Beta-scaling equation

The beta-scaling model is implemented to make it easier to find critical exponents of MCT. The equation is

\[\sigma - \delta t + \lambda (g(t))^2 = \partial_t∫g(t-\tau)g(\tau)d\tau.\]

Here, $\sigma$ is the distance from the critical point, $\lambda$ is the relevant eigenvalue of the stability matrix. $g(t)$ describes the deviation of the order parameter from the plateau. $\delta$ is an optional hopping parameter, defaulting to 0 if not specified. Each of the parameters have to be floating point numbers.

Example

In order to solve the beta-scaling equation, we have to specify the parameters defining the equation and a time-scale t0 that shifts the results.

using ModeCouplingTheory
using MCTBetaScaling, Plots
λ = 0.7; ϵ = -0.1; t0 = 0.001
equation = BetaScalingEquation(λ, ϵ, t0)
sol = solve(equation, TimeDoublingSolver(t_max=10^4.))
plot(log10.(sol.t), log10.(abs.(sol.F)), ylabel="log_{10}(|g(t)|)", xlabel="log_{10}(t)", label="g(t)")

In the figure, the slopes of the straight lines are given by the parameters $-a$ and $b$, which describe the relaxation towards and away from the plateau value of the correlator. These exponents are automatically computed, and are stored in equation.coeff.a and equation.coeff.b.

References

Götze, J Phys Condens Matter 2, 8485 (1990)