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https://github.com/olivierverdier/spinsys
Symplectic midpoint solver for spin systems on a product of spheres.
https://github.com/olivierverdier/spinsys
Last synced: about 5 hours ago
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Symplectic midpoint solver for spin systems on a product of spheres.
- Host: GitHub
- URL: https://github.com/olivierverdier/spinsys
- Owner: olivierverdier
- Created: 2014-05-17T20:16:20.000Z (over 10 years ago)
- Default Branch: master
- Last Pushed: 2021-11-01T10:24:09.000Z (about 3 years ago)
- Last Synced: 2023-03-10T23:27:50.299Z (over 1 year ago)
- Language: Python
- Size: 9.77 KB
- Stars: 1
- Watchers: 3
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# spinsys - Spherical midpoint for spin systems
[](https://travis-ci.org/olivierverdier/spinsys)
[](https://coveralls.io/r/olivierverdier/spinsys?branch=master)
## About
This is a symplectic midpoint solver for *spin systems*. These systems are symplectic differential equations derived from the Lie–Poisson structure of a product of spheres. The papers "[Symplectic integrators for spin systems](http://arxiv.org/abs/1402.4114)", "[A minimal-coordinate symplectic integrator on spheres](http://arxiv.org/abs/1402.3334)" and "[Geometry of discrete-time spin systems](http://arxiv.org/abs/1505.04035)" gives the details of the setting and the method, as well as a theoretical background.
The method and the implementation are a joint work of [R. I. McLachlan](https://www.massey.ac.nz/~rmclachl/), [K. Modin](https://klasmodin.wordpress.com/) and [O. Verdier](https://olivierverdier.com).
## How to use it
### What you need
* `m0`: An initial condition: an Nx3 matrix where each row is a vector of length one.
* `gradient`: A gradient which returns the gradient of a Hamiltonian at each such points.
You may also optionally need
* `strengths`: A N vector of strength which determines the symplectic form on the product of spheres.
If such a `strenghts` vector is not provided, the strengths are assumed to be one for every sphere.
### Run the simulation
The code is then as follows:
```python
import midpoint
import sphere
dt = .01 # time step
nb_steps = 10 # number of steps
vector = sphere.radially_constant(gradient, strengths)
generator = midpoint.run(midpoint.get_increment(vector), m0=m0.ravel(), dt=dt, nb_steps=nb_steps)
# run the solver
ms = [m.reshape(-1,3) for m in generator]
```
Now the solution `ms` is a list of Nx3 matrices which correspond to the solutions at each time step.
## Available systems
There are two common spin systems which are already available in this package.
### Spin chain
The Heisenberg spin chain system for N points is initialized as follows:
```python
from spinsys import spinchain
gradient = spinchain.get_gradient(1/N**2)
```
The strength vector may be omitted in the call of `radially_constant`:
```python
vector = sphere.radially_constant(gradient)
```
### Point vortices
The gradient is obtained from the `strengths` vector:
```python
from spinsys import vortex
import numpy as np
strengths = np.array([1., 1., -1.])
gradient = vortex.get_gradient(strengths)
```