Analysis of self-organized criticality in Ehrenfest’s dog-flea model

Abstract

The self-organized criticality in Ehrenfest's historical dog-flea model is analyzed by simulating the underlying stochastic process. The fluctuations around the thermal equilibrium in the model are treated as avalanches. We show that the distributions for the fluctuation length differences at subsequent time steps are in the shape of a q -Gaussian (the distribution which is obtained naturally in the context of nonextensive statistical mechanics) if one avoids the finite-size effects by increasing the system size. We provide clear numerical evidence that the relation between the exponent tau of avalanche size distribution obtained by maximum-likelihood estimation and the q value of appropriate q -Gaussian obeys the analytical result recently introduced by Caruso [Phys. Rev. E 75, 055101(R) (2007)]. This allows us to determine the value of q -parameter a priori from one of the well-known exponents of such dynamical systems.

Evidence of non-extensivity in the seismicity observed during the 2011–2012 unrest at the Santorini volcanic complex, Greece

“…(9) with caution. On the other hand, Bakar and Tirnakli (2009) analyzed numerically the Ehrenfest's dog-flea SOC model and obtained, with high accuracy, the value of τ S = 1.517 for the powerlaw exponent, along with q R = 2.35 for the q-Gaussian function, which is in agreement with Eq. (9) and in good agreement, taking into account experimental errors, to the value of q R obtained for the Santorini seismicity.…”

“…(9) and in good agreement, taking into account experimental errors, to the value of q R obtained for the Santorini seismicity. The result of Bakar and Tirnakli (2009) has been achieved using a simple prototype SOC model (different from that used by Caruso et al, 2007) and as noted by Sarlis et al (2010), who suggested that q R = 2.35 is the value that one should use in the q-Gaussian to check whether P (R) resulting from earthquake catalogs can be approached, emphasizing to the fact that the result of Bakar and Tirnakli (2009) is much more general and the q R value is not a fitting parameter anymore.…”

On the Nature and Dynamics of the Seismogenetic System of South California, Usa: An Analysis Based on Non-Extensive Statistical Physics

“…This includes theoretical studies of emergent q-exponential distributions in critical (e.g. Caruso et al, 2007 andBakar andTirnakli, 2009) and non-critical seismicity models (e.g. Celikoglu et al, 2010), as well as empirical studies in rock fracture experiments (Vallianatos et al, 2012).…”

“…The second point of view proposes that seismicity is an expression of non-equilibrating fractal tectonic grain that continuously evolves toward a stationary critical state with no characteristic spatiotemporal scale (e.g. Bak and Tang, 1989;Sornette and Sornette, 1989;Olami et al, 1992;Sornette and Sammis, 1995;Rundle et al, 2000;Bak et al, 2002 andBakar andTirnakli, 2009, etc.). This concept is known as Self Organized Criticality (SOC) and suggests that all earthquakes evolve towards the same global population and participate in shaping a non-equilibrium state with correlation between background, as well as between background/foreground and foreground/foreground events, so that instabilities arise spontaneously and any small instability has a chance of cascading into a large shock.…”

Some Open Points in Nonextensive Statistical Mechanics

“…Among others we have (i) The velocity distribution of (cells of) Hydra viridissima follows a q = 3/2 probability distribution function (PDF) [Upadhyaya et al, 2001]; (ii) The velocity distribution of (cells of) Dictyostelium discoideum follows a q = 5/3 PDF in the vegetative state and a q = 2 PDF in the starved state [Reynolds, 2010]; (iii) The velocity distribution in defect turbulence [Daniels et al , 2004]; (iv) The velocity distribution of cold atoms in a dissipative optical lattice [Douglas et al, 2006]; (v) The velocity distribution during silo drainage [Arevalo et al, 2007a,b]; (vi) The velocity distribution in a driven-dissipative 2D dusty plasma, with q = 1.08 ± 0.01 and q = 1.05 ± 0.01 at temperatures of 30000 K and 61000 K respectively [Liu & Goree, 2008]; (vii) The spatial (Monte Carlo) distributions of a trapped 136 Ba + ion cooled by various classical buffer gases at 300 K [DeVoe, 2009]; (viii) The distributions of price returns and stock volumes at the stock exchange, as well as the volatility smile [Borland, 2002a,b;Osorio et al, 2004;Queiros, 2005]; (ix) The distributions of returns of magnetic field fluctuations in the solar wind plasma as observed in data from Voyager 1 [Burlaga & Vinas, 2005] and from Voyager 2 [Burlaga & Ness, 2009]; (x) The distributions of returns in the Ehrenfest's dog-flea model [Bakar & Tirnakli, 2009; (xi)The distributions of returns in the coherent noise model [Celikoglu et al, 2010]; (xii) The distributions of returns of the avalanche sizes in the self-organized critical Olami-Feder-Christensen model, as well as in real earthquakes [Caruso et al, 2007]; (xiii) The distributions of angles in the HM F model [Moyano & Anteneodo, 2006]; (xiv) The distribution of stellar rotational velocities in the Pleiades [Carvalho et al, 2008]; (xv) The relaxation in various paradigmatic spin-glass substances through neutron spin echo experiments [Pickup et al, 2009]; (xvi) Various properties directly related with the time dependence of the width of the ozone layer around the Earth [Ferri et al, 2010]; (xvii) The distribution of transverse momenta in high energy collisions of electron-positron, proton-proton, and heavy nuclei (e.g., Pb-Pb and Au-Au) [Bediaga et al, 2000;Wilk & Wlodarczyk, 2009;Biro et al, 2009;CMS1, 2010;CMS2, 2010;…”