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We prove that the gravitational binding energy {\Omega} of a self gravitating system described by a mass density distribution {\rho}(x) admits an upper bound B[{\rho}(x)] given by a simple function of an appropriate, non-additive Tsallis' power-law entropic functional Sq evaluated on the density {\rho}. The density distributions that saturate the entropic bound have the form of isotropic q-Gaussian distributions. These maximizer distributions correspond to the Plummer density profile, well known in astrophysics. A heuristic scaling argument is advanced suggesting that the entropic bound B[{\rho}(x)] is unique, in the sense that it is unlikely that exhaustive entropic upper bounds not based on the alluded Sq entropic measure exit. The present findings provide a new link between the physics of self gravitating systems, on the one hand,...
The MaxEnt solutions are shown to display a variety of behaviors (beyond the traditional and customary exponential one) if adequate dynamical information is inserted into the concomitant entropic-variational principle. In particular, we show both theoretically and numerically that power laws and power laws with exponential cut-offs emerge as equilibrium densities in proportional and other dynamics.
We obtain time dependent $q$-Gaussian wave-packet solutions to a non linear Schr\"odinger equation recently advanced by Nobre, Rego-Montero and Tsallis (NRT) [Phys. Rev. Lett. 106 (2011) 10601]. The NRT non-linear equation admits plane wave-like solutions ($q$-plane waves) compatible with the celebrated de Broglie relations connecting wave number and frequency, respectively, with energy and momentum. The NRT equation, inspired in the $q$-generalized thermostatistical formalism, is characterized by a parameter $q$, and in the limit $q \to 1$ reduces to the standard, linear Schr\"odinger equation. The $q$-Gaussian solutions to the NRT equation investigated here admit as a particular instance the previously known $q$-plane wave solutions. The present work thus extends the range of possible processes yielded by the NRT dynamics that admi...
Comment: 14 pages, 7 figures, extended version
MaxEnt's variational principle, in conjunction with Shannon's logarithmic information measure, yields only exponential functional forms in straightforward fashion. In this communication we show how to overcome this limitation via the incorporation, into the variational process, of suitable dynamical information. As a consequence, we are able to formulate a somewhat generalized Shannonian Maximum Entropy approach which provides a unifying "thermodynamic-like" explanation for the scale-invariant phenomena observed in social contexts, as city-population distributions. We confirm the MaxEnt predictions by means of numerical experiments with random walkers, and compare them with some empirical data.
On the basis of dynamical principles we derive the Logistic Equation (LE), widely employed (among multiple applications) in the simulation of population growth, and demonstrate that scale-invariance and a mean-value constraint are sufficient and necessary conditions for obtaining it. We also generalize the LE to multi-component systems and show that the above dynamical mechanisms underlie large number of scale-free processes. Examples are presented regarding city-populations, diffusion in complex networks, and popularity of technological products, all of them obeying the multi-component logistic equation in an either stochastic or deterministic way. So as to assess the predictability-power of our present formalism, we advance a prediction, regarding the next 60 months, for the number of users of the three main web browsers (Explorer,...
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