

multipoles is a Python package for multipole expansions of the solutions of the Poisson equation (e.g. electrostatic or gravitational potentials). It can handle discrete and continuous charge or mass distributions.

Background

For a given function $\rho(x,y,z)$, the solution $\Phi(x,y,z)$ of the Poisson equation $\nabla^2\Phi=-4\pi \rho$ with vanishing Dirichlet boundary conditions at infinity is

$$\Phi(x,y,z)=\int d^3r'\frac{\rho(r')}{|r-r'|}$$

Examples of this are the electrostatic and Newtonian gravitational potential. If you need to evaluate $\Phi(x,y,z)$ at many points, calculating the integral for each point is computationally expensive. As a faster alternative, we can express $\Phi(x,y,z)$ in terms of the multipole moments $q_{lm}$ or $I_{lm}$ (note some literature uses the subscripts $(\cdot)_{nm}$):

$$\Phi(x,y,z)=\sum_{l=0}^\infty\underbrace{\sqrt{\frac{4\pi}{2l+1}}\sum_{m=-l}^lY_{lm}(\theta, \varphi)\frac{q_{lm}}{r^{l+1}}}_{\Phi^{(l)}}$$

for a exterior expansion, or

$$\Phi(x,y,z)=\sum_{l=0}^\infty\underbrace{\sqrt{\frac{4\pi}{2l+1}}\sum_{m=-l}^lY_{lm}(\theta, \varphi)I_{lm}r^{l}}_{\Phi^{(l)}}$$

for an interior expansion; where $r, \theta, \varphi$ are the usual spherical coordinates corresponding to the cartesian coordinates $x, y, z$ and $Y_{lm}(\theta, \varphi)$ are the spherical harmonics.

The multipole moments for the exterior expansion are:

$$q_{lm} = \sqrt{\frac{4\pi}{2l+1}}\int d^3 r' \rho(r')r'^l Y^*_{lm}(\theta', \varphi')$$

and the multipole moments for the interior expansion are:

$$I_{lm} = \sqrt{\frac{4\pi}{2l+1}}\int d^3 r' \frac{\rho(r')}{r'^{l+1}} Y^*_{lm}(\theta', \varphi')$$

This approach is usually much faster because the contributions $\Phi^{(l)}$ are getting smaller with increasing l. So we just have to calculate a few integrals for obtaining some $q_{lm}$ or $I_{lm}$.

Some literature considers the $\sqrt{\frac{4\pi}{2l+1}}$ as part of the definition of $Y_{lm}(\theta, \varphi)$.

Examples

Discrete Charge Distribution

As example for a discrete charge distribution we model two point charges with positive and negative unit charge located on the z-axis:

from multipoles import MultipoleExpansion

# Prepare the charge distribution dict for the MultipoleExpansion object:

charge_dist = {
    'discrete': True,     # point charges are discrete charge distributions
    'charges': [
        {'q': 1, 'xyz': (0, 0, 1)},
        {'q': -1, 'xyz': (0, 0, -1)},
    ]
}

l_max = 2   # where to stop the infinite multipole sum; here we expand up to the quadrupole (l=2)

Phi = MultipoleExpansion(charge_dist, l_max)

# We can evaluate the multipole expanded potential at a given point like this:

x, y, z = 30.5, 30.6, 30.7
value = Phi(x, y, z)

# The multipole moments are stored in a dict, where the keys are (l, m) and the values q_lm:
Phi.multipole_moments

Continuous Charge Distribution

As an example for a continuous charge distribution, we smear out the point charges from the previous example:

from multipoles import MultipoleExpansion
import numpy as np

# First we set up our grid, a cube of length 10 centered at the origin:

npoints = 101
edge = 10
x, y, z = [np.linspace(-edge/2., edge/2., npoints)]*3
XYZ = np.meshgrid(x, y, z, indexing='ij')


# We model our smeared out charges as gaussian functions:

def gaussian(XYZ, xyz0, sigma):
    g = np.ones_like(XYZ[0])
    for k in range(3):
        g *= np.exp(-(XYZ[k] - xyz0[k])**2 / sigma**2)
    g *= (sigma**2*np.pi)**-1.5
    return g

sigma = 1.5   # the width of our gaussians

# Initialize the charge density rho, which is a 3D numpy array:
rho = gaussian(XYZ, (0, 0, 1), sigma) - gaussian(XYZ, (0, 0, -1), sigma)


# Prepare the charge distribution dict for the MultipoleExpansion object:

charge_dist = {
    'discrete': False,     # we have a continuous charge distribution here
    'rho': rho,
    'xyz': XYZ
}

# The rest is the same as for the discrete case:

l_max = 2   # where to stop the infinite multipole sum; here we expand up to the quadrupole (l=2)

Phi = MultipoleExpansion(charge_dist, l_max)

x, y, z = 30.5, 30.6, 30.7
value = Phi(x, y, z)

Installation

Simply use pip:

pip install --upgrade multipoles