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Register a new userDamage spreading in the Bak-Sneppen model: Sensitivity to the initial conditions and equilibration dynamics
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The short-time and long-time dynamics of the Bak-Sneppen model of biological evolution are investigated using the damage spreading technique. By defining a proper Hamming distance measure, we are able to make it exhibits an initial power-law growth which, for finite size systems, is followed by a decay towards equilibrium. In this sense, the dynamics of self-organized critical states is shown to be similar to the one observed at the usual critical point of continuous phase-transitions and at the onset of chaos of non-linear low-dimensional dynamical maps. The transient, pre-asymptotic and asymptotic exponential relaxation of the Hamming distance between two initially uncorrelated equilibrium configurations is also shown to be fitted within a single mathematical framework. A connection with nonextensive statistical mechanics is exhibited.
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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 $alpha$ 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 bo
The conventional Hamming distance measurement captures only the short-time dynamics of the displacement between the uncorrelated random configurations. The minimum difference technique introduced by Tirnakli and Lyra [Int. J. Mod. Phys. C 14, 805 (2003)] is used to study the short-time and long-time dynamics of the two distinct random configurations of the isotropic and anisotropic Bak-Sneppen models on a square lattice. Similar to 1-dimensional case, the time evolution of the displacement is intermittent. The scaling behavior of the jump activity rate and waiting time distribution reveal the absence of typical spatial-temporal scales in the mechanism of displacement jumps used to quantify the convergence dynamics.
We consider biological evolution as described within the Bak and Sneppen 1993 model. We exhibit, at the self-organized critical state, a power-law sensitivity to the initial conditions, calculate the associated exponent, and relate it to the recently introduced nonextensive thermostatistics. The scenario which here emerges without tuning strongly reminds that of the tuned onset of chaos in say logistic-like onedimensional maps. We also calculate the dynamical exponent z.
The sensitivity to initial conditions and relaxation dynamics of two-dimensional maps are analyzed at the edge of chaos, along the lines of nonextensive statistical mechanics. We verify the dual nature of the entropic index for the Henon map, one ($q_{sen}<1$) related to its sensitivity to initial conditions properties, and the other, graining-dependent ($q_{rel}(W)>1$), related to its relaxation dynamics towards its stationary state attractor. We also corroborate a scaling law between these two indexes, previously found for $z$-logistic maps. Finally we perform a preliminary analysis of a linearized version of the Henon map (the smoothed Lozi map). We find that the sensitivity properties of all these $z$-logistic, Henon and Lozi maps are the same, $q_{sen}=0.2445...$
Since the Time-Dependent Density Functional Theory is mathematically formulated through non-linear coupled time-dependent 3-dimensional partial differential equations it is natural to expect a strong sensitivity of its solutions to variations of the initial conditions, akin to the butterfly effect ubiquitous in classical dynamics. Since the Schrodinger equation for an interacting many-body system is however linear and (mathematically) the exact equations of the Density Functional Theory reproduce the corresponding one-body properties, it would follow that the Lyapunov exponents are also vanishing within a Density Functional Theory framework. Whether for realistic implementations of the Time-Dependent Density Functional Theory the question of absence of the butterfly effect and whether the dynamics provided is indeed a predictable theory was never discussed. At the same time, since the time-dependent density functional theory is a unique tool allowing us the study of non-equilibrium dynamics of strongly interacting many-fermion systems, the question of predictability of this theoretical framework is of paramount importance. Our analysis, for a number of quantum superfluid any-body systems (unitary Fermi gas, nuclear fission, and heavy-ion collisions) with a classical equivalent number of degrees of freedom ${cal O}(10^{10})$ and larger, suggests that its maximum Lyapunov are negligible for all practical purposes.
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