Self-organized criticality within generalized Lorenz scheme

Abstract

The theory of a flux steady-state related to avalanche formation is presented for the simplest model of a sand pile within framework of the Lorenz approach. The stationary values of sand velocity and sand pile slope are derived as functions of control parameter (driven sand pile slope). The additive noises of above values are introduced to build the phase diagram, where the noise intensities determine both avalanche and non-avalanche domains, as well as mixed one. Being corresponded to the SOC regime, the last domain is crucial to affect of the noise intensities of vertical component of sand velocity and sand pile slope especially. To address to a self-similar behavior, a fractional feedback is used as efficient ingredient of the modified Lorenz system. In a spirit of Edwards paradigm, an effective thermodynamics is introduced to determine a distribution over avalanche ensemble with negative temperature. Steady-state behavior of the moving grains number, as well as nonextensive values of entropy and energy is studied in detail. The power-law distribution over avalanche sizes is described within a fractional Lorenz scheme, where noise of the energy plays a crucial role. This distribution is shown to be solution of both fractional Fokker-Planck equation and nonlinear one. As a result, we obtain new relations between exponent of the size distribution, fractal dimension of phase space, characteristic exponent of multiplicative noise, number of governing equations, dynamical exponents and nonextensivity parameter.