Black swans, extreme risks, and the e-pile model of self-organized criticality

Self-consistent model of the plasma staircase and nonlinear Schrödinger equation with subquadratic power nonlinearity

“…Comparing the NLSE result τ 2.56 (our Table II) to the reference values reported in Table 4 of Ref. [124], one might see that the plasma staircase exhibits characteristics enabling to associate it with the safety class A3 (moderate tail risk). The class is two grades less than the highest grade achievable for dynamical systems with self-regulation (class A5) and three grades less than the safest class (class X ), which requires external (forced) regulation.…”

“…The task is relatively straightforward and refers to an extremerisk classification scheme already developed in Ref. [124] to which the τ value is key. Comparing the NLSE result τ 2.56 (our Table II) to the reference values reported in Table 4 of Ref.…”

“…The class is two grades less than the highest grade achievable for dynamical systems with self-regulation (class A5) and three grades less than the safest class (class X ), which requires external (forced) regulation. While the assignment of the different safety grades might appear to be an academic exercise, we note that the grades are justified from a topological perspective [124] and as such might be used to categorize the various dynamical phenomena resulting in extreme events.…”

“…That said, the exponent τ 2.56 (safety class A3 [124]) appears to be numerically very different from the respective values deriving from sand-pile SOC models [29,30] and their generalizations (e.g., the Zhang model [31], the e-pile model [124], and other models alike): If only in one dimension, which is a very special case of SOC [32] (safety class A1: high risk of extreme avalanches) as well as in the mean-field limit [132] for which a crossover to the diffusive transport could be expected (safety class A5: moderate to low risk of extreme events).…”

“…At a kinetic level, the avalanches correspond to a Lévy fractional FFPE (33) with self-consistent potential and must be considered as coherent structures. Conversely, sand piles lead to a different description as they rely on next-neighbor-like interactions which are local in space and which may generate some non-Markovianity for t → +∞, but not really a nonlocal behavior in the sense of the Lévy-Gnedenko generalized central limit theorem [111] [remark that the e piles [124] bring directly to a non-Markovian diffusion equation with memory and the fractional differentiation over time, while the space derivatives are integer and correspond to the familiar Laplacian operator, in contrast to FFPE (33)].…”