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https://github.com/cchandre/quasiperiodic_frenkel-kontorova
Analyticity breakdown for Frenkel-Kontorova Models in quasiperiodic media
https://github.com/cchandre/quasiperiodic_frenkel-kontorova
dynamical-systems frenkel-kontorova hamiltonian numpy python3 scipy
Last synced: about 1 month ago
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Analyticity breakdown for Frenkel-Kontorova Models in quasiperiodic media
- Host: GitHub
- URL: https://github.com/cchandre/quasiperiodic_frenkel-kontorova
- Owner: cchandre
- License: bsd-2-clause
- Created: 2021-03-26T14:09:16.000Z (almost 4 years ago)
- Default Branch: main
- Last Pushed: 2022-05-01T07:43:48.000Z (almost 3 years ago)
- Last Synced: 2024-11-10T21:11:46.500Z (3 months ago)
- Topics: dynamical-systems, frenkel-kontorova, hamiltonian, numpy, python3, scipy
- Language: Python
- Homepage:
- Size: 343 KB
- Stars: 1
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# Analyticity breakdown for Frenkel-Kontorova models in quasiperiodic media
- [`qpfk_dict.py`](https://github.com/cchandre/Quasiperiodic_Frenkel-Kontorova/blob/main/qpfk_dict.py): to be edited to change the parameters of the qpFK computation (see below for a dictionary of parameters)
- [`qpfk.py`](https://github.com/cchandre/Quasiperiodic_Frenkel-Kontorova/blob/main/qpfk.py): contains the qpFK classes and main functions defining the qpFK map
- [`qpfk_modules.py`](https://github.com/cchandre/Quasiperiodic_Frenkel-Kontorova/blob/main/qpfk_modules.py): contains the methods to execute the qpFK map
Once [`qpfk_dict.py`](https://github.com/cchandre/Quasiperiodic_Frenkel-Kontorova/blob/main/qpfk_dict.py) has been edited with the relevant parameters, run the file as
```sh
python3 qpfk.py
```
___
## Parameter dictionary
- *Method*: 'line_norm', 'region'; choice of method
- *Nxy*: integer; number of points along each line in computations
- *r*: integer; order of the Sobolev norm used in `compute_line_norm()`
####
- *omega*: floats; frequency *ω*
- *alpha*: array of *n* floats; vector **α** defining the perturbation
- *alpha_perp* (optional): array of *n* floats; vector **α⊥** perpendicular to **α**
- *Dv*: function; derivative of the *n*-d potential along a line
- *CoordRegion*: array of floats; min and max values of the amplitudes for each mode of the potential (see *Dv*); used in `compute_region()`
- *IndxLine*: tuple of integers; indices of the modes to be varied in `compute_region()`
parallelization in `compute_region()` is done along the *IndxLine*[0] axis
- *PolarAngles*: array of two floats; min and max value of the angles in 'polar'
- *CoordLine*: 1d array of floats; min and max values of the amplitudes of the potential used in `compute_line_norm()`
- *ModesLine*: tuple of 0 and 1; specify which modes are being varied (1 for a varied mode)
- *DirLine*: 1d array of floats; direction of the one-parameter family used in `compute_line_norm()`
####
####
- *AdaptSize*: boolean; if True, changes the dimension of arrays depending on the tail of the FFT of *h*(*ψ*)
- *Lmin*: integer; minimum and default value of the dimension of arrays for *h*(*ψ*)
- *Lmax*: integer; maximum value of the dimension of arrays for *h*(*ψ*) if *AdaptSize* is True
####
- *TolMax*: float; value of norm for divergence
- *TolMin*: float; value of norm for convergence
- *Threshold*: float; threshold value for truncating Fourier series of *h*(*ψ*)
- *MaxIter*: integer; maximum number of iterations for the Newton method
####
- *Type*: 'cartesian', 'polar'; type of computation for 2d plots
- *ChoiceInitial*: 'fixed', 'continuation'; method for the initial conditions of the Newton method
- *MethodInitial*: 'zero', 'one_step'; method to generate the initial conditions for the Newton iteration
####
- *AdaptEps*: boolean; if True adapt the increment of eps in `compute_line_norm()`
- *MinEps*: float; minimum value of the increment of eps if *AdaptEps*=True
- *MonitorGrad*: boolean; if True, monitors the gradient of *h*(*ψ*)
####
- *Precision*: 32, 64 or 128; precision of calculations (default=64)
- *SaveData*: boolean; if True, the results are saved in a `.mat` file
- *PlotResults*: boolean; if True, the results are plotted right after the computation
- *Parallelization*: tuple (boolean, int); True for parallelization, int is the number of cores to be used (set int='all' for all of the cores)
####
---
For more information:
**References:**
1. R. Calleja, R. de la Llave, *Fast numerical computation of quasi-periodic equilibrium states in 1D statistical mechanics, including twist maps*, [Nonlinearity 22, 1311 (2009)](https://dx.doi.org/10.1088/0951-7715/22/6/004)
1. R. Calleja, R. de la Llave, *Computation of the breakdown of analyticity in statistical mechanics models: numerical results and a renormalization group explanation*, [Journal of Statistical Physics 141, 940 (2010)](https://dx.doi.org/10.1007/s10955-010-0085-7)
1. X. Su, R. de la Llave, *KAM theory for quasi-periodic equilibria in one-dimensional quasi-periodic media*, [SIAM Journal on Mathematical Analysis 44, 3901 (2012)](https://doi.org/10.1137/12087160X)
1. T. Blass, R. de la Llave, *The analyticity breakdown for Frenkel-Kontorova models in quasi-periodic media: numerical explorations*, [Journal of Statistical Physics 150, 1183 (2013)](https://dx.doi.org/10.1007/s10955-013-0718-8)
**Example: Figure 3(A) of Ref.[4]**
