multipoles

KO | EN

gitlite — search

#javascript #python #typescript #golang #android #rust #deep-learning #nodejs #windows #cli #vue #pytorch #openai #mcp

(★ 46)

A Python package for multipole expansions of electrostatic or gravitational potentials

multipoles 최신버젼 다운로드

최종 버전 다운로드 (.zip)

// repository documentation

Was this content helpful?

★ 0 (0 ratings)

# 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.

## 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 <i>
l</i>. 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)
```

multipoles

multipoles 최신버젼 다운로드

오늘의 추천 프로젝트

Was this content helpful?

Recent Feedback

multipoles

multipoles 최신버젼 다운로드

오늘의 추천 프로젝트

Was this content helpful?

Recent Feedback

Download README