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.

A recent study of coherent noise model for the system size independent case provides an exact relation between the exponent $\tau$ of avalanche size distribution and the $q$ value of appropriate $q$-Gaussian that fits the return distribution of the model. This relation is applied to Ehrenfest's historical dog-flea model by treating the fluctuations around the thermal equilibrium as avalanches. We provide a clear numerical evidence that the relation between the exponent $\tau$ of fluctuation length distribution and the $q$ value of appropriate $q$-Gaussian obeys this exact relation when the system size is large enough. This allows us to determine the value of $q$-parameter \emph{a priori} from one of the well known exponents of such dynamical systems. Furthermore, it is shown that the return distribution in dog-flea model gradually approaches to $q$-Gaussian as the system size increases and this tendency can be analyzed by a well defined analytical expression.Comment: 9 pages, 4 eps figures and 1 bbl file, accepted for publication on Physica A, in press (2010

A recent study of nonextensive phase transitions in nuclei and nuclear clusters needs a probability model compatible with the appropriate Hamiltonian. For magnetic molecules a representation of the evolution by a Markov process achieves the required probability model that is used to study the probability density function (PDF) of the order parameter, i.e. the magnetization. The existence of one or more modes in this PDF is an indication for the superparamagnetic transition of the cluster. This allows us to determine the factors that influence the blocking temperature, i.e. the temperature related to the change of the number of modes in the density. It turns out that for our model, rather than the evolution of the system implied by the Hamiltonian, the high temperature density of the magnetization is the important factor for the temperature of the transition. We find that an initial probability density function with a high entropy leads to a magnetic cluster with a high blocking temperature.

We implement the damage spreading technique on 2-dimensional isotropic and anisotropic Bak-Sneppen models. Our extensive numerical simulations show that there exists a power-law sensitivity to the initial conditions at the statistically stationary state (self-organized critical state). Corresponding growth exponent α for the Hamming distance and the dynamical exponent z are calculated. These values allow us to observe a clear data collapse of the finite size scaling for both versions of the Bak-Sneppen model. Moreover, it is shown that the growth exponent of the distance in the isotropic and anisotropic Bak-Sneppen models is strongly affected by the choice of the transient time. PACS. 05.65.+b Self-organized systems -64.60.Ht Dynamic critical phenomena -87.23.Kg Dynamics of evolution 1 Introduction In 1993, Bak and Sneppen (BS) introduced a simple model to describe the biological evolution of an ecology of interacting species [1]. Since then, it has attracted quite some attention among the natural scientists as well aseconomists. This wide range of interest comes from the fact that BS model is the simplest model that exhibits self-organized criticality [1,2]. The self-organized criticala ity feature of the BS model is revealed in its ability to naturally evolve towards a scale invariance stationary state [3]. That is, the correlation length in the BS model is infinite and an initial local perturbation might lead to a global effect. Therefore, it is worth to study the sensitivity to the initial conditions in the BS model.The technique that we use to study the sensitivity to the initial conditions in the BS model is known as the damage spreading technique in dynamical systems theory and can be described as follows: If we consider two copies

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