Analyticity breakdown for Frenkel-Kontorova models in quasiperiodic media

• qpfk_dict.py: to be edited to change the parameters of the qpFK computation (see below for a dictionary of parameters)

• qpfk.py: contains the qpFK classes and main functions defining the qpFK map

• qpfk_modules.py: contains the methods to execute the qpFK map

Once qpfk_dict.py has been edited with the relevant parameters, run the file as

python3 qpfk.py

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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)

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For more information: cristel.chandre@cnrs.fr

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)
2. 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)
3. 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)
4. 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)

Example: Figure 3(A) of Ref.[4]