Self-organized criticality within fractional Lorenz scheme

Abstract

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.

Field theory of self-organization

“…We have shown for the simplest example of sand ow on the inclined surface [6] that the above scheme represents the avalanche formation in self-organized criticality (SOC) phenomena [20][21][22][23][24][25][26][27][28][29][30]. The theory of SOC explains spontaneous (avalanche-type) dynamics, unlike the typical phase transitions and self-organization processes that occur only when a control parameter is driven to a critical value.…”

“…It has allowed to explain the self-organization (the ordering) of an open system subjected to the environment disorder [1]. However it has not been used for description of the nonergodicity property manifesting, for e.g., at formation of structural and spin glasses [2,3], tra c jams [4,5], ux steady states (avalanches) [6]. In connection to this the standard synergetic approach requires nontrivial extension-while studying of the ordering self-organizing system, we have to consider not only a symmetry breaking, but an ergodicity loosing that induces the clusterization of phase space.…”

Self-Organization of Condensed Matter in Fluctuating Environment

“…By this, an increase in the noise intensities causes avalanche emergence even in nondriven systems, where the control parameter noise plays a crucial role. A fluctuation regime of this type corresponds to the case, where a distribution of the order parameter appears in an algebraic form with integer exponent [49]. In order to not being restricted to such a particular case, we introduce a unified Lorenz system with a fractional feedback.…”

“…However, the system under consideration is now parameterized by a set of pseudo-thermodynamical variables, which describes the avalanche ensemble in the spirit of the famous Edwards paradigm [13,14] generalized to nonstationary system [ 49]. With this method, we study time dependences of the avalanche size (density of equipped sites in the avalanche), nonextensive complexity and nonconserved energy of equipped sites in the process of crystallization.…”

“…In accordance with the above estimation, the fractional Lorenz system is nonextensive, essentially if the exponent 2 3 a > / . It is worth to remember that above we have assumed the superdiffusion process only to be related to Lévy flights at discrete time instant Self 49 with arbitrary displacements, including infinite ones [60]. Related to the Fokker-Planck equation (3.28), such processes are characterized by exponents 1 ω = and 1 ϖ < , the first of which is constant, whereas the second one characterizes the fractal time-sequence of the Lévy flights and leads to the dynamical exponent 2 2 z ≡ ϖ/ω < (see the last of equations (3.30)).…”