ModeCouplingTheory.jl

A generic and fast solver of mode-coupling theory-like integrodifferential equations. It uses the algorithm outlined in Fuchs et al. to solve equations of the form $$\alpha \ddot{F}(t) + \beta \dot{F}(t) + \gamma F(t) + \delta + \int_0^t d\tau K(t-\tau)\dot{F}(\tau) = 0, $$ in which $\alpha$, $\beta$, $\gamma$, and $\delta$ are (possibly time-dependent) coefficients, and $K(t) = K(F(t), t)$ is a memory kernel that may nonlinearly depend on $F(t)$. This package exports some commonly used memory kernels, but it is straightforward to define your own. The solver is differentiable and works for scalar- and vector-valued functions $F(t)$. For more information see the Documentation.

Installation

To install the package run:

import Pkg
Pkg.add("ModeCouplingTheory")

In order to install and use it from Python, see the From Python page of the documentation.

Example usage:

We can use one of the predefined memory kernels

julia> using ModeCouplingTheory
julia> ν = 3.999
3.999

julia> kernel = SchematicF2Kernel(ν)
SchematicF2Kernel{Float64}(3.999)

This kernel evaluates K(t) = ν F(t)^2 when called.

We can now define the equation we want to solve as follows:

julia> α = 1.0; β = 0.0; γ = 1.0; δ = 0.0; F0 = 1.0; ∂F0 = 0.0;
julia> equation = MemoryEquation(α, β, γ, δ, F0, ∂F0, kernel);

Which we can solve by calling solve:

julia> sol = solve(equation);
julia> using Plots
julia> t = get_t(sol)
julia> F = get_F(sol)
julia> plot(log10.(t), F)

Please open an issue if anything is unclear in the documentation, if any unexpected errors arise or for feature requests (such as additional kernels). PRs are of course also welcome.