Mechanism in Leading to Richardson's Four-Thirds Law

Multiplicative Renormalization of Stochastic Differential Equations for the Abelian Sandpile Model

“…that is similar to the well-known Richardson "four-thirds" phenomenological law for the diffusion of passive admixtures in an ambient turbulent flow [50][51][52], viz.…”

“…In our work, we consider the original ASM [1]; formulate it as a problem of stochastic nonlinear dynamics following [39,41] whilst keeping all infinite numbers of interaction terms in the equation; prove the multiplicative renormalization of the model, including an infinite number of coupling parameters; calculate an infinite number of renormalization constants following the technique proposed in [48,49]; find a plane of fixed points in the infinite dimensional space of coupling parameters; discuss their stability and critical scaling in the model; and formulate a simple law relating the asymptotic size of an avalanche to a model exponent quantifying the time scale separation between the slow energy injection and fast avalanche relaxation processes in the ASM. The obtained law resembles the famous "four-thirds" phenomenological law of Richardson known in turbulent transport [50][51][52].…”