mctpy: Mode-Coupling Theory Solver in Python
This module provides a python implementation of numerical routines for the mode-coupling theory of the glass transition (MCT). In short, MCT is a theory that predicts dynamical correlation functions for highly viscous fluids, as solutions of certain integro-differential equations that are solved numerically. mctpy supports this numerical solution alongside a number of utility helpers that are often used in conjunction with MCT.
More information on MCT can be found in the standard references.
The core functionality of mctpy is the numerical treatment of evolution equations of the form
solved as an initial value problem for the unknown \(\phi(q,t)\) at a set of given indices \(q\) (in physical terms: the density correlation function to a given wave number), where the memory kernel \(m(q,t)\) is itself a (usually nonlinear) functional of the sought-after solution,
The nonlinearity of this self-consistent closure causes a bifurcation scenario, where the long-time limit \(f(q)=\lim_{t\to\infty}\phi(q,t)\) can show discontinuities upon a smooth change of the parameters entering the equations. Close to such a bifurcation, the relaxation times of both the solution and the memory kernel become arbitrarily large, and hence adapted numerical integration schemes are needed to treat the MCT equations.
At the moment, mctpy is not yet a fully performant solver. It makes heavy use for the numba just-in-time compiler, but this leads to relatively long warm-up times when running even simple code. This code base is mostly intended to provide an easy entry point for people wanting to understand, use, modify and extend the core numerics behind MCT.
Overview
The code is structured in two major groups of classes: “correlators” and “models”. Think of the correlators defining their equation of motion, and the models defining the specific form of the memory kernel.
In particular this means that the correlator classes implement the suitable solvers for their equations, while the model classes implement the numerics to evaluate the mode-coupling functionals. Often, there is a close correspondence between a correlator/solver and a specific model, but in principle, the correlator class is more generic.
Most MCT functionals take as input the static structure factor. While these are not strictly part of MCT, we also introduce a group of corresponding “structurefactor” classes that implement some commonly used approximations for these. Also, some utility methods, for example to perform Fourier transforms of the time-domain correlators to obtain their spectra, or the calculation of MCT’s exponent parameter, are implemented.
Contributors
While this specific implementation is derived from code developed initially during the PhD thesis of Thomas Voigtmann, it is the collected effort of many people working on MCT over many decades. Contributions are in particular acknowledged from
Matthias Fuchs: initial idea for the “moment algorithm” (together with Ivo Hofacker), a lot of building blocks taken from old Fortran code, and code for sheared systems.
Thomas Franosch: many discussions around the matrix code, non-uniform wave-number grids and the two-dimensional version of the theory.
Liesbeth Janssen and her group: for pushing the development of modern code through their development of Julia-based MCT and GMCT code.
Mathias Mayr: for valuable cross-checking, non-Gaussian parameter code.
Tommaso Rizzo: for the development of multi-dimensional SBR code.