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https://github.com/maroba/multipoles

A Python package for multipole expansions of electrostatic or gravitational potentials
https://github.com/maroba/multipoles

electrostatic-potential mathematics physics scientific-computing

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A Python package for multipole expansions of electrostatic or gravitational potentials

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README 

        
# multipoles



 PyPI version 

 build

 




*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.



## Installation



Simply use `pip`:



```

pip install --upgrade multipoles

```



## Documentation



The documentation is available [here](https://maroba.github.io/multipoles/).



## Theory



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:



```python

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:



```python

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)

```