The long time and large scale asymptotic behavior for a stochastic problem related to self-organized critical (SOC) models is studied in the framework of advanced quantum field theory renormalization group (RG) method. The threshold condition and time scale separation between the slow dynamics of energy injection and the fast dynamics of avalanches (relaxation) are taken into account in the model. Herewith, the reciprocal correlation time at wavenumber k scales as tc(k) α k-2+2η with some phenomenological parameter η > 0 corresponding to the anomalous diffusion coefficient z = 2(1 - η). The quantum field theory corresponding to the nonlinear stochastic problem is multi-plicatively renormalizable and has an infinite number of coupling constants. The RG equations have a two-dimensional manifold of fixed points. Some of them relate to the stable asymptotic solutions and stipulate a general scaling with the critical dimensions of time Δ[t] = -2 + 2η and the energy field Δ[E] = d/2 - 3(1 - η). Possible corrections to the leading asymptotic behavior are discussed.
- Quantum field renormalization group
- Self-organized criticality