CTRW

A proof-of-concept library for simulating continuous time random walks (CTRW) in one and two dimensions. The library contains a set of probability distributions, Laplace and Fourier transform routines and a framework for simulating and analyzing CTRWs.

Ensembles of random walk trajectories can be compared with the fundamental solution of the fractional diffusion equation. Trajectories can be characterized using implemented displacement measures, local time and velocity autocorrelation. Application examples can be found in the testing routines.

The program code in this repository is made available under the GPLv3 license (GPL-3.0-or-later).

Usage

• After cloning this repository, run git submodule update --init to install external libraries.
• In ext_mittagleffler/mittag_leffler.py change the import line to from .ml_internal import LTInversion!
• The test scripts generate pdf figures in the results directory.

Literature and references

See the REFERENCES files in the subdirectories:

• distributions/REFERENCES
• stochproc/REFERENCES
• transforms/REFERENCES

Required libraries

• numpy/scipy
• mpmath [T2]
• hankel [T1]
• multiprocessing (partial)
• matplotlib (for testing)
• mittag-leffler: A Mittag-Leffler function implementation. The mittag-leffler source code is provided here. See also references [D2]. The files must be copied to the ext_mittagleffler directory (see Usage).
• PyLevy: Univariate Lévy-stable distributions. The source code of PyLevy can be found here. See also references [D5]. A copy must be placed in the ext_levy directory (see Usage).

Bivariate distributions

For isotropic bivariate distributions, pdf_radius returns the density of the amplitude distribution (the distribution of |X|). pdf_distance returns the density of bivariate distributions (i.e. of X), but takes the absolute value |x| as an argument. This function is not a density on R but on R^2.

CTRWs

Standard spatial increments are configured with a stability parameter 0 < alpha <= 2 and a scale 0 < DC. Standard time increments are configured with a memory parameter 0 <= beta <= 1 and a scale 0 < TC. By convention beta == 0 results in a degenerate increment distribution and constant time steps. The parameter delta_t scales both distributions simultaneously, retaining analytical properties.

Trajectory or particle objects have the data members dx, dt, x, t. Trajectories always start at t_{0} = 0, x_{0} = 0 and for i > 0 the trajectory is defined by t_{i} = dt_{0} + ... + dt_{i-1}, x_{i} = dx_{0} + ... + dx_{i-1}.