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

No abstract

No abstract

We consider both theoretically and experimentally self-organization process of quasi-equilibrium steadystate condensation of sputtered substance in accumulative ion-plasma devices. It has been shown that the self-organization effect is caused by self-consistent variations of the condensate temperature and the supersaturation of depositing atoms. Two possible types of self-organization process have been found out on the basis of the phase-plane method. The aluminium condensation experimental data confirming the self-organization nature of quasi-equilibrium steady-state condensation are discussed.