Information geometry in a reduced model of self-organised shear flows without the uniform coloured noise approximation

We investigate information geometry in a toy model of self-organised shear flows, where a bimodal PDF of x with two peaks signifying the formation of mean shear gradients is induced by a finite memory time γ⁻¹ of a stochastic forcing f . We calculate time-dependent probability density functions (PDFs) for different values of the correlation time γ⁻¹ and amplitude D of the stochastic forcing, and identify the parameter space for unimodal and bimodal stationary PDFs. By comparing results with those obtained under the uniform coloured noise approximation (UCNA) in Jacquet et al (2018 Entropy 20 613), we find that UCNA tends to favor the formation of a bimodal PDF of x for given parameter values γ⁻¹ and D. We map out attractor structure associated with unimodal and bimodal PDFs of x by measuring the total information length L∞ = L(t → ∞) against the location x₀ of a narrow initial PDF of x. Here L(t) represents the total number of statistically different states that a system passes through in time. We examine the validity of the UCNA from the perspective of information change and show how to fine-tune an initial joint PDF of x and f  to achieve a better agreement with UCNA results.